Advances in Atomic, Molecular, and Optical Physics, Volume 30

Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

VOLUME 30

EDITORIAL, BOARD

P. R. BERMAN New York University New York, New York

K. DOLDER The University of Newcastle-upon-Tyne Newcastle-upon-Tyne England

M. GAVRILA F. O.M. Instituut voor Atoom- en Molecuulfysica Amsterdam The Netherlands

M. INOKUTI Argonne National Laboratory Argonne, Illinois

S. J . SMITH Joint Institute for Laboratory Astrophysics Boulder. Colorado

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS

Edited by

Sir David Bates

DEPARTMENT OF APPLIED MATHEMATICS AND THEORETICAL PHYSICS THE QUEEN’S UNIVERSITY OF BELFAST BELFAST, NORTHERN IRELAND

Benjamin Bederson

DEPARTMENT OF PHYSICS NEW YORK UNIVERSITY NEW YORK, NEW YORK

VOLUME 30

ACADEMIC PRESS, INC.

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Contents

CONTRIBUTORS vii

Differential Cnws Sections for Excitation of Helium Atoms and Heliumlike Ions by Electron Impact

Shinobu Nakazaki

I. Introduction 11. Theory

111. Excitation of Helium Atoms IV. Excitation of Heliumlike Ions V. Concluding Remarks

Acknowledgments References

Cross-Section Measurements for Electron Impact on Excited Atomic Species

S. Trajmar and J . C . Nickel

I. Introduction 11. General Remarks

111. Production of Excited Species IV. Detection of Excited Species V. Cross-Section Measurements

Acknowledgments References

1 3

14 33 40 41 41

45 47 48 60 66 98 99

The Dissociative Ionization of Simple Molecuks by Fast Ions

Colin J . Latimer

1. Introduction 105 11. The Dissociative Ionization Process 107

Ill. Energy Distributions of Fragment Ions 112 IV. Energy Distributions of Fragment Ion Pairs: Coulomb Explosions 121 V. Angular Distributions of Fragment Ions: Orientated Molecules 129

VI. Partial Dissociative Ionization Cross Sections 132 References 136

V

V l CONTENTS

Theory of Collisions Between Laser Cooled Atoms

P. S. Julienne. A . M . Smith and K . Burnett

I . Introduction 11.

111. Cold Collisions in the Absence of Light Cold Collisions in a Light Field Acknowledgments References

Light Induced Drift

E. R . Eliel

I . 11.

111. I v. V.

VI. VII .

v111.

Introduction Gas Kinetic Effects of Light Models for the Drift Velocity Techniques for Measuring the Drift Velocity Drift Velocities for Na Light Induced Drift in Astrophysics Other Light Induced Kinetic Effects Conclusions Acknowledgments References

Continuum Distorted Wave Methods in Ion-Atom Collisions

Derrick S . F. Crothers and Louis J . Dub&

1. Introduction and Overview 11. Notation

I l l . Time-Dependent Impact Parameter Formalism IV. Time-Independent Wave Formalism V. Conclusions and Future Perspectives

Acknowledgments Appendix A: Recent Reviews of Ion-Atom Scattering Appendix B: Subject Oriented Index References

141 i43 157 195 195

199 208 213 234 244 267 279 280 28 1 28 1

287 290 296 314 32 1 323 323 3 24 329

INDEX

CONTENTS Of PREVIOUS VOLUMES

337 349

Contributors

Numbers in parentheses indicate the pages on which the authors’ contributions begin

K . Burnett ( 141), Clarendon Laboratory, Department of Physics, University of

Derrick S. F. Crothers (287), Department of Applied Mathematics and Theo-

Oxford, Parks Road, Oxxford, OX1 3PU, United Kingdom

retical Physics, School of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 INN, Northern Ireland

Louis J. Dub6 (287), Departement de Physique, Universite Laval, Quebec,

E. R. Eliel (199), Huygens Laboratory, University of Leiden, P.O. Box 9504,

P. S. Julienne (141), Molecular Physics Division, National Institute of Stan-

Colin J. Latimer (105), School of Mathematics and Physics, The Queen’s Uni-

Shinobu Nakazaki ( I ) , Department of Applied Physics, Faculty of Engineering,

J . C. Nickel (45), Department of Physics, University of California, Riverside,

Canada G I K 7P4

2300 RA Leiden, The Netherlands

dards and Technology, Gaithersburg, MD 20899

versity of Belfast, Belfast, Northern Ireland

Miyazaki University, Miyazaki 889-2 1, Japan

CA 9252 1

A. M. Smith (141). Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom

S. Trajmar (45), Department of Physics, University of California, Riverside, CA 92521

This Page Intentionally Left Blank

ADVANCES IN ATOMIC . MOLECULAR. AND OPTICAL PHYSICS. VOL . 30

DIFFERENTIAL, CROSS SECTIONS FOR EXCITATION OF HEHUM ATOMS AND HELIUMUKE IONS BY ELECTRON IMPACT SHINOB U N A M K1 Department of Applied Physics. Faculty of Engineering. Miyazaki University. Miyazaki. Japan

1 . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ]].Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

A . BasicTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 B . Expansion of Total Wave Function . . . . . . . . . . . . . . . . . . 4 C . Differential Cross Section . . . . . . . . . . . . . . . . . . . . . . 5 D . The Close-Coupling Equation . . . . . . . . . . . . . . . . . . . . 6 E . The R-Matrix Method . . . . . . . . . . . . . . . . . . . . . . . 8 F . The Region of Intermediate Energy . . . . . . . . . . . . . . . . . . 10 G . The Optical Potential Methods . . . . . . . . . . . . . . . . . . . . 11 H . The Distorted-Wave Methods and Related Methods . . . . . . . . . . . . 12

111 . Excitation of Helium Atoms . . . . . . . . . . . . . . . . . . . . . . . 14 A . Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 B.Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 C . Comparison of Theoretical Results and Experiments . . . . . . . . . . . 22 D . Differential Cross Section for the I ’S - 2’s Transition in the Forward

Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 IV . Excitation of Heliumlike Ions . . . . . . . . . . . . . . . . . . . . . . 33

A . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 B . Comparison of the Results . . . . . . . . . . . . . . . . . . . . . . 34

V . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 40 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . 41 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

I . Introduction

Excitation of atoms and ions by electron impact plays an important role as an elementary process in such fields of science and technology as. for example. plasma physics and chemistry. astrophysics. gaseous discharge. laser development. and atmospheric physics . Many theoretical and experimental studies have

Copyright 0 1993 by Academic R o o . Inc . All righls of rcpduclion in any form rcscrved .

ISBN 0-12-003830-7 1

2 Shinobu Nakazaki

been done to obtain the total and the differential cross sections for various transitions in atoms and ions. This chapter is particularly focused on those studies of differential cross sections.

The differential cross section (DCS) is very useful in gaining insight into the details of the collision mechanism. DCS, for instance, reflects more clearly the characteristics of the interaction potential than the total (integrated) cross section (TCS). A long-range interaction generally dominates in small-angle scattering and a short-range one in the large-angle scattering. In a theoretical study of electron-atom collisions, it is customary now to compare DCS to experiment or other theory to test the validity of the theory. DCS is more critically dependent than TCS on the target wavefunction and the approximate method employed in the calculation. Sometimes DCSs obtained in different calculations differ very much, while the corresponding TCSs agree with each other. Furthermore DCS is of practical importance in applications, especially when a spatial anisotropy is involved.

Helium is an abundant element, almost as abundant as hydrogen in the uni- verse. Helium has been the subject of many laboratory experiments, because it is fairly easy to handle. Theoretically helium is the simplest multi-electron atom. Calculations ranging from the most elaborate, using the R-matrix method, to a wide variety of simpler calculations based on perturbation theory have been applied to the e - + He collision.

In contrast to the electron-atom collision, very limited number of works have been reported on the DCS for electron-ion collisions. It is very hard to obtain experimentally DCS for ions. Recently, however, a few attempts to measure DCS for ions have begun to appear. This appearance of experimental data has stimulated theoretical studies of DCS for ions. In fact, several papers have already been published on the theoretical calculation of DCS for He-like ions. Because it would be interesting to compare cross sections for He and He-like ions, the DCS reported by these papers are included in this chapter.

Many review articles have been published on electron-atom or -ion collisions. Some of them discuss the general aspect of DCS and include numerical data on DCS (Gerjuoy and Thomas, 1974; Byron and Joachain, 1977; Bransden and McDowell, 1977, 1978; Walters, 1984; Itikawa, 1986; McCarthy and Wei- gold, 1991). Those reviews can be referred for the DCS of atoms and ions other than He and He-like ions.

After describing several theories for electron impact excitation of atoms in Section 11, we review and discuss the DCS for electron excitation from the ground state to the n = 2 levels of helium atoms and heliumlike ions in Sections 111 and IV. Finally, concluding remarks are given in Section V. Atomic units (a.u.) will be used, unless otherwise stated.

DIFFERENTIAL CROSS SECTIONS FOR EXCITATION OF HELIUM ATOMS 3

II. Theory

A. BASIC THEORY

We consider the scattering of an electron by an atom (ion) of nuclear charge Z and having N electrons. The total Hamiltonian for the (N + 1)-electron system is

where r,, = (r, - r,l and r, = (r,l, r, and r, being the position vectors of the ith and jth electrons from the target nucleus. The Schrodinger equation for the complete system is

(2)

where E is the total energy. We introduce the target eigenstate Qq , which satisfies the equation

(3)

where HN is the target Hamiltonian defined by Eq. ( I ) with (N + 1) replaced by N, and E, is the target eigenenergy.

Consider the excitation process where the electron is incident upon the target in the state Qq and scattered leaving the target in the state Q4,. We solve Eq. (2) with the asymptotic form of the wave function

( H N + i - El* = 0,

( H N - E,)Qq = 0,

* q - @ q X q ( f l N + I ) l k r ( r N + I ) + 2 Qy'X,'(u,+I)fq'q(PN+I)Sty.(rN+I), (4) rNtl-" 4'

where

Ikq(r) = exp{i[k, * r - yq ln(kr - k, - r)]}, ( 5 )

(6 ) 1 r Sky.(r) = - exp{i[k,,r + y,. In2kqtr]},

and

yq = (Z - N)/k,. (7)

In Eq. (4), X,,(u) and X,.(u) are the spin eigenfunctions for the incident and scattered electrons, and f4,,(PN+ I) is the excitation amplitude for the transition

4 Shinobu Nakazaki

q + 9'. The wave numbers for the incident and scattered electrons, k, and k,, , are related to the total energy E of the system by

Finally we have the differential cross section for the transition q + qf in the form

The total cross section is obtained by integrating this over all scattering angles.

B. EXPANSION OF THE TOTAL WAVE FUNCTION

In order to obtain the excitation amplitude f,(,(PN+ I), we expand the wave function in Eq. (2) in terms of a set of target wave functions @, and (N + 1)- electron correlation functions x, as follows:

* q ( x l , * . . 9 X N + I ) = d 2 @ ( ( X I , . . . 9 X N ) I (10)

x e , , ( X N + l ) + 2 X , ( X l , . * . , XN+I)CJ,' I

Here the function e,, and coefficients cJq are to be determined. We adopt the LS coupling scheme of angular momenta. The operator sd antisymmetrizes the total wave function, and the symbol x, denotes the space and spin coordinates of the nth electron.

The function el, may be written as

e,,(r) = r -'~lq(r)~,m,~P)~(~~,lg), (1 1)

where &,,, are spherical harmonics. The subscript q in Eq. (10) implies that the solution has the asymptotic form (4). Introducing function 4,, which couples the target wave function @, with the spin-angle functions for the scattered electron, the total wave function *, is rewritten as

* q ( x l , . . * 9 X N + I ) = 2 $ , (XI , * * * 9 X N ; I

We consider the radial function E,, which obeys the condition

Eq(0) = 0, (13)

KJr) - - [sinO,S, + cosO,KJ, kf > 0 (14) 1

,-m fi


E,(r) - 0, kf < 0 r-w

where

(16) 1 2

8, = k,r - - 1 , ~ + y , ln(2k,r) + cr/,(y,),

and

cI,(yJ = arg r(1, + 1 - iy,). (17)

The K,, in Eq. (14) are the elements of the reactance matrix, K. Instead of the real solutions specified by the conditions (13)-(15), it is often convenient to introduce complex solutions defined by the boundary conditions

Eq@) = 0, (18)

EJr) - - [a,, exp(-ie,) - S,, exp(i&)I, kf > 0, (19) 1

,-bw fl &(r) - 0, kf c 0.

r-=

The quantities S, are the element of the S-matrix, which is related to the K- matrix by

(21) S = (1 + iK)/(l - iK).

The transition matrix T is given by

T = l - S , (22)

or

- 2iK 1 - iK

T = -

It can be proved that K is real and asymmetric, and S is symmetric and unitary.

c. DIFFERENTIAL CROSS SECTION

With the use of the T-matrix defined previously, the scattering amplitude in Eq. (4) for excitation from an initial state a,&,SqML,MS, to a final state a , ~ L , ~ S q ~ M L Y M ~ , . in LS coupling is given by

6 Shinobu Nakazaki

where L , and S, and T are the total angular momentum, spin and parity of the system, and C the Clebsch-Gordan coefficient.

Substituting Eq. (24) into Eq. (9), averaging over the initial states, and sum- ming over the final states, the differential cross section is obtained by

1 8(2L,, + 1)(2S, + I ) k :

- - d d q + 4 ' ) dfl (25)

X c (2A + l)AA(q+ q')Ph(COSO). A = O

Here the coefficient A, is given in terms of the 3-j and 6-j symbols as

x M;, f /;4q + q')MS, f,(q -+ 4 ' ) .

where j , is an angular momentum transfer (j, = k,, - k,.) and M;,( (4 + 4 ' ) is defined by

MZ;,(q + 4 ' ) = c ( - l ) I q + I q [(21,, + 1)(21,. + 1) L W

D. THE CLOSE-COUPLING EQUATION

In order to obtain equations for the functions F,., we introduce the projection operator formalism of Feshbach (1958, 1962). We suppose that the wave function is expanded as in Eq. (12). We define P as the operator that projects onto the target eigenstates $, in Eq. (12), and Q as the operator that projects onto the quadratically integrable functions x,. The function x, may be taken to be orthogonal to the functions $, .

Then, we have

Assuming that the expansion in Eq. (12) is complete, we have

P + Q = l . (29)

Operating P and Q on the Shrodinger equation (2), we obtain

P(H - E)(P + Q)" = 0, Q(H - E)(P + Q)" = 0,


where the subscript ( N + 1) is omitted. These may be interpreted as two coupled equations for the two functions P" and Q". They are formally uncoupled in such a way that first we solve Eq. (31) to give

QHP'P, 1

'* = ' Q ( E - H ) Q

and then put this into Eq. (30) to derive

P(H + HQ QH - E}P* = 0. Q ( E - H ) Q

If an optical potential is defined as

(33)

(34)

then we have an equation of usual form

( H , + Kp, - E)P" = 0, (35)

where H , = PHP. The explicit form of the coupled equations (35) is

& ( r h ' + I ) + 1; W , ( r N + I , r N ) & ( r N ) d r N

+ 1; K y ( r N + I 9 r N ) 4 q ( r N ) d r N 1 . (36)

where I, is the orbital angular momentum of the scattered electron, V, and W, are the direct and nonlocal exchange potentials, and K,, is the nonlocal optical potential (34).

We may express explicitly these potential in the form

x ( H - E)$j(XiI; f N c N ) e q ( r N ) d T *

The integration in Eqs. (37) and (38) includes all coordinates and spins except the radial coordinate of particle N + 1. The x i denotes coordinates and spins omitting the Mth coordinate and spin from a set of ( N + 1) coordinates and spins. The potential V,(r)(i = J ) represents the static or nonexchange interaction of an electron with the target, while Wj represents the exchange interaction of

8 Shinobu Nakazaki

the electron with the target. The potentials x ( r ) for i # j fall off faster than r - I

at a larger r. This has the asymptotic form

where

The nonlocal potentials Wj and K, decrease exponentially for large r, which is determined by the target state +i and the quadratically integrable functions x i .

E. THE R-MATRIX METHOD

Originally the R-matrix method was introduced by Wigner (1946a, 1946b) and Wigner and Eisenbud (1947) in a fundamental paper concerned with the theory of nuclear reactions. Burke er al. (1971) first applied the R-matrix method to electron-atom collisions. A review of the use of the R-matrix method was given by Burke and Robb (1975). The basic idea in the R-matrix method is that the dynamics of electron-atom(ion) system is different depending on the relative distance r of the incident electron and the atomic nucleus. The space surrounding the target is separated into two regions, an inner one ( r € a) and an outer one ( r > a). The radius a is chosen so that the charge distribution of the target states is contained within the sphere r = a. In the inner region ( r < a), electron exchange and correlation between the scattered electron and the N-electrons in the target atom(ion) are important and the (N + 1) electrons in the total system behave as bound electrons. In the outer region ( r > a), the electron exchange between the scattered electron and the bound electron in the target atom(ion) can be neglected. Then, for r > a, the collision is described as the scattered electron moving in a long-range multipole potential. The inner and outer regions are treated using different theoretical approaches, and the wave functions in these two regions are matched on the boundary r = a. The inner region is studied using the configuration-interaction type description similar to the standard approach for the bound state. The outer region is studied using a close-coupling method without the electron exchange effect. By analogy with Eq. (12), the total wave function in the inner region is expanded in the form


In Eq. (41) an additional index j is introduced in the first summation to indicate that the radial function representing the scattered electron, E(r ) , is expanded in a complete set of basis orbitals ui j ( r ) . Those orbitals are defined only in the inner region and have nonzero value on the boundary.

The first expansion in Eq. (41) is taken over all possible scattering channels but is usually truncated to include only a finite number of target eigenstates and some additional pseudo-states. The pseudo-states are introduced to allow for polarization effect. The functions x j are (N + 1)-electron functions constructed from the same bound orbitals and pseudo-orbitals as those used for the N - electron target states I,II~. This takes accounts of short-range correlation effects between the scattered electron and the target ones.

The continuum basis orbitals uii are chosen as the solution of the second-order differential equation:

with the boundary conditions

u l J ( o ) = O, (43)

(44)

The A,,, on the right-hand side of Eq. (42) are the Lagrange multipliers. They

-4/ a = b e

u,(d dr r = y

are determined so that

1: u,,(r)pv(r)dr = 0, if 1, = /,, (45)

where pY(r) are the radial bound orbitals describing the target states. The potential, V ( r ) in Eq. (42) is suitably chosen to represent the static charge distribution of the atom(ion).

The coefficients c]k and d,], in Eq. (41) are determined by diagonalizing the Hamiltonian H,+ I as

(pklHN+IIqk’) = Ef+16kk’, (46)

where the radial integral is taken over the finite range 0 5 r 5 a. The radial functions occurring in q k can be written as an expansion in terms of the basis orbitals:

wk(r) = 2 utJ(r)dt)k* (47) I

We assume that we can expand the total wave function qL in the inner region at any energy E in terms of the basis function q k as

10 Shinobu Nakazaki

We now define

E(r) = 2 AhAwdr), (49)

which is the radial wave function of the scattered electron in channel i in the total wave function qL. It can be shown after some manipulation (Burke ef al., 197 I ; Burke and Robb, 1975) that the radial function E ( r ) at r = a can be related to its logarithmic derivative on the boundary by

where

is called the R-matrix. The amplitudes WA(a) and the poles E f + l of the R-matrix are obtained from the eigenvectors and eigenvalues of the Hamiltonian matrix in Eq. (46). The most important source of error in the R-matrix method is the truncation of Eq. (51) to a finite number of terms. The contribution due to this has been proposed by Burke et a f . (1971).

In order to obtain the K-matrix we solve the equation for r > a at energy E. The solution has the expansion (12). This follows directly from the representation of the R-matrix basis by Eq. (41) and the expansion of the total wave function in terms of this basis by Eq. (48). For r > a, as mentioned previously, the exchange effect between the scattered electron and the target ones vanishes. Thus, in this region, the close-coupling equation has the form, L -

d' f,(l, + 1) + 2(Z - N ) r 2 r

The K-matrix is obtained by matching at r = a the inner region solution (Eq. (50)) to the outer region solution of the equation (52).

In the R-matrix method the effect of the optical potential V,,, introduced in Section 1I.D can be approximated by including pseudo-states in the first sum in Eq. (41) and suitably choosing ( N + 1)-electron configurations in the second sum.

F. THE REGION OF INTERMEDIATE ENERGY

Now we consider electron-atom(ion) scattering at intermediate energies, which is commonly but somewhat loosely defined as the region just below the ioniza-

DIFFERENTIAL CROSS SECTIONS FOR EXCITATION OF HELIUM ATOMS 1 1

tion threshold to several times that energy. The expansion (12) or (41) describing the collision system has been successfully applied to the calculation of the cross section in the lower-energy region. In fact, the general-purpose computer pack- age based on the R-matrix method (Berrington et al., 1974, 1978) or a close- coupling scheme (IMPACT: Crees et al., 1978; NIEM: Henry et d., 1981) has been very widely used so far. At low energies, where only a few channels are open, the expansion (12) or (41) provides the most appropriate description of the collision. All open channels, as well as a few closed ones, can be retained in the first expansion in the equation (12) or (41) and the second expansion allows for short-range correlation effects.

However, at intermediate energies, there are a large or infinite number of open channels to include in the expansion of the wave function. Several approaches have been attempted to extend the low-energy methods to the intermediate energy range. One possibility is to represent the large number of states excluded in the expansion by a small number of pseudo-states of the target. Those pseudo- states can be determined to represent the long-range polarization. In so doing a part of the continuum can be included in the expansion. When pseudo-states are included, however, we usually have unphysical thresholds and pseudo- resonances. In some cases their effect is remarkably large.

Recently, the intermediate energy R-matrix (IERM) theory was introduced by Burke et al. (1987) to proceed to the modeling of electron scattering by atomic and molecular systems at intermediate energies. In the IERM theory, the basis states in the internal region consist of terms that include not only target states coupled to continuum orbitals but also target states of the singly ionized ion, coupled to two continuum orbitals. The latter terms have projections onto the exact intermediate energy scattering wave function. In the outer region, the wave function is approximated by a close-coupling expansion over the channels of interest, as in the standard R-matrix mentioned in Section 1I.E. The IERM method has been applied to electron-hydrogen scattering by Scholz et al. (1988, 1991), Scott et al. (1989), and Scholz (1991).

G. THE OPTICAL POTENTIAL METHODS

Use of the optical potential described in Section D has been reviewed by Brans- den and McDowell (1977, 1978), Byron and Joachain (1977), and Walters (1984) so far.

McCarthy and Stelbovics ( 1980, 1983) developed the coupled-channel optical (CCO) method in momentum space. They applied the method to calculate DCS in the e - + H collision and obtained good agreement with experiment. The CCO method was then applied to the calculation of TCS and DCS for hydrogen, helium, sodium, and magnesium atoms by McCarthy and his coworkers (e.g., McCarthy et al., 1989; Bray et al., 1989, 1990; Brunger et al., 1990).

A detailed description of the CCO method in momentum space can be found

12 Shinobu Nakazaki

in McCarthy and Stelbovics (1983), and McCarthy and Weigold (1990). Here we briefly review the essential features of the CCO method. The CCO calculation is based on the solution of the coupled integral equations (McCarthy and Stelbovics, 1983)

where

T,*, (k,.q’(Tlqk,) = (k,,q’IVI*Fb+fk,)), (54)

is the T-matrix element for the transition from the channel state Iqk,) to Iq’k,,) of the ( N + 1)-electron system. The ket 1°F)) is the exact solution of the (N + 1)-electron Shrodinger equation for entrance channel q.

The potential V includes an appropriate exchange operator. The complex polarization operator V(Q) is given by (see Eq. (34) in Section D)

1 V‘Q) = PVQ E l + - ) -

QHQQ“ ( 5 5 )

The matrix elements of VQ for a two-electron atom can be expressed as (Mc- Carthy et a f . , 1988)

(k,q’lV(Q’lqk,) = ] d3k ( a , + b,PJ (k,q’IVI*q..’-l(k)) d’EQ

1

E(+’ - E,,, - -k2 X (*,J-)(k)IVIqk,). (56) 1

2

Here ‘P>:)(k) is the three-body wave function for a final target state q” with an appropriate boundary condition and P, is the space-exchange operator. Direct and exchange terms have coefficients a, and b, that depend on the total spin S. For the continuum target states q” the summation becomes a momentum integration.

So far, various approximations to practically evaluate the complex polarization matrix elements (53) have been introduced by McCarthy and his coworkers (see the review article of McCarthy and Weigold, 1991).

H. THE DISTORTED-WAVE METHODS AND RELATED METHODS

The distorted-wave (DW) method has been proved useful in modeling electron- atom(ion) excitation at intermediate energies. The advantage of this method is


its simplicity and flexibility. We describe one of the simplest derivations of the DW method based on the standard first-order perturbation theory.

Introducing a distortion potential, VDW, we rewrite the total Hamiltonian HN+ I in Eq. (1) as follows:

(57) -

HN+l = H + V - VDw,

where

(59)

Taking the difference, V - U D W , as a perturbation, we employ the first-order perturbation theory. The transition matrix for the excitation q + q' is given by

(60)

v=--+c-. N N 1

r N + l i = l r i N + I

T!$ = ( x b ~ ' ( V - VDwlxb").

(7T - E)X'*' = 0,

The unperturbed wave function xF)(xb~)) is a solution of the wave equation

(61)

with the outgoing (incoming) boundary condition. Taking the distortion potential UDW as dependent only on the coordinate of the scattering electron, i.e., VDW = UDW(rN+ ,), we can separate the wave function xj') into the target wave function o i (x1 , . . . , x,) and the distorted wave function e!*)(xN+l). Then, the transition matrix T$Y is written in the form

= TDWld) 4 4 + TDWCe), 4 4 (62)

T$YId' = (@4'(XN:I)e$;)(XN+I)iV - vDwl@q(XN:I)eb+)(XN+I))r

T B Y = - N (@q'(X, ' ) e k ; ) ( X N ) I V - uDWI@4(X,:l)8 b+ ' (XN+ 1)).

(63)

(64)

Separating out the spin part from the distorted wave function as 81"(x) = Gj*)(r)ij(m3(cr), we have from Eq. (61)

- 2UDW]G!*)(r) = 0. 2(Z - N ) [V* + k f +

r

It should be noted that, in this case, both the incoming and the outgoing distorted waves are obtained in the same distortion potential VDW.

Another DW method has been derived from an approximation to the two-state close-coupling method by Massey and his colleagues in the 1930s. In the method, the incoming and the outgoing distorted waves are obtained in the potentials chosen separately for the initial and the final target state, respectively. That is, Vtw = V,, for GF) and V?" = Vsq, for Gb;), where the potential Vi is defined in Eq. (37).

Many other kinds of the distorted-wave approximation have been introduced

14 Shinobu Nakazaki

so far. A number of text books and review articles (Bransden and McDowell, 1977, 1978; Walters, 1984) include some account of the distorted-wave method. Itikawa recently (1986) summarized the theories of the distorted-wave method proposed so far and made a review of the results obtained for electron-atom(ion) collisions.

III. Excitation of Helium Atoms

A. EXPERIMENT

Experimentally the differential cross section is obtained by the technique called electron spectroscopy. With this technique, the energy and the scattering angle of the electron are analyzed after collision. Table I lists those experiments that obtained any absolute value of DCS for the excitation of 2'S, 23P, 2'S, and 2'P states of He from its ground state for below 200 eV.

B. THEORY

Fon et al. (1979, 1980) calculated the DCS for the 1's --$ 2'S, 1's + 2'S, 1's + 23P, and 1's + 2lP transitions in the energy range 21.4-29.6 and 81.63-200 eV using the five-state R-matrix (RM5) method. In the same year, Bhadra et al. (1979) reported the DCS for the excitation of the to 2'S, 2'S, 2'P, and 2'P states in the energy region from 29.6 to 100 eV, obtained by the five- state close-coupling (CC5) method. The former calculation is an extension of the work of Berrington et a[. (1975) and employs the FORTRAN program RMA- TRX (Berrington et a f . , 1974). The five atomic eigenstates are constructed from four basis orbitals, i.e., Is, 2s, 2p, and 3s, and two pseudo-orbitals, 3p and 3d. Each eigenstate $; in Eq. (41) is formed with the configuration interaction (CI) taken into account (see Table 11). The boundary radius is chosen as a = 16.044. The potential V(r ) in Eq. (42) is V ( r ) = (4/r) exp ( - 1.8r). They adopt 25 continuum basis orbitals uo in Eqs. (41) and (42) to calculate cross sections up to 200 eV. Bhadra et al. (1979) use also five atomic eigenstates of CI type constructed from three Slater-type orbitals, i.e., Is, 2s, and 2p. The coupled integro-differential equations are solved by means of the noniterative integral equation method (NIEM: Smith and Henry, 1973). They use the program pack- age NIEM developed by Henry er al. ( 198 1 ).

Berrington et af. (1987) calculated the DCS for the 1's + 2'P transition at 22, 24, 26.5, and 29.6 eV by using the results of 11- and 19-state R-matrix (RM11 and RM19) calculations. By using the RM5, RMll , and RM19 results, they studied the convergence of the results with respect to the number of the


states included. Furthermore, Fon et al. (1988) made calculations of the DCS for the excitation of the 2'S, 2'S, 2'P, and 2'P states at 29.6 eV using the RM19 calculations. Fon et al. (1991a) obtained the DCS for the excitations 1 's +

n'"P(n = 2, 3, and 4) at energies 22, 24, 26.5, and 29.6 eV using the RMll and RM19 calculations, and studied the convergence for these transitions.

The details of the wave functions of the target and the scattered electron in the RMll calculation are described by Freitas et al. (1984) and Berrington et al. (1985). They include the eleven lowest states (n = 1, 2, 3) of helium, which are represented by six orbitals, i.e., Is, 2s, 2p, 3s, 3p, and 3d, and two pseudo- orbitals & and qp. Use is made of 25 continuum R-matrix basis orbitals and a boundary radius of 38 a.u. Because pseudo-resonances appear beyond 27 eV, the DCS calculation is made only for the energy below 26.5 eV.

The RM19 calculation, in which the first 19 atomic states ( n = 1, 2, 3, and 4) are included in the expansion of total wave function (41), is reported in detail by Berrington and Kingston (1987). Ten orbitals, i.e., Is, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, and 4f, and two pseudo-orbitals 3s and yp are used, so that an R-matrix boundary radius of 60 a.u. is needed to contain the target wave function. As many as 36 continuum basis orbitals are needed in each channel to obtain convergence of the R-matrix at energies up to 30 eV. A modified version of the R- matrix program (RMATRX) of Berrington et al. (1978) has been used both in the RMll and the RM19 calculations.

Also using the same RM19 calculation, Fon and Lim (1990) obtained the energy dependence of the 1's + 2jS DCS at scattering angles 30", 55", 90", 125", and 140" from the excitation threshold up to 23.8 eV. They calculated the corresponding DCS also using the RMl 1 method. A convergence along the RM5, RMll , and RM19 calculations is tested on the 1 ' S --* 2's DCS at each angle.

Very recently Fon et al. (1 99 I b) calculated the energy dependence of the DCS at 90" as a function for the 1 'S + 2's excitation using the result of the 29-state R-matrix (RM29) calculation done by Sawey et al. (1990). They use 17 orbitals including two pseudo-orbitals, i.e., 6s and 6p. In this calculation the radius of the inner region has to be pushed out farther to 83 a.u. because of the introduction of the highly excited n = 5 level. A larger number of continuum basis orbitals, 48, is required in each channel.

Nakazaki et al. (1991) calculated the DCS for the 1's + 2jS and 1's + 2'P transitions at incident energies of 100 and 200 eV using the 1 I-state and polarized pseudo-state R-matrix (RM11 and RM7) methods to compare with the result of a recent experiment by Sakai et al. (1991). Their RMl1 calculation is the same as in Berrington et al. (1983, except that the number of continuum basis orbitals included is much larger (64) in the former than the latter.

The coupled-channel optical method described in Section 1I.G. was applied by Brunger et al. (1990) to calculate the DCS for the e - + He collision at

-

TABLE I

MEASUREMENTS OF THE ABSOLUTE DIFFERENTIAL CROSS SECTIONS FOR THE EXCITATIONS FROM THE GROUND STATE IN HELIUM

Transitions

References I '~-2~s I 1s-2~~ I ls-2's I 's-2'P

Vriens et a/ . ( 1968) E = 100-225 100-400 100-400

Chamberlain ef a/. ( 1970) 50-400 50-400

Truhlar et a/. (1970) 26.5-81.6

0 = 5-20 5-20 5-20

5 5 0 0-

10-80 E

2 25- I50 f i

0 Crooks ef a/. (1972) 40-70

Crooks ( 1972) 50, 100 50. 100 $ 10- 150 10- I50 5

Opal and Beaty ( 1972) 82,200 82 82,200 82,200 30- I50 30- 105 30- 150 30- I50

Rice el a/. (1972) 26.5-81.6 10-80

Hall et a/ . (1973) 29.2, 39.2.48.2 29.2, 39.2, 48.2 29.2. 39.2, 48.2 29.2, 39.2.48.2

Trajmar (1973) 29.6.40. I 29.6.40. I 29.6, 40. I

Truhlar eta/ . (1973) 29.6.40. I

10-125 10-125 10- 125 10- 125

3-138 3- 138 3-138

3- 138

Suzuki era/. (1974)

Chutjian and Srivastava (1975)

Dillon and Lassettre (1975)

Dillon (1975)

Pichou er a/. ( I 976)

Yagishita et a!. (1976)

Cartwright er a/. (1989)

Brunger ef a/ . (1990)

Sakai eta/. (1991)

Pichou e ta / . (1976)"

Phillips and Wong (1981)"

200-500 7.5-35 21.42 10- 125 50-500 4.5- 120 29.6,40. I 2.5- 140 29.6,40. I 2.5-90 200-800 0-12 19.82-23.42 30.60.90, 120 19.82-24.5 55.90

60.80 5-136

22.97 10- 125 50-200 7- 120 29.6, 40.1

29.6, 40.1 2.5-90

2.5- 140

20.97-24.57 30,60,90, 120 20.97- 24.5 55.90

50-500 4.5- 120

200-700 7.5-35

22.22 10-125

29.6,40. I 2.5- 140 29.6,40.1 2-90

20.62- 24.22 30,60,90, 120 20.62-24.5 55,90

50-500 5- 120 60, 80 5- 136 200-700 7.5-35

23.22 10-125

29.6.40. I 2.5- 140 29.6 5- 100

21.22-24.83 30,60,90, 120 21.22-24.5 55,90

E: the incident energy (eV). 0: the scattering angle (degree). 'Energy dependence for each angle.

TABLE I1

TARGET STATES, ORBITALS, A N D CONFIGURATIONS USED I N THE VARIOUS CALCULATIONS FOR HELIUM

CC5“

States Orbitals Is, 2s. 2p Configurations I S lS2S. IS’, 2s’. 2p’

ls2p. 2s2p

n s 2; i.e., I ’S , 2’s. 2’s. 2 ’~ . 2 ’ ~

’s ls2s 1 . 3 ~

RM5’

States Orbitals Configurations ‘ S

n s 2; i.e., as in CC5 Is , 2s. 2p, 3s. jp , j d ls2s, ls3s, 2s3s, 2p3p, Is’, 2s’. 3s’. 2p’, 3p’, jd’

ls2p. I d p , 2s2p. 2s3p. 3s2p. 3s3p. 2p3d. 3p3d ’s ls2s, I s ~ s , 2s3s, 2p3p ‘”P

RMI 1‘

States Orbitals Configurations ‘ S

n s 3; i.e., as in CC5 + 3’s. 3’s. 3’P. 3’P. 3’D, 3’D Is, 2s, 2p, 3s, 3p, 3d, ds, dp ls2s, 1~3s . Is&, Is’, 2s’. 3s’, 2p’, 3p’, dp’, 3d’

ls2p, ls3p. ls4p. 2s2p. 2s3p, 2s4p ’s ls2s, ls3s, Is& ‘.’P

’D 1 s3d ID ls3d. 2p’

RM19d

States

Orbitals Configurations ‘ S

’s

‘D I 3P

’D “’F

n s 4; i.e., as in RMI 1 + 4%. 4’s . 4’P, 4IP, 4’D, 4’D. 4’F, 4’F Is, 2s. 2p, 3s. 3p, 3d, 4s, 4p, 4d, 4f, 5s. 5p ls2s, ls3s, ls4s. Ids. Is’, 2s’. 3s’. 2p’, 3p’, 3d’ ls2s. ls3s, ls4s, Ids ls2p, ls3p, ls4p. l d p , 2s2p. 2s3p. 2s4p. 2s5p ls3d. ls4d. 2p’ ls3d, ls4d 1 s4f

RM29‘

States

Orbitals Configurations I S

’s

‘D 1.3p

’D “’F 1 . 3 ~

n s 5 ; i.e., as in RM19 + 5%. 5 ’ S , 5’P. 5’P. 9 D , 5’D, 5’F. 5‘F, S’G, 5’F IS, 2% 2p. 3s. 3p, 3d, 4s. 4p. 4d, 4f, 5 s , 5p, 5d, 5f, 5g, 6s. bp 1~2s. 1~3s. ls4s, Is%, lsbs, Is’, 2s’. 3s’. 2p’, 3p’, 3d’ ls2s, ls3s. 1~4s. Is%, Isbs ls2p. ls3p. ls4p, ls5p. Isbp. 2s2p. 2s3p, 2 . ~ 4 ~ . 2s5p, 2s6p ls3d, ls4d. ls5d. 2p’ ls3d. ls4d. ls5d ls4f, ls5f 1 s5g

Source: “Bhadra era/. (1979); ’Berrington era/ . (1975); ‘Berrington era/ . (1985);

dBerrington and Kingston (1987); ‘Sawey et a/. (1990).

18


incident energies 29.6 and 40.1 eV. They couple ten atomic states 1 IS, 2jS, 2'S, 2'P, 2'P, 33S, 3'S, 33P, 3'P, and 3'D, each of which is of CI-type formed by Is, 2s, 2p, 3s, 3p, 3d, 4s, and 4p Hartree-Fock orbitals, andS, p, and a pseudo- orbitals. The optical-potential calculation is made with taking account of the

and z3P -+ 2'P channel couplings by the formalism of McCarthy et al. (1988). They calculate DCS with and without the optical potential V(Q' in order to assess the effect of the continuum.

Table 11 summarizes the target states, orbitals, and the configurations used in the CC5, RM5, RM 1 1, RM19, and RM29 calculations. The radial part of the orbital in the table is expanded in the form

1's -+ PS, 11s -+ ~ ' J s , 1's + ~ I J P , 2 ' s +. IS, 2 3 ~ -+ YS, 2 ' ~ -+ ~ I P ,

The Is orbital in all the R-matrix calculations is the ground state orbital of He+, while the Is orbital in the CC5 calculation is the three-parameter ground state orbital of He given by Clementi (1965). The atomic orbital parameters c , , I , , t , except for the Is orbital are determined by optimizing the excited-state energies with Hibbert's CIV3 program (1975).

Table 111 gives a comparison of excitation energies obtained in the calculations mentioned previously. In Table IV we show the length ( L ) and velocity ( V ) forms of oscillator strengths calculated with the target wave functions used in the respective calculations. They are compared to the accurate values determined by Wiese et al. (1966).

Because of the difficulty in representing the closed-shell ground state with the resulting open-shell orbitals, the correlation energy obtained in the ground state in the R-matrix calculation is less accurate than that in the CC5. Therefore, as

TABLE 111

ENERGY LEVELS ( I N A.u.) OF HELIUM TO THE GROUND STATE I N THE VARIOUS CALCULATIONS

State CC5" RM5b RMI 1 ' RM19" Observed

I 's

2 ' s 2'P 2lP 3's 3's 3'P

2's

3'D 3'D 3'P

0.0 0.0 0.0 0.7397 0.71053 0.68402 0.7639 0.73973 0.71476 0.7751 0.751 10 0.7272 1 0.7879 0.76054 0.73581

0.79018 0.79779 0.80065 0.80266 0.80267 0.80351

0.0 0.68267 0.71354 0.72555 0.73548 0.78858 0.79641 0.79928 0.801 30 0.80 132 0.80217

0.0 0.7284 0.7577 0.7705 0.7798 0.8349 0.8423 0.8455 0.8480 0.8480 0.8485

Source: As in Table 11.

20 Shinobu Nakazaki

TABLE IV

OSCILLATOR STRENGTHS BETWEEN THE n c z STATES I N THE VARIOUS CALCULATIONS FOR

HELIUM ( L = LENGTH FORM, V = VELOCITY FORM)

Transition CC5" RMSb RM11' RM19d RM29' Exp'

1'S+2'P L 0.341 0.279 0.215 0.255 0.249 0.276

2 'S+2'P L 0.333 0.366 0.363 0.364 0.376

2's- 2'P L 0.592 0.561 0.554 0.560 0.539

V 0.318 0.278 0.266 0.273 0.268

V 0.438 0.345 0.290 0.325

V 0.466 0.573 0.519 0.557

Source: As in Table 11; 'Wiese er al. (1966).

seen from Table 111, the excitation energy obtained in the R-matrix calculation is worse than that in the CC5 calculation. In contrast, the oscillator strength in the R-matrix calculation gives a good agreement with the experimental result and is better than that of the CC5 because the 3p and 3d orbitals used in the RM5 calculations were adjusted to well reproduce the dipole and quadrupole polarizability of the ground state, respectively.

For the intermediate energy range, the distorted wave (DW) method and an eikonal method have been used so far to evaluate DCS.

Madison and Shelton (1973) proposed a DW approximation in which both the incoming and the outgoing distorted waves are obtained in a distortion potential. The potential is defined to be the spherical average of the interaction of the free electron with the atomic electrons in the excited state (see Section 1I.H). Thomas et al. (1974) applied the first-order form of the many-body theory (FOMBT) to obtain the DCS in the energy range from 29.6 to 500 eV for the excitation of the 2'S, 2'S, 2'P, and 2'P states. The FOMBT is a variation of the DW method. They calculated the distorted waves in the field of the ground state and employed both in the initial and in the final channels.

Meneses et a f . (1978) calculated the DCS for the 1's + 2'P transition at 80 eV by employing a distorted-wave model, which is a simplified version of the FOMBT of Thomas et a f . (1974). That is, they replaced the transition density of the random phase approximation by that of the Hartree-Fock approximation. Their results are in good agreement with those of Thomas et al. The same method was also applied to obtain the DCS for the excitation of the 2% and 2'P states from the ground state at energies from 30 to 200 eV by Baluja and Mc- Dowel1 (1979). Their results are in good agreement with those of Thomas et al. Recently, Csanak and Cartwright (1988) also calculated the DCS for the 1 'S + n'P ( n = 2, 3) transition at 30, 81.63, and 100 eV with the FOMBT of Thomas et al.


Scott and McDowell(l975, 1976) calculated the DCS for the excitation of the 2'S, 2'S, 3'S, 2'P, 2'P, and 3'P states at 29.2-200 eV using the distorted-wave polarized-orbital (DWPO) method of McDowell et al. (1973, 1974) for the e - + H collision, which allows for the effect of distortion in the initial channel, including the effect of target polarization, and also for distortion of the target by the dipole polarization. They do not consider, however, the distortion in the final channel.

Byron and Joachain (1975) obtained the DCS for the 1 'S + 2's transition using the slightly modified version of the eikonal-Born series (EBS) method at 200 eV. In their calculation, the direct amplitude is taken as

fd = fa, + fez + 2 fGtg = fG - fa + fez, (67) n = 3

where thef,, and fez are the first and second Born scattering amplitudes, respectively. The f, andf,, are the full and the nth order Glauber amplitudes. For the 1's + 2'P transition at 200 eV Joachain and Winters (1977) used the EBS method in which the direct amplitude is evaluated asfd = fB, + fez + f G 3 . This equation is correct through order k - 2 for all momentum transfer. The exchange effects in the EBS calculations for both the 1's + 2's and the 1's + 2'P transitions are included by using the Ochkur amplitude. More details of the eikonal-Born series method are described in a number of review articles (Byron and Joachain, 1977; Bransden and McDowell, 1978; Walters, 1984).

Mansky and Flannery (1990) applied the multichannel eikonal theory (MET) to the examination of DCS for the 1's + n'L transition ( n = 2, 3; L = S, P, D) at 40-500 eV. Their approximation is a high-energy small-angle approximation to the solution of a many channel close-coupling problem, neglecting electron exchange effect. They made a 10-channel calculation using the analytical Hartree-Fock frozen-core wave functions. They correct misprints concerning the original MET results of Flannery and McCann (1975a, 1975b).

Singh er al. (1983a) obtained the DCS for the 1 IS + 2's transition at 100 and 200 eV using the variable-charge Coulomb-projected Born approximation proposed by Schaub-Shaver and Staufer (1980). They take into account the distortion due to the static interaction only in the final channel.

Srivastava et al. (1985) evaluated the DCS for the 1 'S + 2 ' s transition at 100 and 200 eV by using the DW approximation. The effect of the distortion of incident electron, including that due to polarization of the target and the exchange effect are appropriately taken in both the initial and the final channels. Furthermore, their DW approximation was extended by Katiyar and Srivastava (1988) to obtain the DCS for the l'S, 2 's + 2'P, 3'S, and 3'P transitions at energies of 50, 80, and 100 eV. In this calculation the distorted wave in the initial channel is the same as used by Srivastava et al. (1985), but the one in the final

22 Shinobu Nakazaki

channel is evaluated with the use of the following two choices of the distortion potential: U , = V?t, + V,,,, and U 2 = Vgq + V,,,,, where V:$, and V,,, are the ground state static and polarization potentials, and V $ j ' , the final state static potential of the target. The first choice is the same as that adopted by Srivastava et al. (1985).

To obtain a reliable DCS, compared with the experiment, Srivastava ef al. (1989) introduced an averaged distortion potential U for both the initial and final channels as U = ( U , + U2)/2. They calculated the DCS for the 1's + 2'P transition at 60, 80, and 100 eV.

Madison (1979) and Stewart and Madison (1981) obtained the DCS for the 1 'S + 2'P transition at incident energies of 40-200 eV by including polarization exchange and absorption potentials in the DW approximation. They found no evidence that these potentials in the DW calculation improve an agreement between experiment and theory for DCS.

Bransden and Winters (1975) obtained the DCS for the excitation of the 2's and 2 ' s states for energies 50-150 eV by using the second-order potential method of Bransden and Coleman (1972). The continuum wave function to describe the incident channel is determined from a one channel equation with the effective second-order nonlocal potential allowing for polarization, absorption, and exchange. The final continuum wave function is determined using the static interaction in the final state. They used also the standard DW approximation, in which the nonlocal potential in the initial channel is ignored.

Madison and Winters (1983) calculated the DCS at 60, 80, 100, and 200 eV for the 1's -+ 2'P transition using the second-order distorted-wave amplitude. To make the evaluation of the nonlocal second-order distorting potential more tractable, they adopted the approximation of replacing it by a local second-order potential obtained in the closure approximation.

Amus'ya ef ul. (1984) took the first- or the second-order approximation with respect to the electron-atom interaction potential and treated the distortion of electrons by the atomic field in the initial, final, and intermediate states using the Hartree-Fock self-consistent field method. They obtained the DCS for the I 'S -+ 2% transition at the energies 30.8, 79, 1 1 1, and 192 eV.

C. COMPARISON OF THEORETICAL RESULTS A N D EXPERIMENTS

Figures 1-8 show theoretical and experimental differential cross sections as a function of scattering angle at the incident energies of 29.6, 40.1. 100, and 200 eV for the excitations of 2 9 (Figs. 1 -2), 23P (Figs. 3-4), 2 ' s (Figs. 5-6), and 2'P (Figs. 7-8) states. The range of the scattering angle shown is limited to


I 14 20 40 sb so Id0 1:o 110 ANGLE(deg1

(a)

I I I I I I J

He l tS-23S 40. 1 e V : r

FIG. 1. Differential cross sections for the 1 'S + 2% transition in He at (a) 29.6 eV; (b) 40. I eV. Theory: . . . . . . . . . , FOMBT (Thomas et al., 1974); - - - -, FOMBT (Amus'ya et al., 1984) at 30.8 eV; - - -, DWPO (Scott and McDowell, 1975); - -, second-order many-body theory (Amus'ya et al., 1984) at 30.8 eV; - .. -, CC5 (Bhadra et al., 1979); - . -, RM5 (Fon et al., 1979); ~ , RM19 (Fon er al . , 1988); - - -, CCOlO (Brunger et al., 1990). Experi- ment: V, Crooks et al. (1972); 0, Hall et al. (1973) at 29.2 eV and 39.2 eV; v, Trajmar (1973); + , Cartwright et al. (1989); 0, Brunger et al. (1990).

0"- 140°, because almost all the experiments so far have been done in this region, and a detailed comparison can be made in such a widened figure.

As seen from these figures, the experimental results are in good agreement with each other, except in a few special cases (e.g., the forward scattering at 200 eV for the 1's + 2's transition). Therefore, the following discussion is concentrated on the mutual comparison of the theoretical results and how well those theoretical results can reproduce the experimental data.

First we compare the elaborate calculations of close-coupling type. Figures l(a)-8(a) show a comparison between the five-state close-coupling (CC5) calculation of Bhadra et al. (1979) and the five-state R-matrix (RM5) calculation by Fon er al. (1979, 1980). The relative angular dependences of the DCS obtained by those two calculations agree well with each other. The absolute magnitudes of the DCS, however, disagree. The largest discrepancy (by a factor of three) occurs for the 1's + 2's transition at 100 eV.

24 Shinobu Nakazaki

16zt\ He l'S-Z3S 200eV

ANGLECdeg) ANGLE(deg1

FIG. 2. Differential cross sections for the I 'S -+ 2's transition in He at (a) 100 eV; (b) 200 eV. Theory: . . . . . . . . .. FOMBT (Thomas et a/.. 1974); - -, DW (Bransden and Win- ters, 1975); - - - -, DW (Baluja and McDowell, 1979); - .. -, CC5 (Bhadra er a/ . . 1979); _ . - , RM5 (Fon et a/. , 1979); - - -, RM7 (Nakazaki et a/.. 1991); ~ , RMll (Nakazaki e t a / . . 1991). Experiment: *, Vriens eta/. (1968); 0, Crooks (1972); 0, Opal and Beaty (1972); x , Dillon (1975); A, Yagishita eta/ . (1976); 0, Sakai eta/ . (1991).

(a) (b)

16'- 1 I I I I I _

He l 'S -Z3P 29.6eV :

-3- " 0 20 40 60 80 100 120 140

ANGLE(deg1

(a) FIG. 3. Differential cross sections for the 1 'S

Other details are as for Fig. I .

20 40 60 80 100 120 140 ANGLE(deg;

(b) + 2'P transition in He at (a) 29.6 eV; (b) 40.1 eV.


lo2. I I I I 1 I . 1 I I I 1 I _

He l ' S - 2 3 P 200eV:

1°: 20 40 6b /O Id0 I;O 140 l o \ 2b 40 60 -90 I i O . 1;O 140 ANGLE(deg1 ANGLE(deg1

FIG. 4. Differential cross sections for the 1's .--) 23P transition in He at (a) 100 eV; (b) 200 eV. (a) (b)

Theory: - - - -, DWFQ (Scott and McDowell, 1976). Other details are as for Fig. 2.

loo I I I I I I

He l 'S-2 'S 2 9 . 6 ~ V

t 1 fi5-

0 20 40 60 80 100 120 140 ANGLE(de9)

I I I I I

-5- " 0 20 40 60 80 100 120 140

ANGLE(deg1

(a) (b) FIG. 5. Differential cross sections for the 1's + 2's transition in He at (a) 29.6 eV; (b) 40.1 eV.

Theory: - . -, RM5 (Fon et al., 1980). Other details are as for Fig. 1.

26 Shinobu Nakazaki

l o o I I I I I I -

He 1 '5-2 's 200eV- -

;\ 10' -

L m \

;I6'? ;102r 7 -

%

-

; lo2 - - c - D - "\ 2 - '"4% 0 ~ ~ 3 8

li4:

% \ D 0

A .\ - 0 -2 ,p.\ .'

I o3 A D >+---: 3

"b 20 40 60 80 100 120 140 lO"0 2b bb sb 8b Id0 I l O 140

I I I 1 I I loo I 1 I I I I ?

He l ' S - 2 ' P 2 9 . 6 e V -

3 -

lo; 20 40 60 I30 Id0 l;O 140 lib 2b 4b 60 eb Id0 l;O 140 ANGLE(deg) ANGLEcdeg)

(a) (b) FIG 7 Differential cross sections for the I 'S -+ 2'P transition in He at (a) 29 6 eV, (b) 40 1 eV

Theory , FOMBT (Thomas eta/ , 1974), --, FOMBT (Csanak and Cartwright, 1988). - - - -, DWPO (Scott and McDowell, 1976). - - - -. DW (Stewart and Madison, 1981). - -, CC5 (Bhadra et a / , 1979). -- . -, RM5 (Fon era/ , 1979). ~ , RM19 (Fon el a1 , 1988); - - -, CCOlO (Brunger et a / , 1990) Experiment 0, Hall et a/ (1973). V, Truhlar et a1 (1973). +, Cartwright e t a / (1989), 0, Brunger et a1 (1990)


FIG. 8. Differential cross sections for the 1 'S -+ 2'P transition in He at (a) 100 eV; (b) 200 eV. Theory: - - -, EBS (Joachain and Winters, 1977); - - - -, DWPO (Scott and McDowell, 1976);

, DW (Stewart and Madison, 1981); - - -, DW (Srivastava er a!.. 1989); - --, FOMBT (Csanak and Cartwright, 1988); - .. -, CC5 (Bhadra et al . , 1979); - . --, RM5 (Fon e t a / . , 1980). Experiment: *, Vriens e ta / . (1968); 0, Chamberlain era/ . (1970); 0. Opal and Beaty (1972); A, Suzuki era/ . (1974); X , Dillon and Lassettre (1975).

The difference in the two methods, CC5 and RM5, is in the target wave function employed and the details of the short-range correction included. As is shown later, the difference in the target function does not result in such a large discrepancy (see Fig. 9). In fact, the oscillator strengths obtained by the respective wave function in the CC5 and the RM5 method differ only by about 20% (see Table IV). This suggests that the difference in the DCS is ascribed mainly to the different way of incorporation of the short-range correlation effect. This can be supported by evidence that the difference is more pronounced for spin-forbidden transitions, which are excited only through the short-range exchange interaction.

The 10-state coupled-channel optical (CCOlO) method by Brunger et al. (1990) is compared with other calculations at 29.6 and 40.1 eV. This method produces very good results, as compared with the experiment, for the excitation of 2 ' s (Fig. 5) and 2'P (Fig. 7) states. For these transitions, the CC010 results show the best agreement with the experiment. On the other hand, the CCOIO results for the 1 ' S + 2'S, z3P transitions are very poor, generally worse than the CC5 calculation.

The essential ingredient of the CCOlO is its optical potential. Brunger et al.

28 Shinobu Nakazaki

He l'S-Z3S 200eV '621

1 C i 5 6 0 20 40 60 80 100 120 140

ANGLE (deg) FIG. 9. Differential cross sections for the three cases of the target wave function in the 5-state

R-matrix calculations for the 1 ' S + 2's transition in He at 200 eV. Theory: - -. the wave function of Bhadra et a/ . (1979); - - -, the wave function of Fon ei a/. (1979); ~ , the wave function of Fon e t a / . (1988).

tested the importance of the optical potential and found that without the potential they cannot get the good agreement with the experiment. An optical potential, which effectively takes account of the coupling to higher states, can correct the direct interaction. It is rather difficult, however, to properly correct the exchange interaction to give a good result for the singlet-triplet transition.

In the last ten years, the R-matrix method calculation of the cross section for the e - + He collision has evolved as shown in Table 11. The convergence of the resulting DCS in the RM5, RMl 1 , and RM19 calculations was discussed by Berrington er al. (1987) for the 1 'S + 2'P transition and by Fon er (11. (1991a) for the 1 ' S --., 23P transition. Comparisons of the DCS were made at the energies below 26.5 eV, and in this region, the results of the RMll and RM19 are in better agreement with each other than with that of RM5. This suggests that the RM19 results are almost converged with respect to the number of the states included in the close-coupling expansion. As for the total cross section, for the 1 ' S + 2'S, 2 's transitions, this has been confirmed with the 29-state calculation by Sawey er al. (1990).

When comparing various R-matrix calculations, one should note that the pseudo-resonances appear at different energies depending on the different calculations. In the case of RM l l , a pseudo-resonance affects seriously the cross


section at the energies above about 27 eV. The result of RM19, however, is free from the pseudo-resonance at least at the energies below 30 eV.

Now we compare the best DCS obtained at 29.6 eV by the RM calculation (i.e., RM19) to experimental results. This was originally done by Fon et al. (1988). We reproduce that in Figs. l(a), 3(a), 5(a), and 7(a). Fon et al. concluded that the theory (RM 19) is qualitatively good but quantitatively does not well reproduce the experimental results. The discrepancy is large (up to a factor of two) in the region of large angles (8 > 60") for the 1 'S + 2 9 , 23P transitions and in the region of 60"-80" for the 1 'S -+ 2 ' s transition. The agreement is better for the 1's -+ 2'P transition.

Another kind of comparison to test for the 1 'S + 2's transition was done by Fon and Lim (1990). They compared their DCS (obtained by RM19) at 30", 55", 90", 125", and 140" to the experiment as a function of collision energy from threshold to 23.8 eV. The agreement is much better than in the case of RM5. There is, however, still 20% discrepancy between the theory and the measurement of Pichou et al. (1976) at 90". Very recently the comparison at 90" was repeated with the RM29 by Fon et al. (1991b). They obtained a complete agreement between the DCS of RM19 and RM29 from the threshold to 23.5 eV. Thus the 20% discrepancy between the RM calculation and the experiment at 90" remains unsolved.

For the higher energies (100 and 200 eV), the RM5 calculation by Fon et al. (1979, 1980) generally reproduces the experimental result both qualitatively and quantitatively. One exception is the DCS for the 1 'S + 2 9 transition at 100 eV. In this case, a reasonable agreement is obtained only in the small angle region (8 < 20"). It is to be noted, however, that the experimental data scatter widely in this case. Another discrepancy is found between the RM5 and the experiment in the forward direction in the excitation of 2% state at 200 eV. This will be discussed later.

Now we return to the results of the calculations other than those of the close- coupling type. The FOMBT calculation of Thomas et al. (1974) gives the DCS at 29.6-100 eV. Their DCSs are generally in poor agreement with the experiment, except for the excitation of 2'P state. For the 1's + 2'P transition, the FOMBT gives quite good results even for the lowest energy (29.6 eV).

Amus'ya et al. (1984) made a calculation similar to the FOMBT for the excitation of 23S state. The angular dependence of the resulting DCS is very similar to that of the DCS obtained by Thomas et al., but there is a large difference in magnitude between the two DCS. This difference comes from the fact that Amu- s'ya et al. calculated the final-channel distorted wave in the field of the final state of the target but Thomas et al. in the field of the initial target. Amus'ya et al. calculated the DCS also taking the second-order correction into account in their perturbative method. They found that the second-order effect is very large at 30 eV (see Fig. l(a)), but relatively small at higher energies.

30 Shinobu Nakazaki

The DWPO method is a variation of the distorted wave method but partially includes the effect of target polarization. In the original version of the DWPO (Scott and McDowell, 1975, 1976), no distortion is considered in the final channel. The DWPO calculation gives a poor DCS, compared with the experiments, for the transition and incident energy considered. The revised DWPO method by Srivastava et al. (1985, 1989) takes into account the distortion both in the initial and final channels. Their calculation at the higher energies usually results in good agreement with the experiment (see Figs. 6 and 8).

In a distorted-wave method, the distortion potential can be chosen rather arbitrarily (Itikawa, 1986). Madison (1979) and Stewart and Madison (1981) studied the dependence of the DCS for the 1 's + 2'P transition on the distortion potential chosen. Good results were obtained by using the static potential constructed from the target wave function of the 2'P state with 1s core frozen. The resulting DCS is shown in Figs. 7 and 8. Later Madison and Winters (1983) found that better results can be obtained with the potential formed as a sum of one-third of the ground-state potential and two-thirds of the excited-state one. They investigated also the second-order effect in the DW calculation, but found it gave little further improvement.

At 200 eV, the eikonal-Born series (EBS) calculation of Byron and Joachain (1973, and Joachain and Winters (1977) can give a good DCS for the 1 'S + 2 ' s and 1 'S + 2'P transitions, respectively. This is a high-energy approximation but corrected elaborately with higher-order terms. Their results, shown in Figs. 6(b) and 8(b), reproduce well the experiment at 200 eV. They obtained DCS also with the Glauber approximation, but the results are good only for the small angle region (0 < 40").

In the calculation of DCS, different wave functions of the target state lead to different results of DCS, even if the same method is employed to treat the collision dynamics. Fig. 9 shows this. In the figure, we compare the DCS for the transition 1's + 2% at 200 eV, calculated in the RM5 method with three different wave functions:

( i ) The wave function used in the CC5 calculation by Bhadra et a f . (1979). (ii) The wave function used in the RM5 calculation by Fon et al. (1979).

(iii) The wave function used in the RMI 1 calculation by Fon et al. (1988) and

The details of each function are shown in Table 11. Figure 9 shows that the relative angular dependence of the DCS changes little, depending on the target function used. The absolute magnitude, however, differs by as much as 40%. This is the same result which Joachain and Van Den Eynde (1970) investigated for the 1 'S + 23S transition at 225 and 500 eV using the Born Oppenheimer approximation. Scott and McDowell (1975) also reported that the DCS is very sensitive to the approximation target wave function employed in the DWPO calculation for the same transition at 29.6 eV.

Nakazaki er al. (1991).


D. DIFFERENTIAL CROSS SECTION FOR THE 1 'S + 2's TRANSITION IN THE FORWARD DIRECTION

Recently Sakai era/. (1991) measured the DCS for the 1 'S + 2's transition with a high angular resolution of about 1". Their measurement was done at the incident energies of 200 and 500 eV for the angle from 0" to 12". The DCS at 0" was obtained also for 200 to 800 eV. In Fig. 10, the RMI 1 and RM7 calculations of Nakazaki et al. (1991) and the RM5 one by Fon et al. (1979) are compared with the experiments of Sakai et al. and others at 200 eV. There is a fairly good agreement between theory and experiment in the region 4"- 15". Toward the forward direction, however, the experimental values obtained by Sakai et al. increase sharply and deviate very much from the theoretical ones. At 0", the experimental DCS is larger by a factor of seven than the RMll result. As is discussed in the previous subsection, the RM 1 1 calculation should be much more accurate than the RM5 one. The improvement achieved by the RMll calculation, however, is very small in this case compared to the experimental data.

Let us discuss in more detail the DCS at 0". Skerbele et al. (1973) and later Klump and Lassettre (1975) determined the DCS (0") for the first time for the energies 100-500 eV. In contrast to their anticipation, the value was found to be very large. Furthermore they found a peculiar feature in the energy dependence

He l 'S-23S P O O e V 4

ANGLE(deg1

FIG. 10. Differential cross sections for the I'S + 2% transition at 200 eV for low angles. Theory: 0, RM5 (Fon et al.. 1979); - - -, RM7 (Nakazaki er al.. 1991); --, RMI I (Nakazaki e t a / . . 1991). Experiment: *, Vriens et al. (1968); X , Dillon (1975); A, Yagishita e t a / . (1976); 0, Sakai e t a / . (1991).

32 Shinobu Nakazaki

FIG. 1 I . Differential cross sections for the I 'S + 2's transition in He at 0 = 0". Theory: - - - -, second order (Huo, 1974); - .. --, RM2 (present. the target wave function used in the CC5 calculation of Bhadra et a / . , 1979); - - -, RM3 (present, same as the RM2); - - -, RM5 (present, same as the RM2); . . . . . . . . . , RM5 (present, same as the RM I 1 ); ---, RM I 1 (Nakazaki el al . , 1991. the target wave function used in the RMI I calculation of Fon er a / . . 1988). Experiment: 0, Skerbele ef a/. (1973); A, Klump and Lassettre (1975); 0, Sakai etal. (1991).

of the DCS (0"); i.e., a minimum at around 225 eV. In order to explain this result, Huo (1974) calculated the DCS using a high energy approximation. In her calculation, she took into account the second-order term in the exchange T-matrix, analogous to the Ochkur approximation in the first-order theory. She showed that the second-order effect dominates in the calculation of DCS (0") at the higher energies. She obtained a minimum at around 150 eV.

The recent experiment by Sakai et af. gave a DCS (0") much higher than the previous one (see Fig. 1 1. The results of Skerbele et al. and Klump and Lassettre in the figure are estimated from the results of their experimental intensity ratio (1's + 23S)/(l 'S 2'S), by using the 1 'S + 2 ' s DCS for the RM5 results of Fon et al. (1980) and the experimental results of Sakai et a f . for 100 and 150 eV, and above 200 eV, respectively.) To compare with the experiment, Nakazaki et a f . also calculated the energy dependence of DCS (0") with a simpler method (i.e., RM2, RM3, and RM5 with the rather simple target wave function of Bhadra et al., 1979). The resulting theoretical energy dependence is completely different from the measurement of Sakai et a f . In particular, the R-matrix calculations show no minimum in the energy dependence of the DCS (0"). I t should be noted here that, if we take into account an extrapolation (to 0") of the


DCS measured by Trajmar (1973) and Brunger et al. (1990) at 40.1 eV (see Fig. l(b)), the measurement by Sakai et al. suggests a minimum somewhere between 100 and 200 eV, supporting the calculation of Huo.

Coupling to continuum states, which is not included in the RM theory, may solve the discrepancy between the RM calculation and the experiment. An exact second-order Born calculation would be helpful to show such a possibility, as suggested by Sakimoto er al. ( 1 990).

IV. Excitation of Heliumlike Ions

A. THEORY

Although a lot of theoretical and experimental studies of DCS have been done for the e - + He collision, there are no experimental and a limited number of theoretical results for the e - + He-like ions. Experimental DCS for the e - + ion collision have been reported so far only for Mg +, Ar7+, Zn +, Cd + , and Ba46+ (Chutjian and Newell, 1982; Chutjian, 1984; Williams et al., 1985, 1986; Marrs et al., 1988; Huber er al., 1991). These experiments, however, have stimulated theoretical studies of DCS for ions.

Sural and Sil (1966) calculated the DCS for the 1 'S + 2 ' s transition at threshold in Li + using the Coulomb-Born approximation.

Bhatia and Temkin (1977) obtained DCSs for the 1's + Z3S, 1's + 2'S, 1's + 23P, and 1's + 2'P transitions in Li', 06+, and Si12+ using the DW approximation in which only the initial continuum wave function is distorted. '

Singh et al. (1983b) calculated the DCS for the I 'S-2's transition in various He-like ions ( Z = 3-26, Z being nuclear charge) using the variable charge Coulomb-projected Born approximation, which is the same as used in e - + He collision (see Section 1II.B).

Srivastava and Katiyar (1987) obtained the DCS for the 1 IS + 2's transition in 06+ at 50, 60, and 75 Ryd using the DW approximation that was applied to the e - + He collision (see Section I11.B). In this method, target polarization is included in both the initial and the final channels, and the exchange transition matrix is calculated using the Bonham-Ochkur approximation (see, Srivastava et al., 1985). They compared their DCS to those of Bhatia and Temkin (1977) and Singh et a/ . (1983b) and found that the angular dependence depends significantly on the approximation employed.

Itikawa and Sakimoto (1988) and Sakimoto and Itikawa (1989) calculated the DCS for the 1 'S + 2%. 1 'S + 2'S, 1 'S +. Z3P, and 1 'S -+ 2'P transitions in L i+ , Oh+, and Si i2+ using the DW method developed by themselves (Itikawa and Sakimoto, 1985). They compared the DCS along the He-like isoelectronic

34 Shinobu Nakazaki

sequence at the same energy in threshold units, and showed that the DCS multiplied by Z4 are quite similar both in magnitude and angular distribution all along the sequence.

Griffin and Pindzola (1990) studied the DCS for the transition from the 1's state to the 2'S, 2'S, 2'P, and 2'P states in Li+ by using the distorted-wave and the close-coupling approximations. Comparison was made between the non- unitarized distorted-wave (DW), 5-state unitarized distorted-wave (UDW), 5- state close-coupling (CC5), and 1 I-state close-coupling (CC 1 1) calculations. They showed that the shape of DCS can be significantly different for various levels of approximation, even when the total cross sections are in reasonably close agreement.

Very recently Nakazaki and Berrington (1991) calculated the DCS for the transition from the ground state to n = 2 states in Li + at the energies of around 4.5 and 5.8 Ryd using the 19-state R-matrix (RM19) method. The wave functions for the 19 lowest target states are the same as in the e - + He collision calculation (see Table 11, and Berrington and Nakazaki, 1991). They calculated the DCS also using the DW code of Itikawa and Sakimoto (1985) with the same target wave functions used in the RM19 calculation.

B. COMPARISON OF THE RESULTS

1 . Excitation of Li +

The RM19 results at incident energies 5.6 and 6.0 Ryd, and the DW results at 6.0 Ryd of Nakazaki and Berrington (1991) for the transitions from the 1 'S state to the 2'S, 2'P, 2'S, and 2'P states are shown in Figs. 12- 15. Also the CCl1 results of Griffin and Pindzola (1990) are shown at 5.2 Ryd. In these figures, a scaled DCS, Z4dr/dfl, is plotted against the scattering angle. As can be seen from these figures, the angular dependence of the RM19 result is the same as the CCl 1 result. Furthermore, the DCS at 5.2 Ryd of the CCI 1 calculation is consistent in magnitude as a function of energy with the DCS at 5.6 and 6.0 Ryd of the RM 19 one.

It is noted that the DW results sometimes disagree both in magnitude and in shape with the RM19 and the CC11 calculation. A similar conclusion was reached by Griffin and Pindzola when they compared the DW result with the CCI 1 , CC5, and UDW results in their paper. This conclusion is acceptable when we consider the discussion for the e - + He collision in Section 1II.C.

It is interesting.to note that the UDW result is closer to the CC5 and the CCI 1 one than the DW. The UDW takes into account partially the coupling among the states considered.


2 2 10, 30 60 90 120 150 180

ANGLE(deg1

FIG. 12. The scaled differential cross sections, Z'duldn for the 1's + 2's transition in Li'. Theory: - - -, CCI I at 5 .2 Ryd (Griffin and Pindzola, 1990); . . . . . . . . ., DW at 6.0 Ryd, ____ , RM19 at 5 .6 Ryd, -- . -, RM19 at 6.0 Ryd (Nakazaki and Berrington, 1991).

1 6 ' 6 0 30 60 90 120 150 180

ANGLE(deg)

FIG. 13. Same as Fig. 13, but for the 1's -+ 2'P transition.

36 Shinobu Nakazaki

3 2 10, 30 60 90 120 150 180

ANGLE(deg1

FIG. 14. Same as Fig. 13, but for the 1's + 2's transition.

FIG. 15. Same as Fig. 13, but for the l ' S + 2 'P transition.


2. Comparison berween Ions and Neutrals

In Figs. 16-19, we compare the RM19 results for He (Fon et al., 1988) at 29.6 eV, and for Li + (Nakazaki and Berrington, 1991) at 76.2 (5.6) and 81.6 eV (6.0 Ryd) with experimental data for He at 29.6 eV. It is interesting to note that the RM19 results for Li + are very similar in shape to both the experimental and the theoretical DCS for He for all the transitions. It should be noted that the target wave functions used in the RM19 calculation for Li' are similar to those for He (Fon et al., 1988) in the choice of orbitals and configuration interactions.

Now we compare the DCS along the He-like isoelectronic sequence. Figures 20 and 21 show the DCS calculated by Itikawa and Sakimoto (1988), and Saki- mot0 and Itikawa (1989) for the ions with 2 = 3, 8, 14, for the 1's + 23S at X = 2.0 and for the 1's + 2's at X = 1.8, respectively ( X being the incident energy in the threshold units). Again the scaled DCS, Z4dvldQ, is shown in the figures. They employed a DW approximation to obtain the DCS. To complete the comparison along the isoelectronic sequence, the DCS for He obtained by Thomas et al. (1974) (see Figs. l(b) and 5(b)) are plotted there. The FOMBT used by Thomas et al. is essentially the same as the DW method of Itikawa and Sakimoto.

1 ANGLE(deg1

FIG. 16. The scaled differential cross sections, Z'dr id f l for the I 'S + 2's transition in He and Lit in the 19-state R-matrix calculation at incident energies in threshold units, X . Theory: -___ , He at X = 1.49 (Fon el a/ . , 1988); -- .. --, Li+ at X = 1.32 (Nakazaki and Ber- rington, 1991); ---, Lit at X = 1.41 (Nakazakai and Berrington, 1991). Experiment: +, He at X = 1.49 (Cartwright e ta / . , 1989).

38

,-. L u) , ? - - ' D L

'D k * N

Shinobu Nakazaki

P s - ~ ~ P

/-> _ - c . -. A ,',

/

\ >-/- - - -, \

He at X = 1.44;


l ' S - 2 ' P

FIG. 19. - .. -

1 1 s - 2 3 ~ x=2.0 1

-1

He at X = 1.40;

40 Shinobu Nakazaki

FIG. 21. and Itikawa,

Same as Fig. 20. but for the I I S + 2's transition at X = 1 . 8 , DW 19891, and He at X = I .95.

method (Sakimoto

As seen from Figs. 20 and 21, the scaling of the DCS appears satisfactory in presenting DCS for different ions. The scaled DCS for 06+ and Si '?+ almost coincide with each other and are very close to the limiting value at Z = =. The DCS for the 1 'S + 2% transition in ions has the same angular dependence as that for He. This indicates that the mechanism of the excitation process in the ion is not much different from that for He, at least at X = 2.0 . This was also shown for the 1 'S + 23P transition (Itikawa and Sakirnoto, 1988).

For the spin allowed transition 1 'S + 2 ' s at X = 1.8, the angular dependence for Li + is rather similar to that for He. The DCS has a minimum at around 60". This minimum turns to be a shoulder as Z increases, so that excitation mechanism in the ions with large Z is different from that in He. This was also shown for the 1 'S + 2'P transition at X = 2 . 0 . This kind of analysis of the DCS for other transitions along other isoelectronic sequence. would be helpful in understanding the mechanism of the e - + ion collisions.

V. Concluding Remarks

We have made comparisons among theoretical results and experiments for differential cross section for electron impact excitation from the ground state to the n = 2 levels of He and He-like ions.

The 19-state R-matrix calculation, which is probably the most elaborate theo-


retical calculation of the DCS to date, gives qualitatively good agreement, but there are some quantitative discrepancies with the experimental results in He at low energy. At 200 eV for the 1's + 2's transition, the experimental DCS at 0" of Sakai et al. (1991) lies higher by a factor of about seven than the 1 1-state R- matrix result.

In the close-coupling calculation, to get exact results one has to include an infinite number of bound states and also an integral over the continuum in the wave function expansion. The discrepancy between the best R-matrix calculation and the experiment could be due to the neglect of the continuum state in the expansion.

The intermediate energy R-matrix (IERM) method has been derived by Burke et al. (1987) to take account of coupling to the continuum. This new R-matrix approach has given very good results for the e - + H collision.

The coupled-channel optical method, which gives good agreement with the experiments for the singlet states of helium at low energy, takes account of coupling to the continuum in the optical potential.

IERM and CCO and other methods that take into account the continuum should be developed and applied to various atoms and ions to improve the reli- ability of the theoretical predictions.

It is shown that the scaled DCS, Z4daldl l , against scattering angle at a given incident energy in threshold units, appears satisfactory in presenting DCS for He and He-like isoelectronic sequence for the transitions. The comparison and analysis along the electronic sequence in the scaled DCS are helpful in understanding the mechanism of the electron-atom(ion) collision. For the other transitions and incident energies this kind of analysis would be of interest. Finally, it would be desirable to have an experimental DCS in the He-like ions to compare with the experiments of He and the theoretical data of He and He-like ions in the scaled DCS.

Acknowledgments

The author would like to express his sincere thanks to Professor Yukikazu Iti- kawa for his encouragement and his valuable discussion and suggestions. He also is indebted to Dr. Keith A. Berrington for a critical reading of the manuscript.

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1063.

ADVANCES IN ATOMIC . MOLECULAR . AND OPTICAL PHYSICS. VOL . 30

CROSS-SECTION MEASUREMEWS FOR ELECTRON IMPACT ON EXCITED ATOMIC SPECIES S . TRAJMAR * and J . C . NICKEL Department of Physics University of California Riverside. California

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 I1 . General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

I11 . Production of Excited Species . . . . . . . . . . . . . . . . . . . . . . 48 A . Excitation in a Discharge . . . . . . . . . . . . . . . . . . . . . . 49 B . Electron-Beam Excitation . . . . . . . . . . . . . . . . . . . . . . 51 C . Near-Resonant Charge Exchange . . . . . . . . . . . . . . . . . . . 53 D . Laser Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 56 E . Altering Excited Atom Compositions in Mixed Beams . . . . . . . . . . 60

IV . Detection of Excited Species . . . . . . . . . . . . . . . . . . . . . . 60 A . Thermal Detectors . . . . . . . . . . . . . . . . . . . . . . . . . 61 B . Optical Detectors . . . . . . . . . . . . . . . . . . . . . . . . . 61 C . Secondary Electron Ejection . . . . . . . . . . . . . . . . . . . . . 64 D . Ionization Detection . . . . . . . . . . . . . . . . . . . . . . . . 65 E . Superelastic Electron Scattering . . . . . . . . . . . . . . . . . . . 65

V . Cross-Section Measurements . . . . . . . . . . . . . . . . . . . . . . 66 A . Total Electron Scattering Cross Sections . . . . . . . . . . . . . . . . 66 B . Ionization of Excited Atoms by Electron Impact . . . . . . . . . . . . . 73 C . Line Excitation, Apparent Level Excitation and Integral Electron-Impact

Excitation Cross Sections-Optical Methods . . . . . . . . . . . . . . 84 D . Differential Cross Sections . . . . . . . . . . . . . . . . . . . . . 91

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 98 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

I . Introduction

Electron collisions with excited atoms play a prominent role in high-density gas discharges (Krivchenkova and Khakaev. 1975; Delcroix et al., 1976; Massey et al., 1982a). astrophysical plasmas (Allen. 1984). ionospheric and auroral processes of planetary atmospheres (Massey et al., 1982b). and electron-beam- and discharge-pumped lasers (Massey et al., 1982c) . Particularly. electron collision

* Permanent Address: Jet Propulsion Laboratory. California Institute of Technology. Pasadena .

Copyright 0 1993 by Academic Press . Inc . All rights of reproduction in any form reserved .

ISBN 0-12-003830-7 45

46 S . Trajmar and J.C. Nickel

TABLE I ENERGIES A N D LIFETIMES FOR METASTABLE

RARE GAS SPECIES (FROM DELCROIX Er A L . , 1976)

Atom Level Energy (eV) Lifetime (sec)

He 2 'SI 19.82 2 SI 20.61

Ne 3 'P? 16.62 3 'P" 16.72

Ar 4 'P? 1 I .55 4 'PI1 1 I .72

Kr 5 JPI 9.92 5 'PI1 10.56

Xe 6 'Pp? 8.32 6 'Po 9.45

6 X lo5 2 x 10-2 >0.8 >0.8 >1.3 >1.3 > I > I ? ?

processes involving metastable atoms are very important in partially ionized systems because of their long lifetimes and the large amount of energy they can transfer to the electrons. As an example, a summary of these lifetimes and energies for the rare gases is given in Table I. So far only a few experimental or theoretical studies have been reported on this subject. While a large body of cross-section data is available for electron collision processes involving ground- state atoms, our data base for electron collisions with excited species is very limited. The main reason for the scarcity of this type of data is the difficulty experienced in generating these species in the relatively high concentrations necessary for collision cross-section measurements. Our aim is to summarize and review differential and integral (DCS and Q, respectively) cross-section measurement techniques and available cross-section data for electron collisions with excited neutral atoms. We will also briefly discuss methods for production and detection of excited atoms with emphasis on recent methods that relate to cross- section measurements.

The earliest attempts to study electron collisions with metastable atoms was summarized up to 1969 by Massey, Burhop, and Gilbordy (1969). A number of improved electron-impact excitation and charge-exchange schemes have been reported in recent years for production of metastable atoms, and with the avail- ability of lasers, it is now also possible to produce large populations in levels with short radiative lifetimes as well as in metastable levels through cascade processes. The latter method is, however, not yet fully exploited. These production methods are discussed in some detail later. Quantitative cross-section measurements require the knowledge of excited atom densities (at least relative to ground state species) and, in cases where more than one excited species is present, the relative densities of these species. The methods for determining these densities will also be briefly described. A summary of cross-section measure-

CROSS-SECTION MEASUREMENTS FOR ELECTRON IMPACT 47

ment techniques and cross-section data for various electron collision processes will be discussed in some detail in this chapter.

II. General Remarks

Electron collision cross-section measurements on excited atoms require somewhat different methods than those encountered with ground state species. The fact that the initial state is excited implies a method of preparation of the target atoms. One deals, therefore, with a two-step process: preparation of the target and electron scattering. Electron-impact excitation and ionization in these cases are often referred to as stepwise excitation and ionization. A general review of two-step processes involving electron collisions and laser excitations has been given by Mac Gillivray and Standage (1988).

The target preparation step necessarily leads to mixed beams (or static targets) containing ground and excited atoms. Electron-impact excitation, while simple and effective, is highly nonspecific, and the characterization of the mixture is rather difficult. Preparation of excited atoms by laser pumping, however, can be very well defined. Specific fine and hyperfine levels of individual isotopes can be excited, and the magnetic sublevel populations can be controlled. When laser excitation is utilized for preparing the target, collision cross-section measurements that distinguish levels separated by as little as eV become possible, compared to the eV resolution achievable in general. Depending on the method of preparation, the distribution of populations in the magnetic sublevels of the target atoms may be uneven, some degree of polarization (alignment or orientation) may, therefore, be present, and the DCS may become dependent on the azimuthal scattering angle (4).

In conventional electron scattering cross-section measurements (ground level atomic beams), the measured cross section is an average for the experimentally undistinguished processes (sum over final, average over initial unresolved states) with the assumption of even populations in the ground magnetic sublevels. In the case of excited targets, the method of preparation may introduce uneven populations in the ground and excited levels. The cross sections in these cases are different in nature from those obtained in conventional measurements, and they are meaningful only if the averaging is precisely defined. This is a very important matter when comparison is made between experimental or theoretical cross sections.

Electron collisions with excited atoms can lead to deexcitation of the atoms (superelastic scattering). The superelastic cross sections can be related to the inelastic cross sections (associated with the appropriate inverse process) by the principle of detailed balance. The application of this principle to transitions between quantum mechanically pure states is straightforward. When the transition involves

48 S . Trajmar and J .C. Nickel

degenerate states or magnetic sublevels, the appropriate statistical weights have to be introduced. As indicated previously, special care has to be exercised, however, in defining the statistical weights for the case of aligned or oriented atoms.

III. Production of Excited Species

A list of excited atom production methods utilized in electron collision cross- section measurements is given in Table 11. Electron-impact excitation (direct excitation, cascade, and recombination processes) is the most frequently utilized method for producing excited atoms either in a discharge (dc, RF, or microwave) or in a beam arrangement (transverse or coaxial). In these schemes, thermal, or in some cases superthermal (0.3 < Eo < 10 eV), excited species are produced in a mixture of overwhelmingly higher (by about a factor of lo4) concentration of ground state species. Charge-exchange processes are frequently utilized for converting ions to metastable atoms in vapor cells (mainly alkali vapors), but solid state converters have also been tried. In these methods fast (typically 1 keV) metastables are produced that represent from a few to about 50% of the total flux. Laser excitation is the least utilized method so far but can be expected to gain considerably more importance in the future. This approach of excitation

TABLE I1 PRODUCTION OF EXCITED ATOMS

~

Rel. Density Method Process [Flux, sec-Isr- I ] Remarks

1. Discharge (dc, RF, microwave)

2. Electron beam (transverse, coaxial)

3. Charge-exchange a. Gas

b. Solid

4. Laser

(Recomb. + dir. exc. + cascade)

(Dir. exc. + cascade)

Recomb

Recomb.

Direct exc. or (dir. exc. + cascade)

-0.5 [ 10'0- 10161

-0.5 [1019]? -0.25 [<10171

He (DCS; ioniz.; opt. ; tot. ) Ne (opt.) TI (opt.) He (ioniz.) Ar (opt.; tot.) Kr (opt.) Sr (ioniz.)

H (ioniz.) He (ioniz.) Ne (ioniz.) Ar (ioniz.) He (opt.)

Na (DCS; tot.) Ba (DCS; ioniz.)


is well suited for preparation of short-lived species by direct pumping via optically allowed (or partly allowed) transitions or by various stepwise excitation schemes. Due to the very high monochromaticity and the polarized nature of the laser radiation, the selectivity of excited species can be extremely high. The method is also suitable for preparation of metastable species by optical pumping and subsequent cascade. A comprehensive review of these methods is not given here but only references to recent developments and a few general remarks.

A. EXCITATION IN A DISCHARGE

In this method, the desired gaseous species is included in an electrical discharge. The random electron current of the discharge, through various collision processes (direct excitation, cascade, recombination), produces the excited atoms. One faces then the difficult problem of extracting a beam from the source region without deactivating the excited species. Obviously the only excited species that can be expected are the metastable ones. During the expansion at the orifice of the source, the quenching rate of the metastables will still be high while no more production is taking place. The discharge schemes, therefore, suffer from the problem that the metastable fractions in the beam are typically only or less, which is prohibitively low for most cross-section measurements. The situation can be somewhat improved by sustaining the discharge through an expansion nozzle as described by Searcy (1974), Leasure et al. (1975), Fahey et al. (1978, 1980), Verheijen et al. (1984), Muller-Fiedler et al. (1984), Hotop et al. (1981), and Brand et al. (1991). Searcy (1974) developed a simple metastable atom source by combining a corona discharge with a sonic nozzle. Meta- stable fluxes of about 1Olo atoms sr-lsec - I were obtained with a velocity distribution similar to a nozzle beam with Mach numbers between 10 and 11. Leasure et al. (1975) further improved this source. In their experiments the electrons in the discharge and the atoms in the expanding beam move in opposite directions. Both thermal and superthermal metastables are produced with fluxes of about lo '* to 10I4 atoms sr-Isec-l, depending on the source pressure and the type of rare gas. Fahey et al. (1980) reported a modification of this source that resulted in further simplifications, enhanced beam flux and species-independent energies, In their scheme, the electrons and atoms move in the same direction. This source generated metastable fluxes of about 10l4 atoms sr -Isec-' at thermal energies (66, 72 and 74 meV for He, Ne and Ar, respectively). A similar source was applied by Hotop et al. (1981) and Ruf et all (1987) for producing metastable Ne and He atoms, respectively, for Penning ionization mass spectrometric studies and by Muller-Fiedler et al. (1984) in their DCS measurements on He (23S). Muller-Fiedler et al. were able to produce a density of 6 X lo7 cm-3 and I x lOI3 cm-3 of metastable and ground state species, respectively, at a distance of

to

50 S. Trajmar and J.C. Nickel

1.2 cm above the exit aperture of their source structure. Verheijen et al. (1984) made minor modifications of the source used by Fahey et al. (1980) and studied its performance. They obtained centerline intensities in the supersonic beam of He, Ne and Ar metastables corresponding to 7.3, 2.0 and 1.4 x loL3 atoms sr-lsec-', respectively. Very recently Brand et al. (1991) utilized this type of source, and with some improvements on the original Fahey et al. (1980) design, they obtained a metastable neon flux of 3.6 X loL4 atoms sr-'sec-l, which is higher than those reported by Veiheijen er al. (1984) and Fahey er al. (1980) by about a factor of 20 and 2.7, respectively, and comparable to that of Hotop et al. (1981). It is interesting to note that Hotop et al. (1981) found the 'PZ and 3P0 metastable ratio in a neon beam produced by this type of discharge to be 5.0 (27%). Rall et al. (1989) applied a hollow-cathode discharge for producing He (2%) species for their optical measurements of electron-impact excitation. At the exit of their source, they estimated the metastable density to be 5 x lo9 cm-', corresponding to a metastable to total He density fraction of 3 x Theuws et al. (1982) described a magnetically stabilized hollow-cathode arc source for the production of superthermal (0.5 to 1.5 eV) rare gas beams containing a mixture of ground and metastable atoms. The method was based on production of the beam from an approximately 50% ionized plasma with densities at such a value that the ion temperature approached the electron temperature. Fast ground state atoms were produced in collisions between ions and atoms (charge exchange), and fast metastable atoms were produced in collisions between electrons and fast ground state atoms (excitation) and between ions and slow metastable atoms (charge exchange). They obtained, for Ar, typical center line intensities of 2 x loL8 atoms sr-'sec-' with metastable intensity of 2 x lo i4 atoms sr-lsec-I.

A simple source for producing metastable alkaline-earth atoms ( '8, 3& levels) was described by Brinkmann et al. (1967, 1969). They applied a low-voltage discharge to the metal vapor effusing from a heated container. Similar sources have been used for producing metastable atoms of Pb (Garpman et al., 197 I) , Bi (Svanberg, 1972), Ba (Ishii and Ohlendorf, 1972), Mg (Giusfredi er al., 1975; Kowalski and Heldt, 1978; Kowalski, 1979), Ca (Giusfredi er al., 1975; Dagdigian, 1978; Pasternek and Dagdigian, 1977; Kowalski, 1979) and Sr (Ko- walski and Heldt, 1978; Kowalski, 1979; Wilcomb and Dagdigian, 1978). Fur- ther improvements and characterization of this source was discussed by Brink- mann et al. (1979, 1980). Urena et al. (1990) also described a low-voltage discharge source for producing metastable atoms of low vapor-pressure metals. This source could be operated in cw or pulsed mode, and its operation was demonstrated for producing 'P and ID metastable Ca atoms. They found that the 'P and ID population ratio was strongly dependent on the oven temperature. Microwave discharge sources with extended cavity design (as described by Murphy and Brophy, 1979, and Arnold, 1986) have been used in scattering stud-


ies for the production of unstable ground state atoms (H, C, N, 0, S), but have not yet been applied to the production of metastable species in connection with electron-impact cross-section measurements.

Production of highly polarized, thermal He (23S) metastable atoms was described by Slobodrian et al. (1983a, 1983b). They used a hollow-cathode discharge and achieved state selection (separation and focusing of the Zeeman components) with a sextupole magnet and reported a metastable flux of 6 x lOI5 atoms sr - Isec - at 0.07 and 0.1 eV kinetic energies. More recently, Baum ef al. (1988) described a polarized He (23S) beam source based on the design principles of Fahey et al. (1 980) and Ruf et al. (1 987) and the application of a sextu- p l e magnet for polarization. The polarized metastable atom flux in a 0.6 cm diameter (FWHM) beam was 5 X 10” atoms sec-’ corresponding to a density of lo7 cm-3. The ground state He component of the beam flux was 5 x lo f3 sec-I.

B. ELECTRON-BEAM EXCITATION

Some of the earliest attempts to generate metastable atomic beams utilized an electron beam transverse to an atomic beam and relied on the direct excitation of some fraction of the atoms to the metastable state by electron impact. Unfortu- nately, this method yielded typically a metastable fraction of only in the resulting beam. Additional problems were presented by the nonspecificity of the electron-impact excitation and recoil effects on the metastable portion of the beam. Some improvements were introduced by coaxial electron-beam excitation but the relative concentration of the metastables could not be significantly improved.

Transverse electron-beam excitation schemes for production of thermal metastable rare gas beams have been described by, e.g., Olmsted et al. (1969, Fry and Williams (1969), Freund (1970), Chen et al. (1974). More recently, a pulsed metastable source utilizing a low-energy, sheet electron beam and a fuel injec- tor-generated atomic beam was designed for photo ionization studies by Cze- chanski et al. (1989). They estimated the peak metastable atom density to be about 1.7 x lo5 ~ m - ~ for He 2’s and 5.2 x lo5 for He 23S compared to a peak neutral atom density of lOI4 cm-3. The effect of recoil on the velocity distribution of the metastable atoms in the transverse arrangement was studied by Pearl et al. (1969).

Rundel et al. (1974) described a scheme that utilized a magnetically collimated coaxial electron beam to minimize transverse recoil of the atoms and thus increase the metastable beam intensity. The electron and atom beams propagated the same direction. Typical metastable flux over a solid angle of 4 X sr was 8 x lo7 atoms sec-I (corresponding to 2 X lolo atoms sr-lsec-I) for He and several times larger for Ne and Ar. They studied the velocity distribution functions for metastable He, Ne and Ar and compared them to those of transverse


excitation sources. Using this type of source for producing a metastable Ne beam, Dunning et al. (197517) found that the 'P, to 'Po metastable flux ratio depended on the electron energy.

A high-intensity, well-collimated (order of 0.4" FWHM) metastable He source with good velocity resolution (3% FWHM) was constructed by Brutschy and Haberland ( 1977). They produced a ground-state beam with high-velocity resolution by a supersonic nozzle source (which could be cooled to 80" K with liquid N,) and utilized a coaxial electron beam (parallel or antiparallel) to produce exited He atoms. Electrons were extracted from a spherically shaped, indirectly heated, nickel-alkaline-earth, sinter cathode that had a hole at the center to pass the atomic beam. The electron beam was magnetically confined (-60 mT). Un- der typical operating conditions (150 to 200 eV) the electron beam current was in the 50 to 150 milliampere range. The design also incorporated a He gas- discharge lamp [to quench He (2IS)], an electrostatic deflector (to get rid of ions) and an electric field (to field ionize highly excited atoms). With 150 eV electrons they obtained a metastable beam flux of 3 X 1O'O atoms sec-' over a solid angle of sr (corresponding to 3 x 10l4 atoms sr-lsec-I) about one- third of which was in the 23S and two-thirds in the 2IS state, respectively. The combined metastable flux represented only a fraction of lo-' of the total beam flux. This is the price paid for the high degree of collimation and high-velocity resolution. Both the metastable flux and velocity distribution, however, represented improvements over previous electron beam-generated metastable He sources (Freund, 1970; Chen et al., 1974; Rundel et al., 1974). Johnson and Delchar (1977) designed a compact source similar, in principle, to that of Rundel et al. (1974) that produced a metastable He beam with a flux of 4 X 10l4 metastable atoms sr - Isec - l and less than l % ultraviolet photon content at an electron beam energy of 100 eV. The stability of the source was about 2%. In their design, a single-turn, thoriated-tungsten-wire filament, with its plane perpendicular to the electron (and atom) beam axis was set symmetrically around the beam axis in a Pierce electrode arrangement to serve as the cathode. The emitted electrons were focused onto the atomic beam by an einzel lens. In their arrangement the electron and atom beams traveled in opposite directions. The collimating magnetic field was between 0.01 and 0.02 T. Onellion et al. (1982) found that beam sources applying coaxial electron beam arrangements contain principally fast (superthermal) neutral atoms instead of thermal metastable atoms if high source pressure or large focusing voltages are applied to the electron lens elements. Kohlhose and Kita (1986) described a metastable beam source for time-of-flight applications utilizing a coaxial electron beam. The electron beam was controlled and accelerated by a combination of three hemispherically shaped tungsten grids and could be operated either in pulsed or continuous mode. With continuous operation the source produced a metastable flux of about lo i5 atoms sr-lsec-l for He or Ar. Applying short electron pulses, they found a velocity distribution for the supersonic metastable beam as low as 3.8% for He and 6.5% for Ar


(FWHM) with 100% modulation, making this source applicable for velocity- analyzed scattering experiments relying on time-of-flight techniques.

Srigengan and Hammond (1991) are in the process of developing a pulsed metastable beam for use in electron scattering experiments. They begin with a skimmed, pulsed supersonic gas beam whose velocity distribution has a small spread around a central nonthermal velocity. The gas beam is crossed with a magnetically collimated, rectangular electron beam carrying a current of 100 ma. The electron beam will produce metastable atoms and momentum transfer during the electron impact excitation will separate the metastable atom beam from the ground state atom beam, yielding a relatively pure metastable beam. They hope to develop a beam with metastable densities 2 lo7 cm - 3 in the interaction region and have achieved a lower limit of 2 x lo5 cm-3.

Riddle et af. (1981) designed a polarized He (23S) thermal metastable beam source based on earlier coaxial electron-beam excitation schemes (Rundel et af., 1974; Brutschy and Haberland, 1977; Johnson and Delchar, 1977). From the mixed beam, they removed the 2's components via the 2 ' s += 2'P += 1 IS transitions by illuminating the beam with light from a dc-excited, flowing, helium- discharge lamp wound coaxially around the beam. A transverse electric field ionized the high-Rydberg components and removed the ions. The 23S beam was polarized by optical pumping with circularly polarized 23P 4 z3S radiation incident along the magnetic field direction. Depending on the sense of circular polarization, He ( 2 3 S ) atoms with M, = + 1 or M, = - 1 (with respect to the magnetic field) were produced. They obtained a beam with about lOI4 metastable atoms sr- lsec- ' with 50% polarization. Giberson et af. (1982, 1984, 1985) utilized similar electron-beam excitation and subsequent optical pumping to produce polarized metastable beams of He (2)S), Ne ('P2) and Ar (3P2). Production of a polarized metastable Ar (3P2) beam was also described by Lynn et af. (1986). They applied a coaxial electron-beam excitation and subsequent optical pumping with a frequency modulated dye laser to achieve orientation.

C. NEAR-RESONANT CHARGE EXCHANGE

In this method, an ion beam of the desired species is produced and made to recombine into metastable species by resonant charge transfer in a gaseous medium or on a solid surface. The requirements for resonant charge exchange are that the ionization energy of the gas medium or the work function of the solid match the ionization energy of the metastable. The recombination can be made very efficient and specific as far as the metastable state is concerned. The large majority of the applications of this technique has been for the production of metastable rare gases with alkali vapor converters. In Table 111, we summarize the pertinent parameters concerning the rare gas conversions.

The gaseous conversion scheme has been extensively utilized since the 1960s


TABLE I11 SUMMARY OF PARAMETERS PERTINENT TO NEAR-RESONANT CHARGE EXCHANGE

BETWEEN RARE GAS IONS AND ALKALI ELEMENTS

A E (eV)

Designation E (eV) I.P. (eV) Na K Rb c s

He 'SO 2 9 , 2'So

Ne 'Sn 3'P?

3'Po

'Sn 43Pl

43Po

'SO 5'P?

5'Po

'SO 6'P?

6'Pn

Ar

Kr

Xe

Na K Rb c s

3s[3/2]8

3s' [ 1/21:

4s[3/2]8

4s' [ 1/21;

5s[3/2]:'

5s' [ 1/21;

6s[ 3.21 y 6s' [ 1/21;

0.00 24.48 19.82 4.66 20.62 3.86

0.00 21.56 16.62 4.94

16.72 4.84

0.00 15.76 11.55 4.21

11.72 4.04

0.00 14.00 9.92 4.08

10.56 3.44

0.00 12.13 8.32 3.81

9.45 2.68

5.14 4.34 4.18 3.89

+ 19.34 -0.48 - 1.28

+ 16.42 - 0.20

- 0.30

+ 10.62 -0.93 - 1.10

+8.86 - 1.06

- 1.70

+ 6.99 - 1.33

- 2.46

+ 24.14 +0.32 -0.48

+ 17.22 +0.60

+0.50

+ 1 I .42 -0.13 -0.30

+9.66 -0.26

-0.90

+7.79 -0.53

- 1.66

+ 20.30 +0.48 -0.32

+ 17.38 +0.76

+0.66

+ 11.58 + 0.03 -0.14

+ 9.82 -0.10

-0.74

+7.95 - 0.37

- 1.50

t 20.59 +0.77 - 0.03

t 17.67 + 1.05 +0.95

t11.87 +0.32

+0.15

t10.11 +0.19

- 0.45

+8.24 - 0.08

- 1.21

AE = (1P)mCl - (IP),II

to produce metastable atoms (Peterson and Lorentz, 1969; Dixon et a f . , 1976; Defrance el a l . , 198 1). The ions are extracted from a plasma source, selected by a charge to mass ratio analyzer and transported to the conversion cell by one or more stages of acceleration-deceleration. The ions then pass through the cell at kinetic energies usually in the range of few hundreds to few thousands eV. Re- sidual charged particles are electrostatically removed, and a fast neutral beam containing ground and metastable excited species is obtained that can be used in various collision studies. In the vast majority of cases applied so far, the exchange medium in the exchange cell was an alkali vapor and the metastable atoms produced were atomic hydrogen (2s) and rare gases representing a mixture of 2's and 2's species for He and 'P2 and 'Po species for the other rare gases, mixed in all cases with fast ground state species. Ion to metastable conversion efficiencies typically range from few to about 50%. It has been found (Neynaber


0-

:-

I Ne : Transpori

. Chamber 1 Chamber 2

X Interaction

Region

UHV Chamber3 ‘ , ~~

L - - - - - - - - - - - - _ - - - - - - - - - - - . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * - - - - - - - - - - - - - - - - - . l

FIG. I . Fast metastable beam source of Fujii era/ . (1991). A fast (1 keV) ion beam is extracted from the plasma source and transported to a charge exchange cell with periodic focusing ion optics. After unwanted charged particles are removed, the fast neutral (metastable and ground state) beam travels to the interaction region and is detected by a thermoelectric detector. Chambers 1, 2 and 3 are differentially pumped, and the interaction region is in the UHV pressure range.

and Magnuson, 1976; Coggiola et al., 1979; Gaily et al., 1980) that the charge exchange produces metastable (3Pz and and short-life (3P,) rare gas atoms with a composition that corresponds to statistical weights (5 : 1 : 3).

Fujii et al. (1991) reported the development of a high-density metastable beam source based on near resonant charge exchange of a fast ion beam. The apparatus configured for metastable ionization experiments is shown in Fig. 1. In most fast beam sources, a relatively small ( - lp ampere) ion current is delivered to the charge exchange cell. In this new source, a dense ion beam is extracted from a Penning ion source and transported to the charge exchange cell by periodically focusing ion optics. The motion of the dense ion beam is governed primarily by space charge forces, and the extracted beam expands. The einzel elements simply flip the slope of the ion trajectories, and the beam compresses to approximately its initial diameter and then expands again. Appropriately placed lens elements with “wavelenths” L repeat this process. The ion density at the waist of the periodically focused beam is given by

For E , = 1 KeV and L = 1.5 inches (probably as short as can be constructed practically) we can expect ion densities of the order of lo9 ions/cm3. Immedi- ately after the charge exchange cell, one might expect metastable densities on the order of 2-5 x lo8 cm-). The fast neutrals can easily be detected by a thermoelectric detector. Preliminary results have produced 300 na of neutral cur-


rent at 0.8 KeV beam energy that corresponds to neutral densities of 6 x lo6 neutrals/cm3 at the interaction region, which is located 9 inches from the conversion cell. This probably corresponds to about 4 X lo8 neutrals/cm3, 1 inch from the conversion cell.

A method for producing a fast metastable He beam by utilizing a solid capillary converter was described by Khakaev and coworkers (Gostev et al., 1980a, 1980b, 1982; Khakaev et a l . , 1982). They used a graphite capillary array as a converter. Ions in the few hundred to few keV energy range making a single collision on the inside surfaces of the capillaries were converted into metastables. (Multiple collisions deactivate the metastable to ground fast state species.) This approach avoids the need to transport ions from the source to the converter since the graphite converter can be located very close to the exit aperture of the ion source and can act as the extraction element. Very high fluxes of He metastables (1019-1020 atoms sec-'sr-I) were reported (Gostev et a l . , 1980a, 1982). Sub- sequent investigations (Khakaev, I989), however, revealed that the high flux was mainly due to slow metastables diffusing out of the ion source over a very small solid angle and a more diverging but weaker fast metastable flux was superim- posed on it. High beam densities could be achieved only very close to the exit of the capillary array and only with high background pressure Torr) and large ion intensity (- 10 milliampere), which destroyed the graphite converter in about 1 hour. Attempts to produce metastable beams for Ne and Kr by this method were unsuccessful (Khakaev, 1989). This was attributed to field broadening experienced by the metastable atoms near the solid surface, which caused an overlap of the metastable and resonance levels and made possible a radiative decay to the ground state. Investigations in our laboratory before and after our discussions with Khakaev and coworkers ( 1989) confirmed their findings (Nickel et a l . , 1989).

D. LASER EXCITATION

Preparation of specific, short-lived, excited atoms is achieved almost exclusively by utilization of lasers. Extensive literature is available on this subject (Cohen- Tannoudji, 1975; Walther, 1976; Hertel and Stoll, 1977; Scoles, 1988 and 1990; Shore, 1990). Here we only briefly comment on laser excitation methods and principles as they relate to the preparation of excited atoms for electron collision cross-section measurements. The topics of interest to us are the specificity of the excitation process, the state of polarization (orientation, alignment) of the atomic target ensemble and the population of the excited levels. Although the one-photon laser excitation, considered here, produces short-lived species, it also has to be pointed out that spontaneous decay from these excited levels to intermediate levels, which are forbidden to decay to the ground level, results in an efficient and convenient production of metastable species. In addition, we


will briefly discuss the utilization of laser excitation of initially metastable atoms (two-step process) for the purpose of altering (quenching) and determining population densities of individual metastable components (laser induced fluorescence) in a mixed beam. These are important matters when cross sections for individual metastable species are desired.

The four properties of lasers, which make them uniquely suitable for preparation of excited atomic beams, are

1. Well-defined wave length (level selectivity); 2. Well-defined polarization (magnetic sublevel selectivity); 3. High power density (high excited level population); and 4. High degree of coherence (important, e.g., for short time-scale excitations).

The high monochromaticity of the laser light makes possible the selective excitation of specific fine and hyperfine levels of specific isotopes. This high selectivity is desirable in certain situations but not needed in conventional cross- section measurements in general since the dependence of these cross sections on nuclear properties can be neglected to a very good approximation.

The polarization of the laser light determines the magnetic sublevel populations. Optical selection rules govern and the angular momentum vector coupling coefficients determine the relative populations of magnetic sublevels in the photon frame of reference. A transformation to the collision frame will, in general, result in uneven populations, and as far as the electron collision process is concerned, the target atomic ensemble is polarized to some degree and the scattering cross section becomes dependent on the azimuthal scattering angle. The utilization of the laser polarization opens up the possibility of a more detailed investigation of the collision process (determination of individual or specific average of magnetic sublevel cross sections, coherences among magnetic-sublevel- specific scattering amplitudes). At the same time, polarization effects complicate the characterization of the cross section being measured. A careful specification of the experimental conditions and the target polarization is necessary to define the cross sections and compare them to other experimental or theoretical results. In conventional cross-section measurements, where polarization effects are ab- sent, the average cross sections are obtained that represent summation over final and averaging over initial magnetic sublevels with the assumption of even populations in the intial magnetic sublevels. For practical purposes one usually requires only these conventional cross sections. In such cases the polarization effects have to be eliminated either by depolarizing the laser beam or by carrying out the excited atom preparation at high beam densities where radiation trapping dominates. Alternatively, one can carry out the cross-section measurements on targets prepared by a sufficient number of different polarizations such that the magnetic sublevel cross sections involved and their average can be determined.

For the purpose of cross-section measurements, high excited level densities are required, and the precise knowledge of these densities is needed. The critical


parameters determining these densities are the laser wave length, the laser power density, pumping time, lifetime of the excited level, branching ratios and the match between the atomic level width and the laser band width. Because the laser light is coherent, under appropriate conditions the atomic excitation may be coherent (the atomic response maintains a well-defined phase relative to the electric vector of the laser light). These coherences in the level populations (Rabi oscillations) may be observed, but they usually are completely damped out by spontaneous decay at times larger than the radiative lifetime at moderate laser power densities. When coherence effects in the laser-excitation process can be neglected, the level populations can be calculated by the rate-equation method instead of the more rigorous (and more involved) density matrix approach. For two-level systems (or nearly two-level systems), the two approaches become equivalent and a cw laser pumping of an atomic beam will result in a steady state population in the two levels. For three-level systems, leakage to the third level (which is not available for optical pumping) occurs, and the system is gradually pumped over to this level. The rate, at which this happens, depends on the laser power density and the branching ratios. The branching ratio is the probability for spontaneous decay of the laser-excited level to a specific level relative to decay to all possible levels. If the branching ratio to the leakage channel is very small, the three-level system can be treated as a nearly two-level system.

In order to be able to discuss some specific questions related to laser-excitation for electron scattering measurements, we define a set of typical conditions in Table 1V. Some general remarks for cases deviating from these typical conditions will also be made. For the case when the typical conditions of Table IV apply we can make the following remarks. The overall atomic level width is about 100 MHz, which is small compared to fine-structure and, in some cases, even to hyperfine- and isotopic-structure spacings. With a laser of 5 MHz width these levels can be selectively excited. For conventional cross-section measurements this high selectivity may not be required because of the independence of cross sections on nuclear parameters, as mentioned previously. From the practical point of view, pumping of individual hyperfine and isotopic levels can be more efficient and this procedure does not introduce any complication into the interpretation of the cross sections. (The individual hyperfine and isotopic level cross sections are equal to the averaged cross sections.)

The polarization of the laser beam causes some degree of orientation or alignment of the target atoms when the atomic density is less than about l o i 2 cm-3 (Hertel, 1982). The measured electron collision cross section, therefore, does not correspond to the conventional averaged cross section (as discussed previously), and the nature of the initial level and the cross section have to be precisely defined. At higher target densities polarization effects disappear, the influence of coherence effects on level populations can be disregarded and the rate-equation approach predicts the proper level populations. For small branching ratios, the two-level limit is approached, the atoms undergo a large number of pumping


TABLE IV A TYPICAL SET OF EXPERIMENTAL CONDITIONS APPLIED IN LASER EXCITATION OF ATOMIC

TARGET BEAMS FOR ELECTRON SCATTERING CROSS-SECTION MEASUREMENTS

Laser Beam: Diameter Line Width (single mode) Power (cw) Energy Density Polarization

Atomic Beam: Diameter Averaged Speed (thermal) Collimation Density Spontaneous Radiative Lifetime Transit Time through Laser Line Broadening (FWHM): Natural (or lifetime) Power (or saturation) Collision (or pressure) Doppler

Electron Beam: Diameter Current Flux (electrons) Energy

0.2 cm 5 MHz (FWHM) 10 mW 8 X 10-5ergscm-3 Linear or circular

0.1 cm los cm s e c t 50 1011-10t3 cm-’

sec 2 x 10-bsec

Lorentzian 16 MHz Lorentzian 40 MHz Lorentzian negligible Gaussian 40 MHz (collimated thermal beam)

0.1 cm 10-y-10-4 Amp 10l2- 10” sec-I c r r 2 10-1-103 eV

cycles as they travel through the laser beam and high (20-50%) populations in the excited level can be achieved. For larger branching ratios to the leakage channels, the atoms are fairly rapidly lost to the unpumped levels and the population in the laser-pumped level is drastically reduced.

Special considerations apply to laser excitation of atomic beams when the laser interaction time is comparable to the natural life time (fast atomic beams, long lifetimes). In this case, Rabi oscillations become important. When line widths overlap or the laser width covers several levels, interference effects cannot be neglected and to predict level populations correctly one must use the density matrix approach. In thermal static targets and slightly collimated thermal beams, the Doppler width is on the order of a few Ghz and 100 MHz, respectively. Pumping with a single-mode laser in these cases affects only a small section of the Doppler distribution (hole burning). Multimode laser pumping achieves the same effect at several frequencies. A better excitation condition could be achieved by matching the Doppler distribution with the laser frequency distribution. This can be achieved by modulating the laser frequency as demonstrated by Giberson et al. (1985) and Lynn et al. (1986). Also, sperical considerations apply to pulsed and high-power lasers but these situations are of no concern to us here.


Laser excitation, for electron-excited atom scattering measurements, was applied so far to Na and Ba. Extensive literature on laser pumping of Na (e.g., Fischer and Hertel, 1982; Dreves et a f . , 1983; Cusma and Anderson, 1983) and of Ba (e.g., Bradley et a f . , 1973; Carlsten, 1974) exists.

Hertel and coworkers excited Na to the 3*E,* level and studied superelastic electron scattering by these atoms (in various degrees of polarization) for the purpose of determining electron impact coherence parameters. The present chapter is not concerned with this topic but interested readers are referred to the reviews by Hertel and Stoll (1977) and Anderson et al. (1988). Laser excitation of Na has been utilized by Bederson and coworkers in their measurements of electron collision cross sections on Na (3*Si2) atoms (Bhaskar et a f . , 1977; Jaduszliwer et a f . , 1980, 1985; Vuskovic et a f . , 1989b; Zuo et a f . , 1990; Jiang et a f . , 1990b, 1991a, 1991b). Some details on their schemes will be described in connection with the review of the cross-section results in Section V.

Trajmar and coworkers applied laser pumping to excite Ba into the 6'P; level for subsequent electron scattering measurements (Register et a f . , 1978, 1983; Trajmar et af., 1986; Zetner et a f . , 1989, 1990). Again some details on their procedures will be given in connection with the discussion of cross-section measurements in Section V.

E. ALTERING EXCITED ATOM COMPOSITIONS IN MIXED BEAMS

As it was pointed out earlier, atomic beams with excited atom components in many cases contain more than one excited species. In order to gain information on individual excited species, methods for eliminating components of the beam or altering the composition of the beam are required. This problem has been especially faced and investigated in connection with rare gas beams containing two metastable (and of course ground state) components. Most of the work along this line has been concerned with laser pumping of specific metastable atoms, and this matter will be discussed in Section 1V.B. in connection with laser induced fluorescence as in both cases the same principles and laser pumping techniques apply.

IV. Detection of Excited Species

For the purpose of absolute cross-section measurements, it is required that the density (or flux and velocity) of excited atoms in the target'beam be measured, and in cases where only relative cross sections are sought, detectors for monitoring beam intensities are required. Atomic beam detectors fall into four main categories: thermal, optical, secondary electron emission and ionization detec-


tors. Detailed discusions of these detectors for beams containing only ground state atoms are available (e.g., Scoles, 1988). Here we are going to consider detectors for excited atoms and only schemes that have been used in electron collision measurements. A few general remarks and pertinent references will be given for these detectors.

A. THERMAL DETECTORS

In thermal detectors the atoms impacting on the detector element deposit thejr kinetic (and internal) energy in the form of heat and cause a temperature rise, which in turn results in a change in the resistance (thermistor detectors) or in the potential (thermocouple, thermopile, pyroelectric detectors; Sharp et al., 1974; Bernker et al . , 1968; Geis et al . , 1975; Tiffanny, 1975) across the detector element. This change can be conveniently measured and calibrated as the function of deposited energy. The calibration is based on the fact that the energy deposition and detector response are independent of species. One can, therefore, utilize ion beams with known intensities for calibration.

Thermal detectors have been used most extensively for measuring intensities of fast metastable beams either as preliminary detectors or for calibrating secondary electron emission detectors, which are more convenient to use. Un- fortunately, thermal detectors cannot distinguish between fast ground and excited atoms. Applications of these detectors in cross-section measurements on metastable H (2s) and rare gases have been described by, e.g., Dixon et al. (1975, 1976) and Defrance et al. (198 1 ) .

B. OPTICAL DETECTORS

Detection of short-lived excited species is conveniently achieved by measuring the radiation emitted by the decay of these species. Although this approach has been extensively used, in general it has not been applied in studies concerned with cross-section measurements for electron collisions with excited atoms. We, therefore, omit the discussion of this topic.

Optical detection of metastable atoms is based almost exclusively on laser inducedjourescence (LIF) techniques. A general review on this topic is given by Kinsey (1977). Application of the LIF technique for determination of cross sections for electron impact excitation of metastable levels of rare gases (from the ground state) was described by Phillips et al. (1981, 1985). Zetner et al. (1986) utilized this technique for studying resonances in the excitation of the 2's level (from ground state) in He. In this chapter our concern is limited to the determination of excited (mainly metastable) level populations.

In the LIF method the laser beam, with appropriate wave length and polariza-


tion, is utilized to excite the metastable atoms into a higher level, which in turn decays by spontaneous radiation to the initial level and level(s) other than the initial one. Under proper laser pumping conditions, the fluorescence signal is proportional to the number of metastable atoms in the illuminated region. One measures the intensity of fluorescence, and if the experimental parameters, detection efficiency and optical transition probabilities are known, the metastable atomic density can be deduced. The proper laser pumping condition means that the laser power is low enough that the fluorescence signal is directly proportional to the laser power. In addition, since the analysis of the LIF results is usually based on the rate equations, one has to make sure that these equations are applicable. One can eliminate the need for determination of detection geometry and efficiency by measuring the fluorescence from the same level (excited by another process for which the cross section is also known) with the same experimental arragement (e.g., excitation of the same fluorescing level by electron impact from the ground state). In the expression for the ratio of these two fluorescences, the effective interaction volume and detector efficiency cancel out, and from the measured intensity ratio and corresponding excitation cross sections, one can deduce the target density of the excited atoms with respect to the ground state atoms. To the laser-excitation step the same general principles apply as those discussed in Section 1II.D. Specific application of the LIF method to metastable rare gas beams has been described by Coggiola et al. (1979), Gaily et al. (1980) and Rall et al. (1989). The rate equations for these processes have been given by Bussert (1986). The fluorescence distributions for the various rare gases was discussed by Bondybey and Miller (1977) and for the case of metastable (63P2 and 60P0) Hg atoms by Hanne et al. (1985).

If the laser-atom interaction time is comparable to the lifetime of the laser- excited level, and if the laser power is not low, Rabi oscillations in the level populations cannot be disregarded and the rate equation approach cannot be applied to predict the population developments in time. Such a situation was demonstrated in the case of laser pumping of a metastable Ne beam with a focused laser beam (Kroon et al . , 1985). Although in their experiments the atomic beam was thermal (V = 2 x lo5 cm sec-I), the effect could be observed because the relatively long lifetime of the excited ~ P ’ ~ P , level ( 18 psec) and the small diameter of the focused laser beam (0.04 cm diameter with Gaussian distribution of intensity), corresponding to an interaction time of about 100 psec. They interpreted their observations in terms of calculations based on the density matrix approach. Similar situation arises in connection with fast metastable beams even with un- focused, conventionally used laser beams. For example, for the same metastable Ne case, a laser beam diameter of 0.2 cm and V = lo7 cm sec-I, the pumping time is about 20 psec, which is comparable to the 18 psec lifetime.

Special attention has to be paid in all laser pumping cases to the homogeneous and inhomogeneous line widths and the laser beam frequency distribution. For fast beams, in general, and for uncollimated thermal target atoms the overall


atomic line widths (including Doppler broadening) are of the order of few GHz. For single-mode laser beams the line width is about 1 MHz. For multimode laser beams the overall frequency distribution is a few GHz but, within this overall width, the individual cavity mode spacings are large compared to the width of the individual modes. In either case only a fraction of the atomic distribution is pumped (hole burning). One way to avoid this effect and evenly pump the whole atomic distribution is to sweep the laser frequency with a single-mode laser over the full atomic width profile and measure the integrated flourescence for interpreting the LIF results (see Rall et al . , 1989). Another interesting approach was described by Giberson et al. (1985) and Lynn et al. (1986). They utilized a multimode laser to pump an argon metastable beam by modulating the laser frequency. This procedure allowed them to cover evenly the whole atomic distribution and to achieve a very efficient pumping.

As mentioned previously, altering the excited atom composition (quenching a metastable component) of mixed beams by optical pumping is an important rnat- ter in electron-excited atom scattering cross-section measurements. Extensive studies on this question have been reported in connection with metastable rare gases. The method is based on the same general principles as those described in connection with preparation of excited atomic beams by laser pumping and the LIF detection method. We, therefore, describe here only several applications of laser pumping to quench metastable rare gases.

Dunning et al. (1975b), Hotop et al. (1981) and Brand et al. (1991) applied laser pumping to quench and state select 3P0 and 3P2 metastable neon in a thermal beam. They all utilized multimode lasers ( ~ 1 0 0 mW power) that crossed the atomic beam several times. The multiple reflection applied (and the fiber optics cable in the case of Brand et al . ) caused a depolarization of the laser light and made the quenching more efficient by eliminating coherences (and of course increased the interaction length). The presence of the earth's magnetic field also contributed to the quenching efficiency by mixing the magnetic sublevels (Hotop et al . , 1981; Weissmann et al., 1984). Pumping of both 3P0 and 'P2 was demonstrated and depletion of 95% or more was achieved. Hotop et al. (198 1) also utilized a single-mode laser with power much less than 100 mW and achieved 99% quenching of 'P2. Depending on the transition (wave length) selected for the optical pumping, some of the pumped metastables are transformed into the unpumped metastable level, thus increasing the population of that level. The detection of the metastables was based, in all cases, on secondary electron emission measurements. State purification of fast (-- 1 KeV) metastable beam of Ne was reported by Gaily et al. (1980). In fast beams a significant Doppler narrowing takes place in the beam direction (Kaufman, 1976; Anton et a l . , 1978). Gaily et al. estimated that in their experiments the axial Doppler width was about 80 MHz (FWHM). To take advantage of this effect they used a single-mode laser (-50 mW power) in a coaxial arrangement. The laser and atom beam overlap was about 32 cm. The metastable component of the beam was determined by


monitoring the UV fluorescence resulting from the i.3Pi -+ 'So transition and using rate equations. They estimated that more than 90% of either metastable component of the beam could be removed. In their measurement no effort was made to depolarize the laser, and in their analysis polarization effects were not considered. Weissmann et al. (1 984) described the utilization of a multimode, cw, dye laser for intercavity fine-structure state selection of Ne, Ar and Kr thermal metastables. They applied a magentic field with a direction of 45" with respect to the electric field vector of the laser to achieve efficient mixing of magnetic sublevels and facilitate the complete removal of the J = 2 metastables. They used Penning ionization electron spectrometry for detection of metastables and demonstrated essentially complete (>99%) removal of the 'PZ components (part of it converted to ").

Convenrional phoro-absorption techniques have also been utilized to determine metastable rare gas densities in mixed beams by, e.g., Mityureva and Pen- kin (1975), Gostev et al. (1980b), Shafranyosh et al. (1989 and 1991), and in flowing after glow, e.g., Valezco er al. (1978).

C. SECONDARY ELECTRON EJECTION

Metastable atoms in a collision with metal surfaces (or gaseous species) can cause ionization and liberate an electron with some kinetic energy. The principles of electron ejection from solid surfaces have been discussed (e.g., Conrad et al., 1979, 1980). This process offers a convenient method for the detection of metastable atoms via well-established electron detection techniques. The requirement, that the internal energy of the atom has to be larger than the ionization energy of the target, puts limit on the applicability of the method but this pre- sents no problem for metastable rare gases. For quantitative application of this method, the key factor is the secondary emission coefficient. Extensive measurements concerning these coefficients have been reported (Dunning and Smith, 1971; Dunning et al., 1971, 1975a; Woodward et al., 1978). Very recently, Schohl et al. (1991) described a novel method for the absolute determination of metastable rare gas atoms fluxes based on secondary electron emission measurement and in situ determination of the electron emission coefficient. Continuous dynode electron multipliers can be conveniently used to detect metastables with internal energy larger than about 8 eV (Brunt et al., 1978). To extend the application to lower internal energies, lower work-function materials or heating the surface to a temperature just below that required for thermionic emission is required (Freund, 1971; Anderson and Jostell, 1974; Zubek and King, 1982; Parr et al., 1982).

Secondary electron emission detectors are widely used with both thermal metastable and fast beams because of simplicity and convenient, well-developed technology. However, as mentioned previously, calibration against other primary detectors is necessary.


D. IONIZATION DETECTION

Ionization detectors are the most popular devices for detecting ground state atoms. Electron-impact or photo ionization is applied to produce the characteristic ions, which are then conveniently detected by mass selection and current measuring or counting devices. (For a general description of this scheme see, e.g., Scoles, 1988). Measurement of the ion signal and knowledge of the experimental geometry, ion detection efficiency and ionization cross section are required for quantitative determination of atomic densities. When one deals with a mixed beam containing excited atoms the same methods can be applied if selective ionization is utilized. The electron (or photon) energy is kept below the value required for ionization of ground state species. The application of this approach for quantitative measurements is, however, severly restricted because the ina- vailability of ionization cross sections for excited atoms. The method is very sensitive and convenient to use for monitoring relative densities of excited atoms. Such an application to excited Ba atoms was utilized by Trajmar et al. (1986) and Bushaw et al. (1986). Penning ionization electron spectrometry can also be used to monitor metastable rare gas fluxes as has been demonstrated, e.g., by Weissmann et al. (1984).

E. SUPERELASTIC ELECTRON SCATTERING

Densities of excited atoms in a mixed beam can be deduced from measurements of superelastic scattering intensities IS(&, 0). The differential superelastic scattering signal at a given impact energy ( E , ) and scattering angle (6) is related to the corresponding differential superelastic scattering cross section, DCSS(Eo, 0), the excited state density (n,J and the effective scattering path length. DCSS can be measured by techniques similar to those applied in elastic and inelastic DCS measurements or can be obtained from the corresponding inelastic DCS by utilizing the principle of detailed balance. The effective scattering path length can be determined from the knowledge of scattering geometry, and thus the excited atom density can be deduced. It is more convenient and reliable, however, to eliminate the effective scattering path length by measuring the ratio of superelastic to inelastic scattering signals in the same experiment. Under appropriate conditions, the following equation holds:

wol 8) - - 5 DCSSW~, 6) wo, e) n , DCSW,, e)

where n, and the unindexed quantities refer to the ground state. From this equation the relative excited atom density (with respect to the ground atom density) is obtained, which is all that is needed for the interpretation of certain measurements. To obtain the absolute excited atom density, either the total atom density


or the ground state atom density needs to be determined. This approach has not been utilized so far in electron collision measurements.

V. Cross-Section Measurements

In this section, a complete review of existing electron impact data on excited atoms will be presented and the methods used for their acquisition will be discussed. Table V gives a summary of cross-section techniques based on various detection methods, and Table VI gives a summary of the cross-section measurements that have been carried out so far. Each cross-section type will be discussed separately.

A. TOTAL ELECTRON SCATTERING CROSS SECTIONS

Total electron scattering cross sections ( QToT) can be deduced from beam attenuation measurements (Bederson and Kieffer, 197 1 ; Trajmar and Register, 1984). Attenuation of either the electron beam through the atomic target (usually static gas for ground state species and beam for excited species) or the atomic beam crossing an electron beam can be related to these cross sections. The electron beam attenuation (or transmission) method requires the knowledge of interaction geometry and the measurement of electron beam attenuation relative to the incoming beam intensities as a function of target density, but there is no need for measuring absolute electron beam intensities. The atomic beam attenuation (or recoil) technique relies basically on the same principles as the electron beam attenuation method but requires higher angular resolution because of the small recoil angles. In this case, knowledge of the collision geometry, the absolute electron beam intensity, average atomic velocity and the attenuation of the

TABLE V CROSS-SECTION MEASUREMENT TECHNIQUES

A. Case in which Primary Particles are Detected: 1. Electrons

DCS,(Ei, 8) Qm(Eo) (integration over angles) QTOT ( Eo ) (transmission/attenuation)

(angular and energy distribution)

2 . Atoms (recoil) DCS,,(E,,, e) QTUT (Eo )

B. Case in which Secondary Particles are Detected: 1 . Ions

2. Photons Q4#," (Eli )

Line exc. xns." Branching Ratios Apparent exc. xns. Electron-impact exc. xns.

Cascade

"exc. xns. stands for excitation cross sections.


TABLE VI SUMMARY OF CROSS-SECTION MEASUREMENTS

Type Species References

DCS He (2%) Na (3zP3,2)

He (FS), ( 2 5 ) He (2% + 2lS)

Muller-Fieder et al. (1984) Zuo et a / . (1990); Vuskovic et al. (1989b); Jiang et al. (1990b, 1991a, 1991b) Register et a/. (1978) Mityureva and Penkin (1975, 1989): Gostev et al. (1980b); Rall er al. (1989). Mityureva and Penkin (1975) Mityureva e t a / . (1989 a,b,d) Mityureva er al. (1989 c,d) Mityureva et al. (1991) Sturnpf and Gallagher (1985) Shafranyosh eta/. (1991) Shafranyosh et a/. (1989) Dixon and Harrison (197 1); Dixon et a/. (1975); Defrance e t a / . (1981) Long and Geballe (1970); Dixon er al. (1976) Dixon et a/. (1973); Fite and Brackrnann (1963); Vriens et a / . (1968); Koller (1969); Shearer-Izumi and Botter (1974) Dixon et a/. ( 1 973) Dixon et a/. (1973) Vuskovic (1991) Trajmar et a / . (1986) Aleksakhin and Shafranyosh (1974) Neynaber er al. (1964) Wilson and Williams (1976) Celotta er al. (1971) Bhaskar et a / . (1977); Jaduszliwer er al. (1980 and 1985)

"Opt. Ex. F. stands for Optical Excitation Function.

atomic beam intensity with respect to the incoming beam intensity are required, but there is no need for measuring absolute atomic beam intensities. In both cases corrections for undistinguishable forward scattering and scattering back into the forward beam need to be considered. For ground state species these measurements can be carried out with high precision and total electron scattering cross sections are the most accurately (within a few percent) known electron scattering cross sections. When one applies these methods to excited atoms, a number of additional complications arise. It is, in general, not possible to produce a pure excited atomic beam (or static target), and one has to devise some scheme to separate out or selectively detect the excited species. In some cases only the knowledge of relative densities is required (if the ground state cross sections are known and attenuation measurements with and without the presence of excited species can be carried out). Depending on the method of preparation


of the excited atoms, one may also have to consider the effect of recoil or the polarization (orientation or alignment) of the atomic target due to the preparation process. We describe here in some detail the various schemes that have been applied so far and will summarize the cross-section results.

The first measurement was carried out on He (23S) atoms by Neynaber et al. (1964) in the 0.8 to 8 eV electron-impact energy range. The 23S, (M, = 1 ) component of the mixed thermal beam (prepared by discharge) was deflected out of the original beam by an inhomogeneous magnetic field, and the attenuation of this metastable beam by the action of a transverse electron beam was measured (recoil approach). A detailed description of the procedures and the results is given by Massey et al. (1969). The cross section was found to be about two orders of magnitude larger than those corresponding to ground state He in this energy range. See Fig. 2 and Table VII.

Wilson and Williams ( 1976) reported absolute total scattering cross sections for He (2IS0) and relative cross sections for He (z3S,) species in the 0.45 to 9.4 eV

1000

Metastable

100

f 4

El Y

d?

10

I 1 10

E&V)

FIG. 2 . Total electron scattering cross section for ground and metastable He species. 0 = groundstate, Kennerly and Bonham (1978) and Nickel er al. (1985); 0 = 2)s. Neynaber et a / . (1964); A = 2's. Wilson and Williams (1976) normalized to 145 x cm2 at 7 .84 eV; 0 =

2's. Wilson and Williams (1976).


TABLE VII TOTAL ELECTRON SCATTERING CROSS SECTIONS FOR GROUND

AND METASTABLE HE SPECIES

Q(l0-l6 cm*)

Ea(eV) 1's" 23sb 23s' 2 s d

0.37 0.50 0.71 0.74 0.83 0.87 0.96 1 .oo 1.09 1.33 I .36 1.49 I S O I .83 I .90 2.00 2.50 2.83 2.96 3.00 3.83 3.96 4.00 4.77 5.00 5.85 5.87 6.00 7.94 8.00 8.13 9.92

10.00

"Below 4 eV: Kennedy and Bonharn (1978).

hNeynaber et al. (1964). 'Wilson and Williams (1976); normalized to 145 (A)2 at

dWilson and Williams (1976).

Above 4 eV: Nickel era/. (1985).

7.84 eV.


impact energy range. They utilized a discharge to produce a well-collimated, thermal beam containing these metastable and ground state atoms. The metastable beam intensity was measured by a surface-ionization detector. By application of optical quenching, they could remove the 2'S0 component of the atomic beam both upstream and downstream of the electron beam crossing region. The atomic beam velocity distribution and average velocity were determined by chopping the atomic beam and determining the time-of-flight distribution of metastable He. The electron beam crossed the atomic beam at 90" angle and was produced by a magnetically collimated gun described by Collins et al. (1970). The electron beam was modulated, and the resulting modulation of the metastable atomic beam was measured simultaneously with the electron beam current. Scattering cross sections for 2's species were calculated from the difference of the signals measured on the unquenched and upstream-quenched beams. Upstream quenching eliminated the 2 IS atoms. The difference signal, therefore, represents scattering by 2's atoms. The cross sections were calculated (as in general for recoil measurements) from the equation

( 1 )

where S = lo - I is the rate at which the metastables are scattered out of the atomic beam of initial intensity lo and mean velocity of (v) by a crossed electron beam of intensity I , and height of h. I is the atom beam intensity after passing through the electron beam. The relative cross sections for He (23S, ) were obtained from measurements on the totally (both upstream and downstream) quenched beam. (Downstream quenching was necessary to remove atoms that have undergone the z 3 S , + 2'S0 spin-flip scattering.) The z3S, cross sections could not .be obtained in absolute units because of the sensitivity of the metastable atom detector to resonance radiation present in an unknown amount in the beam. The data were normalized, therefore, at 7.94 eV to the value of 165 .rra; (based on the measurement of Neynaber et al., 1964). These results are also shown in Figure 2 and Table VII.

Celotta et al. (197 1) reported total electron scattering cross sections for metastable Ar based on atomic beam recoil measurements. They produced a thermal beam containing metastable and ground state species by utilizing a magnetically collimated, rectangular electron beam and ground state Ar beam. The two beams crossed each other at 90". A second electron gun was used for the scattering measurements. This gun could be turned on and off in alternating sweeps to obtain the metastable signal attenuation by the electron beam. Both electron guns were of the Collins type (Collins et al . , 1970). The 3P2 and )Po metastable components of the beam were not distinguished and an equation equivalent to Eq. ( I ) was used to obtain cross sections for the unspecified mixture of metastables in the 0.35 to 6.75 eV electron energy range. The velocity distribution of the atoms was determined by time-of-flight technique. The results are summarized and compared to ground state cross sections in Fig. 3 and Table VIII.


1000

100

1

0.1 0 L-, 3 6

E , W

FIG. 3. Total electron scattering cross sections for an unspecified mixture of 3P2 and 'Pa metastable and ground-state argon. 0 = ground state, Jost et al. (1983); 0 = groundstate, Ferch et nl. (1985); 0 = 'P2 + 'Po, Cellota et al. (1971).

TABLE VIII TOTAL ELECTRON SCATTERING CROSS SECTIONS FOR UNSPECIFIED

MIXTURE OF 'P, A N D 'Pl1 METASTABLE A N D FOR IS, GROUND STATE AR ( C M ~ UNITS)

0.35 0.50 1 .o 2.0 3.0 4.0 5.0 6.0 6.75

858 537 305 258 200 I88 175 I72 166

0.31 I 0.32 0.416 0.46

I .49 3.41 5.45 7.18 9.12 11.3

"These results were obtained from Figure 6 of Celotta et nl.

'From Ferch et nl. (1985). "From Jost et al. (1983).

( I97 1).


Total electron scattering cross sections for laser-excited Na (3 2pj,2, M, =

3/2) species were measured by Bhaskar et al. (1977) at 4.4 eV, by Jaduszliwer et al. (1980) in the 0.84 to 6.0 eV range and by Jaduszliwer et al. (1985) in the 6 to 25 eV energy range. These measurements represent a generalization of the conventional atomic beam recoil technique and involve considerations for the recoil caused by the laser excitation. It may be called, therefore, a double-recoil technique. They applied a 785 Gauss magnetic field along the electron beam axis in the interaction region to fully decouple the nuclear and electronic magnetic moments in the excited level and to split the M, magnetic sublevels. A standing wave, single-mode, cw, linearly polarized, dye-laser beam (with its propagation vector perpendicular both to the electron and atomic beams) was utilized to prepare three *e,* excited atoms in the M, = -+ 3/2 magnetic sublevels. The electron beam recoil technique was then applied to the beam containing both ground and excited atoms. The excited to ground state population fraction was determined by utilizing the photon recoil acting on the excited atoms and measuring the scattering-out signal with the laser beam on and off. In addition the known values of the total electron scattering cross sections for ground state Na were used to obtain the excited state cross sections. A velocity selection in the original atomic beam was required to make this approach feasible. This selection was

TABLE IX TOTAL ELECTRON SCATTERING CROSS SECTIONS FOR GROUND

AND 3?P12 SODIUM ATOMS ( CM? UNITS)

E,(eV) Ground 32PIt?

- 0.5 3.52 0.75 2.46 -

0.83 - 5.95 1 .oo I .97 1 . 1 1 - 2.95 I .5 1.52 -

1.68 - 2.30 2.0 1.29 2.31 - 2.25 2.5 1.13 3.0 I .02 2.10 3.5 0.97 - 4 . 0 0.93 4.48 4.5 0.88 - 5.0 0.82 -

5.5 0.79 - 6.0 0.72 1.15 8.0 0.66 -

10.0 0.75 1.21 15.0 - 1.28 20.0 0.65 1.28 25.0 - 1 . 1 1

-

-

-

-

I .25 -


50 t 01 ' " I '

I

1 10

E,(4

1973); 0 = 3 jRi2 , (Jaduszliwer er a / . . 1980 and 1985).

achieved by using a hexapole electromagnet that focused the atoms having M, =

+ 1/2 into only the interaction region. The results of these measurements are summarized in Table IX and Fig. 4. For comparison the ground state cross sections (from Kasdan et al . , 1974) are also given. The difference between cross sections for ground and excited species for Na are not as for He or Ar. This reflects the general trend that as one goes to heavier elements and to larger, more loosely bound electronic structures, the difference between ground and excited species diminishes as far as electron scattering is concerned.

FIG. 4. Total electron scattering cross sections for sodium. 0 = ground state (Kasdan et al . .

B. IONIZATION OF EXCITED ATOMS BY ELECTRON IMPACT

The majority of reported ionization cross-section measurements on excited atoms have involved metastable targets, and the species most widely studied has been helium. Several measurements on metastable atomic hydrogen have also been

74 S . Trajmar and J . C . Nickel

reported and one unpublished study of neon and argon has been made. In addition, ionization measurements of excited barium and strontium have been published. Only integral ionization cross sections (Qi,,) have been reported sofar. It will be convenient to discuss these experiments in terms of the species involved.

1 . Metastable Helium

Using an RF discharge metastable source, Fite and Brackman (1963) were the first to measure the ionization cross section for an unknown mixture of 23S and 2's metastable helium. For the ionization, they applied a crossed electron-atom beam arrangement. Vriens et al. (1968), also using a crossed-beam technique, employed direct electron beam excitation on a helium atomic beam in the hope of producing only 23S metastables (plus ground state species) in their beam. They hoped to eliminate the 2 ' s contribution by adjusting the energy of their excitation electron beam below the 2 ' s threshold. In the end, they reported absolute cross sections for an unknown 23S and 2's mixture, with the absolute scale being determined in a somewhat arbitrary fashion. Long and Geballe (1970) made measurements of the ionization cross section of z3S helium using an electron beam excitation technique. Their apparatus consisted of two closely spaced chambers, the metastable producing (M.P.) chamber and the ionization producing (1.P.) chamber. Both chambers were filled with helium at a pressure of 1 micron. The impact energy of the electron beam in the M.P. chamber was set to 20.4 eV to produce only 23S metastables. A fraction of the metastables produced drifted into the I.P. chamber and was ionized by a second electron beam. A fraction of the ions so produced were collected and focused onto the slit of a 60" magnetic analyzer. The cross sections were placed on an absolute scale by comparing the metastable signal at 12.0 eV electron impact energy with the ground state signal at 42.0 eV electron impact energy. The accuracy of the metastable cross sections obtained by their technique depends upon the metastable production cross section at 20.4 eV, the ground state ionization cross section at 42.0 eV and a kinetic theory calculation used to determine the density of metastables in the interaction (1.P.) region. They estimate their error at the calibration point to be 30%, and the statistical errors at each point appear to be on the order of 15%. The results of Fite and Brackmann, Vriens el al. and Long and Geballe are shown in Fig. 5. Since none of the authors gave data in tabular form, their data in Fig. 5 has been extracted from the published curves. It should be observed that the electron impact energies in their measurements range from metastable threshold to ground state threshold. This energy range limitation is inherent in most experiments utilizing a discharge or direct excitation beam source. The metastable to ground state ratios in these beams ranges from lo-' to and the ground state contribution to the ionization signal swamps the metastable contribution at electron energies above ground state threshold. Shearer- Izumi and Botter (1974) showed the importance of high Rydberg states of helium

CROSS-SECTION MEASUREMENTS FOR ELECTRON IMPACT

8r- - - - - - -7 6 -

5 -

Helium i

I I I I 0

A 0 A Metastable

3

75

10 100 1000

E o ( 4 FIG. 5 . Cross sections for ionization of helium by electron impact. The long dashed line is the

ground state, Krishnakumar and Srivastava (1988); the dotted line is 2's + 2's. Fife and Brackmann (1963); the short dashed line is 2 5 + 2'S, Vriens et a/. (1968); the solid line is 2's. Long and Geballe (1970); 0 = z3S, Dixon et al. (1976); A = 2's with corrections made for slow ion trapping in electron beam, Dixon et a/. (1976).

in ionization experiments. When metastable helium is produced by direct electron impact at electron energies above 24 eV, long-lived Rydberg atoms will be produced and will contribute to the ionization signal. They suggest this effect as a source of disagreement between earlier results.

In order to extend the electron impact energy range above the ground state threshold, one must use an atomic beam in which the ratio of metastable density to groundstate density is about 0.1 or higher. Such beams can be produced by charge exchanging an ion beam of the desired species in an alkali metal vapor cell, the fast beam technique.

Using an atomic beam generated by the fast beam technique in a crossed electron-atom beam configuration, Dixon et al. ( 1973) reported preliminary ionization cross sections for electron impact on metastable helium and final re-


sults in Dixon et al. (1976). Their metastable beam was produced by charge exchanging a fast (2-6 keV) singly charged He+ ion beam in a low-density cesium vapor cell where single-collision conditons prevailed. Using deflectors and field ionization wires following the charge exchange cell, charged particles and high-lying Rydberg atoms ( n 2 14) were removed from the atomic beam. The atomic beam was then crossed with an electron beam and ions produced were collected, passed through a magnetic analyzer and detected. In the ionization region, the atomic beam consisted of fractionsf,, fs, f l andf,, corresponding to ground, 2 'S , 2's and Rydberg atoms (with 8 S n S 13), respectively. Ryd- berg atoms with n 14 were removed by field ionization, and those with n S 7 decayed between the charge exchange cell and the interaction region. While Ryd- berg atoms had small fractions, their effects could be significant, especially at low electron impact energies, because their ionization cross sections are proportional to E , I . The fractions f, , fs, and fi were obtained by assuming that in the charge exchange process, electrons were captured only into the n = 2 manifold (2'S, 2IP, 2's and 2'P) and that these levels were initially populated in the ratios of their statistical weights. If it is assumed that the 23P and 2IP levels radiatively decay to the 23S metastable and I IS ground levels, respectively, one obtainsf, =

0.06, fl = 0.75 and f, = 0.19. If it is assumed that direct capture to the 2 ' s level does not occur, one obtains f, = 0, fi = 0.8 and f, = 0.2. Dixon et al. (1976) made the latter assumption in analyzing their measurements. By measuring absolute electron and atomic beam currents and ion count rates and by measuring the electron beam flux and atomic beam density distributions, they obtained a quantity Qm(Eo) that they call the measured cross section for ionization at electron impact energy E, . The measured cross section is related to the ionization cross sections from the ground (QJ, the 2'S(Q,) , the 23S(Qf) and the Rydberg (Q,,) levels by

14

Q A E J = f3QJ(EJ + f f Q , ( E J + 1 - f> - f f - c f.) n = X

14

X Q,(Eo) + c fnQn(E0) + S ( E d n = 8

where S(E,) is any contribution to Qm from spurious effects. By assumingf, =

0, f f = 0.8, f, = 0.2 and S ( E , ) = 0 and accounting for the Rydberg contribution, EfnQn, they first determined Ql(Eo), which is shown in Fig. 5. The Ryd- berg contribution was determined by fitting CfnQn = m / E , to the ionization data below threshold. Subsequently, the authors make a careful analysis of spurious contributions to the ion signal. In these types of experiments, the largest contribution to the background ionization signal comes from stripping of fast neutral atoms by residual gas in the vacuum chamber. These stripped ions can be removed immediately before the ionization region but not after it. Because the


signal to background ratio was small (-0. l), modulated beam and gated detection methods were employed. When the electron gun was pulsed there was the possibility of enhanced outgassing of the electrodes during the electron beam pulse, causing a modulation of the background. By judiciously adjusting the period of their electron beam pulse ( t , ) relative to the time constant for changes in gas density (7,) the authors (using t , < 7,) were able to show that this pressure modulation effect was not important in their experiment. Another spurious effect, present but accounted for in their experiment, was caused by charge exchange of fast metastables with slow ions trapped in the ionization electron beam. When the electron beam was on, residual gas such as 0, was ionized and the resulting slow ions could be trapped in the potential well of the electron beam. It was argued that the cross sections for processes such as

He(23S) + 0: +- He+ + 0,

are large and could contribute significantly to the ion signal when the electron beam was on. When the electron beam was off, the ions were not present and this effect did not contribute to the background. This effect causes an overesti- mation of ionization cross sections. The authors calculated a correction, S ( E , ) , for this effect and presented corrected cross sections Q,(Eo), which are also shown in Fig. 5 . The effects of slow ion trapping are minimal up to about 50 eV but become increasingly important at higher energies.

In Table X, ionization cross sections for ground and metastable helium are given for comparison.

2 . Metastable Hydrogen

Dixon et al. (1975) reported electron impact ionization cross sections of atomic hydrogen in the metastable 2s state. A 2 keV proton beam was charge exchanged in a cesium vapor cell to produce a fast hydrogen neutral beam consisting of metastable 2s and ground state 1 s species. This neutral beam was crossed by an electron beam in the ionization region to produce the desired ionization signal. The ions passed through a magnetic mass analyzer and were detected. The neutral beam intensity (atomskec), lo, was measured by observing the secondary emission in a Faraday cup with the secondary emission coefficient being determined by a vacuum thermopile.

The basic experimental arrangement for atomic hydrogen ionization was similar to the arrangement used by Dixon et al. (1976) described previously for He ionization but with some important differences. As in their He case, a set of field ionization wires was placed immediately following the charge exchange cell to ionize high-lying Rydberg states ( n = 8) followed by a set of weak field deflectors to remove charged particles from the atomic beam. Unlike the He case, however, the electric fields in the field ionizer and deflector regions could partially

S . Trajmar and J.C. Nickel

TABLE X IONIZATION CROSS SECTIONS FOR GROUND AND METASTABLE HE SPECIES ( 10-lb CM? UNITS)

E d e W Ground" 2 6 + 2]Sb 2 ' s + 2%' 23Sd 235 235'

4 5 6 6. I 6.6 7 7.1 7.6 8 8.6 9 10 10.6 I 1 12 12.6 13 14 15 15. I 16 17.6 19 20.1 22.6 25 27.6 30 32.6 35 37.6 40 45 47.6 50 58 68 78 88 98 100 I23 I48 I50 173 193

4 x 10-3

6.72 x lo-?

,108

,168 ,209

,237

,370

,380

.20

.56 I .2

1.8

2.2

2.5

2.7 2.8

2.8

2.8

2.8

2.8

.I4

.41 1.2

2.0

2.6

3.1 3.5

3.9 4.3

4.6

5.0

5.1

5.1

.20 3.2

5.4

6.6

7.0 6.8

6.7 6.5

6.3

6. I

6.0

4.03 5.09

5.63 5.59

6.20

6.98

7.23

7.15

7.19

6.70 6.43

6.14

4.99

5.02

4.27

4.07 3.50 3.26 2.96 2.72

2.45 2.11

I .93 I .79

6.38

4.89

4.11

3.87 3.28 3.04 2.73 2.49

2.21 1.87

I .70 1.58


TABLE X (continued)

EdeV) Ground" 2 3 + 2'Sh 2 's + 2 5 ' 2?S" 235' 235'

198 1.70 1.49 200 ,360 248 1.44 1.24 298 1.30 1.12 300 .305 348 1.12 .95 398 1.06 .90 400 ,257 498 .88 ,745 500 ,220 598 .736 ,615 600 ,196 698 ,651 ,542 700 ,178 798 ,605 .505 800 ,163 898 ,553 ,460 900 ,149 988 .516 .428 998 ,503 ,414

lo00 ,139

"Krishnakumar and Srivastava (1988); selected energies. bFite and Brackman (1963); data extracted from curve. 'Vriens e: a / . (1968); data extracted from curve. dLong and Geballe (1970); data extracted from curve. 'Dixon e: a/ . (1976). 'Dixon et a / . (1976); corrected for charge exchange of metastables with trapped ions.

quench the metastable component of the atomic beam by Stark mixing the 2s and 2p levels. The fast transit time through the field ionization region and the weak deflector fields ensured that the metastable component in the atomic beam was not totally but only about %%quenched. By applying a strongelectric field, the metastable component could be totally quenched, and this possibility was put to good use in their experiments for two purposes. First, they write for the H(2s) metastable intensity, 12s, in the neutral beam at the ionization region

1 2 5 = f F l o

where f is the fraction of the beam in the 2s state immediately after the conversion cell, and F is the fraction that survives the electric fields of the ionizer and deflectors and reaches the ionization region. They took f to be 0.25, a result expected from a statistical weights argument for charge exchange in a single- collision cesium target. F was determined by a Lyman a detector immediately following the ionization region. The Lyman a detector consisted of tubular elec-


trodes that provided a total-quench electric field and a photomultiplier to observe the resulting Lyman a radiation. F was determined by the ratio of the Lyman a detector outputs with the field ionizer and deflector electric fields on and off and was found to be 0.45. Another application of the total-quench capability was made by placing a set of plates immediately preceding the interaction region. The potential difference between the plates could be pulsed to provide a neutral beam at the ionization region whose 2s metastable content could be modulated.

As with most fast atomic beam ionization experiments, the single largest contribution to background noise was stripping of neutral fast atoms in the background gas. Dixon er al. (1975) had a signal to noise ratio of 1 : 200. To extract the metastable ionization signal, they used pulsed beams (both electron and metastable) and a gated detector data acquisition scheme. Th details of this scheme are described both in their paper and by Harrison (1968). Their data was reduced according to the expression

where AS is the accumulated ionization signal (countshec), v is the electron velocity, V is the atom beam velocity, Tis the time averaged electron current, i2, is the time averaged intensity of the H(2s) beam entering the collision region, e is the electronic charge and R is the detection efficieny. The term h' is a beam overlap function and is approximately the height of the collision region. All parameters except i2, and Q( 1 s) are directly measured, and iZs is determined from Eq. (2). The results for Q(2s) obtained by Dixon er al. (1975) are shown in Fig. 6 .

Defrance et al. (1981) also reported ionization cross sections for electron impact on metastable atomic hydrogen. In many respects, their experimental arrangement was similar to that of Dixon er al. (1975). A mass-selected beam of protons having an energy of 3125 eV was charge exchanged in a cesium vapor cell. Charged particles and high Rydberg states were removed by a set of deflectors and a field ionizer. An additional set of electrodes allowed them to completely quench the metastable component of the beam. The resulting neutral hydrogen beam was detected by observing secondary emission in an appropriately biased Faraday cup with the secondary emission coefficient being determined by a bolometer. Beginning with a metastable fraction of 20% immediately following the conversion cell, they calculated the attenuation by quenching in the subsequent electric field and determined that the metastable fraction in the interaction region was 6%. As in the Dixon et al. (1975) experiment, an electron beam crossed the neutral beam at the ionization region, and the resulting ions were passed through a magnetic analyzer and detected. Unlike the Dixon et al. (1975) experiment, however, the electron beam was ,wept through the neutral beam (in a perpendicular direction) at a frequency of 390 Hz. The swept beam


11

10

9

8

7

h

N! 6 s 2

W-

v

g 5

4

3

2

1

0 d 10 100 1000

E,(eV) FIG. 6. Cross sections for ionization of atomic hydrogen by electron impact. 0 = ground state,

calculated from Lotz formula as given in Defrance ef al. (1981); 0 = 2s. Dixon er al. (1975); A =

2s, Dixon ef al. (1975) with apparatus modification giving higher signal-to-noise ratio; 0 = 2s, Defrance er al. (1981).

technique has two advantages. First, the ionization cross sections can be extracted without resort to beam profile measurements. Second, it is suggested that the swept-electron-beam technique minimizes the effects of slow ion trapping in the electron beam provided the beam is swept with frequencies >20 Hz in their case. This effect was not taken into account in the Dixon et al. (1975) hydrogen paper.

Defrance et al. (1981) show that the number of ionizations, K, produced by an electron beam of intensity I, during one passage across an atomic beam of intensity I, is given by


where v is the ionization cross section, V , and V, are the velocities of the electrons and atoms, respectively, and U is the sweep velocity of the electron beam. In their actual experiment, the metastable fraction is pulsed on and off with a period of 2.56 sec (allowing 100 electron beam passes through the neutral beam). In this fashion they can measure Q2s - Q l s . Using the Lotz formula (Lotz, 1966) for Q l s , they obtain QZs, which is also shown in Fig. 6.

A summary of ionization cross sections for ground and metastable hydrogen is given in Table XI.

3. Metastable Neon and Argon

Dixon et al. (1973), presented ionization measurements on He (2)S, 2'S), Ne ('P2, )Po) and Ar ()P2, 2Po) at the VIIl ICPEAC. A full paper was later published on the helium results but not on the neon and argon results (although these results have been widely used and quoted in the literature). They were reluctant to pub- lish a full paper on Ne and Ar because of uncertanties in the beam fractions and neutral detector efficiency. Very recently Fujii et al. (1991) reported preliminary measurements of the ionization cross sections for metastable neon. These results were in good agreement with those reported by Dixon et al. (1973).

4 . Laser Excited Barium

Trajmar et al. (1986) reported electron impact ionization cross sections for the 6s6p .IP and 6s5d (ID + 3 D ) excited levels of I3*Ba. In this experiment, an electron beam crossed a neutral barium beam of natural isotopic abundance at the ionization region. Ions produced were extracted from the ionization region by a weak electric field, transported to a quadrupole mass spectrometer (tuned to the 138 mass peak) and detected. A single-mode laser beam tuned to the 6s2 IS + 6s6p 'P transition (553.9 nm) crossed the barium beam and could be positioned below the ionization region or in the ionization region. With the laser beam positioned in the ionization region, the ionization region contained a mixture of ground, excited IP, and metastable (ID + )D) species. When the laser beam was positioned below the ionization region, the I P species decayed before the beam reached the ionization region and the beam in this region consisted of a mixture of IS ground states and (ID + 30) metastable species. With the laser off, the ionization region naturally contained only IS ground state species. The beam fractions of the species present for each laser position were determined by optical pumping calculations using a rate equation approach. These beam fractions together with the known ground state ionization cross section of barium allowed the extraction of the ionization cross sections of the 'P and (ID + )D) species. Measurements were carried out in the energy range of threshold to 10 eV. The results together with the ground state cross sections are shown in Fig. 7. Due to a small (-100 Gauss) collimating magnetic field in the electron gun, the 18 level was Zeeman split and the actual data shown in Fig. 7 repre-


TABLE XI IONIZATION CROSS SECTIONS FOR GROUND AND METASTABLE

HYDROGEN (10-l6 C M ~ UNITS)

Eo(eV) Ground” 2sb 2s‘ 2sd

6.3 8.3 8.5

10.3 12.3 13.5 14.3 18.3 23.3 23.5 25.3 31.8 33.3 38.3 38.5 48.3 68.3 68.5 98.3 98.5

148.3 148.5 198.3 198.5 218.3 218.5 248.3 298.3 348.3 348.5 398.3 498.3 498.5 748.3 998.3

. I2

.25

.40

.45

.57

.57

.60

.67

.65

.58

.48

.40

.37

.34

.30

.27

.25

.21

. I5

. I2

7.25

9.1

6.7

5.7

3.58

2.48

2.19

1.83

1.27

9.5

7.3

4.94

3.84

3.11

2.61

2.05

I .63

1.19

5.94 8.75

10.5 7.67

8.06 7.56 6.22

6.63 5.39 4.93

4.08 3.14

2.91

I .93

1.75

1.61

1.54 I .26 1.15

1.04 ,867

,655 ,482

“Calculated from Lotz formula as given in Defrance et a/.

bDixon et al. (1975). ‘ Dixon et al. (1975); using slightly different experimental

apparatus than in b, which yielded a higher signal-to-noise ratio.

( I98 1).

dDefrance e t a / . (1981).

84

IMPACT ENERGY (eV) FIG. 7. Electron-impact ionization cross sections for 'j8Ba (. . . 6s6p1P,), 0, and I3*Ba

(. . . 6s5d1D and >D), 0, species from Trajmar et a / . (1986). For comparison the ground ( I S )

cross sections are also shown (from Dettmann and Karstensen, 1982).

sent ionization of laser excited M , = - 1 level of 13*Ba (chosen by a combination of laser frequency, polarization and mass selection). However, it was shown that the ionization cross sections from all magnetic sublevels were identical and need not be distinguished. It should be also mentioned that although the measurements were carried out on the 138 isotope, the same cross sections apply to other isotopes (and to the naturally occuring isotopic mixture) as nuclear effects on the ionization cross sections can be neglected. The ionization cross sections for the excited species are larger (by about a factor of 2 at their peak) than those in the ground state (see Table XII).

5 . Metastable Strontium

Aleksakhin and Shafranyosh (1974) reported ionization cross sections for 1 ID, metastable strontium. Using a two-electron beam technique (preparation and ionization) they found that the ionization cross section has a peak value of 8 x l O - I 4 cm2 at an electron impact energy of 10 eV. They present the energy dependence of the ionization cross section in threshold energy units.

c . LINE EXCITATION, APPARENT LEVEL EXCITATION AND INTEGRAL

ELECTRON-IMPACT EXCITATION CROSS SECTIONS-OPTICAL METHODS'

The literature on optical methods uses a variety of terminology for reported cross sections. For the purpose of adopting a consistent set of definitions for discussing

"We learned, after the completion of this manuscript, that a chapter was written by Lin and Anderson for Advances in Atomic, Molecular. and Optical Physics (Vol. 29, 1992). This chapter reviews and discusses the application of optical and laser techniques to study electron-impact excitation of rare gases into and out of the metastable levels.

C R O S S - S E C T I O N M E A S U R E M E N T S F O R ELECTRON I M P A C T 85

TABLE XI1

AND ID + 'D EXCITED STATES OF BARIUM, UNITS ARE 1 0 - t 6 c ~ z (TRAJMAR E T A L . , 1986)

E,(eV) Ground State ID + 'D 1P

IONIZATION CROSS SECTIONS FOR GROUND STATE, 'P EXCITED STATE

3.1 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0

I .oo 2.00 2.80 3.90 4.80 5.70 6.60 7.40 8.30 9.20

10.00 10.60 I 1.40 12.1 12.8 13.3 14.0 14.5 15.0

1.66 2.04 3.6 4.62

.45 6.40 1.3 8.42 2.3 11.0 3.3 13.0 4.7 16.2 5.7 21.7 7.5 24.6 9. I

11.4 30.7 13.3 33.0 14.5 35.5 16.2 34.6 15.8 35.4 18.0 39.1 17.6 33.0 17.5 30.8 19.3 32.2 20.3 32.7 20.6 30.2 20.3 30.4 20.7 27.6 21.7 28.5 22.0 28.4 22.6 28.5 23.4 29.4 23.0 26.9 23.9 30.0

these results, we briefly review the various measurement techniques and methods used to deduce cross sections.

Atoms (the interaction region) are illuminated uniformly by an electron beam having energy Eo. Let the atoms have the schematic energy-level diagram shown in Fig. 8, where the lowest level ( i ) is the initial level, usually the ground level or a metastable level. In a typical experiment, the total photon flux produced in the interaction region, @,k (photonshec) for a particular transition between levels j and k having wavelength A,, is measured.

A cross section, the line excitation cross section, can be defined for this process so that


1 ' I

FIG. 8 . A schematic energy-level diagram indicating electron impact excitation (dashed lines), cascade (solid line, t? + j ) and emission (solid line, j -+ k) processes.

where J , is the electron flux, N , is the number of target atoms in level i , I , is the electron current, n, is the target density, e is interaction length and Q, is the line excitation cross section for producing photons of wavelength hJk. The line excitation cross section as a function of electron impact energy is sometimes referred to as the optical excitation function.

The apparent level excitation cross section, Q:, is defined as

where y,k is the branching ratio, defined as

Where AJk are the spontaneous emission coefficients. The summation in the denominator is over all levels to which level j can decay by radiation. Therefore yJk accounts for all undetected radiation. Q: represents the cross section for all excitation processes that contribute to the population of level j . These include both direct electron impact excitation from level i and cascade contributions from all electron-impact excited levels e that are above j and decay by spontaneous emission to level j .

A cross section for direct electron impact excitation of level j from level i , the integral electron-impact excitation cross section, can be defined as


N, = J,N,Qi = I Ln,lQ,9 e

Where N,, is the direct production rate of species j from species i and Q, is the integral electron-impact excitation cross section from level i to level j . Q, is related to, but not generally equal to, the apparent level excitation cross section Q; . It can be shown that

The bracketed term on the right-hand side of Eq. (4) takes into account cascade contributions from levels t? higher than j . Comparing Eq. (3) with Eq. (4), it is seen that Q; is an approximation to Q, when cascade contributions into level j are neglected. If there were no cascade contributions into level j , we would have Q; = Q,. It is convenient to discuss the available optical data in terms of the target species involved.

I . Metastable Helium

Gostev et al. (1980b) reported excitation cross sections (it is not clear to us whether these are integral electron-impact excitation or apparent level excitation cross sections) for excitation from, the 23S metastable helium level to the 33P, 3’0, 4’0 and 3’P levels in the electron-impact energy range from threshold to about 10 eV. Their 2)s metastable beam was generated by charge exchanging He ions (extracted from a Penning discharge source) in a solid capillary converter. The atomic beam was crossed by an electron beam, and the resulting radiation was observed. The 23S metastable density in their interaction region estimated as was 6 X lo9 cm-3. They also observed line radiation (e.g., 388.9 nm) at electron impact energies well below the expected threshold, which they attributed to dissociative recombination of molecular ions formed along the direction of the fast metastable helium beam. It is postulated that electrons interacting with these molecular ions in the interaction region can directly produce excited states that can subsequently radiate. A correction of their data for these effects is made but the interpretation is difficult.

Rall et al. (1989) reported absolute line excitation cross sections for five lines, absolute apparent level excitation cross sections for seven levels and absolute integral electron-impact excitation cross sections for three levels out of the 2’s level of helium in the electron impact energy range of 3- 16 eV. Their apparatus


consisted of a hollow-cathode discharge source, which produced a mixed beam of ground and metastable helium atoms that was crossed by an electron beam at the interaction region. Line radiation generated in the interaction region was observed by a photomultiplier and appropriate interference filters. They estimated that the metastable density at the interaction region was about 5 X

10’ cm-’, which comprised about .003% of the total density. The small metastable fraction restricted their maximum usable electron impact energy to about 16 eV, somewhat below the threshold for ground state excitation. They measured relative line excitation cross sections for the 3’s + 2’P (706.5 nm), 4’s + 2’P (471.3 nm), 3’P 2’s (388.9 nm), 3’0 + 2’P (587.6 nm), 4’0 + 2’P (447.2 nm), 5’0 + 2’0 (402.6 nm) and 6’0 + 2’P (382.0 nm) transitions. To ensure that the relative cross sections were on the same scale, they measured the relative optical efficiency of their system by observing the optical signal for the same lines excited from the ground level and using known line excitation cross sections from the ground level. The relative line excitation cross sections were placed on an absolute scale by using a laser induced fluorescence technique where the 388.9 nm line of the 2’s + 3’P transition was laser pumped. Curves were given for the absolute line excitation cross sections for the 3 ’ 0 + 2’P, 3 ) s + 2’P, 3)P + 2’S, 4’0 + 2’P and 4’s + 2’P transitions from threshold to 16 eV. Using known branching ratios, absolute apparent level cross sections for the 3’S, 3’P, 3’0, 4’s and 4’0 levels are given at 6, 10 and 16 eV and for the 5 ’ 0 and 6)D levels at 10 eV. Finally, by correcting for cascade contributions, they calculated the integral electron impact excitation level cross section for exciting the 3’P level from the 2’s level at 4.5 eV ( Q = 3.1 x cm*), 6 eV (Q = 3.0 X 10-l6 cm2), 10 eV (Q = 2.1 x cm2), and 16 eV ( Q = 1.7 X 1 O - l 6 cm*). All these cross sections are several orders of magnitude larger than the corresponding singlet-to-singlet excitation cross sections from the ground state. It was also observed that the integral electron-impact cross section from the metastable level to the 3’P level was smaller than those from the 3’s and 3’0 levels. This result is contrary to results obtained for corresponding singlet excitations from the ground state.

Mityureva and Penkin (1975) reported a relative line excitation function for the 388.9 nm line excited from an unknown mixture of 2IS and 2 ) s levels of metastable helium as well as estimates of absolute cross sections at the peak of the line excitation functions for the strongest lines of helium. A more complete reporting of this work is given in Mityureva and Penkin (1989). Their apparatus consisted of two parallel, interconnected tubes with a plasma metastable source in one tube and an electron beam in the other tube. Metastables generated in the plasma source drifted into the interaction region through the interconnecting tube, and radiation was observed from this region. Charged particles from the plasma region were prevented from reaching the interaction region by a system of electrodes. Both tubes were filled with helium at a pressure of 0.1 Torr (density of about 3 X I O l 5 cm-)). Using an optical absorption technique, they found


TABLE XI11

EXCITATION OF THE 2's LEVEL OF HELIUM TO HIGHER LEVELS (10-l6 chi* UNITS)

APPARENT LEVEL EXCITATION CROSS SECTIONS FOR ELECTRON-IMPACT

Peak* of Peak* of Excitation Excitation

Level 6 e V a 1 0 e V 16eVa Curveb Curvec

33s 9.5 5.6 2.8 3'P 3.8 3.0 2.3 270 (- 5 eV) 7 0 ) 33D 13 9 .4 5.8 1W3) 43s 1.5 1.5 1 .o 4)D 1.6 I .5 I .2 60 (- 5 eV) 7(4) S D .24 6'D . 1 1

'Rail et al. (1989). bMityureva and Penkin (1989). [Gostev et al. (1980b). *Approximate energy of peak is given in parentheses.

the metastable density in the interaction region to be about 10" cm-3 with the z3S level about five times as populated as the 2IS level. Mityureva and Penkin (1989) present relative line excitation functions in the electron-impact energy range of threshold to about 12 eV for triplet transitions 23S + 33P (388.9 nm) and 23P + 43D (447.1 nm) and singlet transitions 2 ' s + 3IP (501.6 nm) and 2IP + 3ID (667.8 nm). They give absolute cross sections at the peak of these line excitation functions so they can be placed on an absolute scale. Then using known branching ratios, they obtain the apparent level excitation cross sections for the 33P, 3 'P, 3 ' 0 and 43D levels.

There seems to be a great deal of variation between the results reported by the three groups discussed previously. Table XI11 gives the apparent level excitation cross section results for electron impact excitation of the 2's level of helium to higher levels obtained by these three groups. For example, for the peak apparent level excitation cross section of the 3'P level, Rall er al. (1989), Gostev et al. (1980b) and Mityureva and Penkin (1989) report values of 3.8 X cm2, 7 x 10-l6 cm2 and 2.7 x 10-l4 cm2, respectively. In general, Gostev etal . and Mityureva and Penkin report larger cross sections than Rall et al., sometimes by nearly two orders of magnitude. The reasons for these discrepancies are not clear.

2 . Metastable Neon

Mityureva and Penkin ( 1 973 , using the experimental arrangement discussed previously for their work on helium, reported a relative line excitation function for the 640.2 nm line of neon, with the initial level being an unknown mixture


of 3P0 and 3P2 metastable levels. Again they report estimates of the cross sections at the peak of the line excitation function and find it to be on the order of 10 - Is- 10 - I4 cm2.

3. Metastable Argon

Mityureva et al. (1989a) described a new apparatus for obtaining line excitation cross sections and apparent level excitation cross sections from metastable levels, which they use for argon, krypton and xenon. Their new apparatus consists of a single, pulsed electron beam in a chamber uniformly filled with gas at 40- 160 microns. The first pulse (-20 psec long at an energy of 20-25 eV) creates the metastables while the second pulse (-5 psec long at a variable energy of 1-40 eV) excites the metastables. The time delay between the first and second pulses was variable but was typically 10-14 psec. The line radiation produced by the second pulse was observed by a gated photomultiplier system. Metastable densities present during the second pulse were determined by an optical absorption technique. Mityureva et al. (1989b) gave line excitation functions in the electron-impact energy range from threshold to about 14 eV for 14 lines produced in the electronic excitation or argon atoms from the 3p54s metastable levels to the 3p54p levels. They found that the population of the 3P2 metastable level exceeds the population of the remaining 4s levels by more than an order of magnitude, so that it was assumed that the initial level was 3P2. Using known branching ratios, they calculated the apparent level excitation cross sections from the 3P2 level to all of the 2pl through 2p9 (Paschen notation) levels of the 3p54p configuration. At the peak of the apparent level excitation cross section functions, the cross sections are in the range of 10-l4 cm2.

4 . Metastable Krypton

Using the single, pulsed electron beam method described earlier for argon, Mit- yureva et al. (1989~) presented line excitation cross sections for 10 spectral lines originating from 2p2-2pg (Paschen notation) levels of the 4ps5p configuration of krypton. Using known branching ratios, they gave the apparent level excitation cross sections for the 2p2 through 2p9 levels in the electron impact energy range of threshold to about 12 eV. Maximum cross sections were found to be about 10-15-10-14 cm2.

5. Metastable Xenon

Using techniques described earlier for studying metastable argon and krypton, Mituireva et al. (1991) reported apparent level excitation cross sections for exciting the 2pl through 2ps levels of the 5ps6p configuration of xenon from the


3P2 level of the 5p56s configuration. Maximum cross sections were found to be in the range of 10-16-10-14 cm2.

6. Excited Sodium

Stumpf and Gallager (1985) presented apparent level excitation and integral electron impact excitation cross sections for exciting the 3 0 level from the excited 32P,,2 level of sodium. The excited states were produced by crossing a sodium beam with a circularly polarized laser beam tuned to the F' = 2 + F = 3 hyperfine transition of the 32S,,2 + 321$2 line at 589 nm. The laser pumping scheme prepares the excited sodium atoms in the 32P,,2 ( F = 3, M F = 3) hyperfine sublevel. This is a pure spin and angular momentum state with ML = 1 and M, = 1/2 (M, = 3/2). These excited species were excited to the 3 0 level by an electron beam, coaxial with the laser beam, and subsequent fluorescent radiation at 819 nm was observed as a function of electron impact energy. By properly incorporating effects of the polarized radiation anisotropy and by normalizing to the Born approximation at high energies, they obtained absolute apparent level excitation cross sections Q, (3P M L = 1, M, = 1/2 4 3 0 ) for excitation from the 3P(ML = 1 , M, = 1/2) state to the 3 0 levels. This cross section represents excitation to the 3 0 level from a pure M, = 1, Ms = 1 /2 initial state and not from a statistical distribution of M, states. They also presented integral electron- impact excitation cross sections for a statistical distribution of M L states by estimating cascade contribution and performing the proper averaging. The integral electron-impact excitation cross section from the 33& level is considerably larger than those from the ground state to the same levels.

D. DIFFERENTIAL CROSS SECTIONS

Differential (in angle) scattering cross section measurements are more difficult than integral or total scattering cross section measurements. This is partly due to the requirement of more sophisticated instrumentation and techniques but mainly to the reduced signal levels. A typical solid angle, over which the signal is collected in a atomic-beam, electron-beam scattering experiment, is about ste- radian instead of 47r, which means roughly a loss of four orders of magnitude in the signal compared to integral measurements. In the case of excited atoms, one usually has to deal with much smaller target densities than in the case of ground state atoms, which results in additional reduction in the scattering signal. This is the reason why so few DCS measurements for excited atoms have been carried out so far. Electron scattering measurements have been reported for laser-excited Ba (. . . 6s6p'P) and for discharge-excited He Q 3 S ) . The atomic beam recoil technique was applied to laser-excited Na (325,2) atoms. We will describe these measurements here in some detail.


Register et al. (1978) utilized a tunable, single-mode, cw, dye laser to excite 13sBa atoms from the (. . . 6s2 IS,) ground level to the (. . . 6s6p 'PI) excited level. Subsequent cascade processes (or collisions of atoms in the beam) populated the lower lying 3P, ID and 3D levels to such a degree that they were also able to observe electron scattering processes associated with these species. The measurements were carried out under high atomic-beam-density conditions so that radiation trapping washed out the effect of polarization in the laser beam and no alignment or orientation was present either in the ground or the excited levels. An energy-loss spectrum for the mixture of these excited and ground state atoms is shown in Fig. 9. The ground state species include also isotopes other than 138. In Fig. 10, the observed processes are summarized in a matrix form. The diagonal squares correspond to elastic scattering by the various species. Individual

LASER ON

X 5 0 0

. x l .

I

1

'S I I lo I

I

1 1

-2 -1 0 1 2 E N R C Y LOSS (eV)

FIG. 9. Energy-loss spectra of Ba obtained at 30 eV impact energy and 5" scattering angle. Top spectrum was obtained from electron scattering by ground state Ba atoms and contains the elastic and inelastic scattering features. The lower spectrum was obtained from electron scattering by a beam containing ground (IS), laser-excited (. . . 6s6p'Pl) and cascade (or collision) populated ID, 'D and 'P Ba species. In addition to the features appearing in the top spectrum, a large number of features appear that correspond to excited level transitions and superelastic scattering. (From Register et al., 1978).


FIG. 10. Summary of various electron impact processes observed by Register er al. (1978). The diagonal squares correspond to elastic scatterings, squares with numbers above this diagonal represent observed inelastic processes and below this diagonal superelastic processes. The number desig- nations are those of Register et a / . (1978).


TABLE XIV SUMMARY OF DIFFERENTIAL INELASTIC SCATTERING CROSS SECTIONS FOR EXCITED BA ATOMS

(FROM REGISTER E T A L . , 1978; SEE TEXT FOR EXPLANATION)

DCS cm2/sr)

Eo = 30eV %, = 100 eV

Transition Loss (eV) No. 5" 10" 15" 20" 5" 15" Energy Peak

6s6p 'PI +

6s' ' S o 6s5d ID2

6s5d ID, 5d' ID2

5d6p 'Dl 5d' 'Pa 6s7s IS, 6s7s IS,, 6 ~ 6 d ID2 6 ~ 7 ~ 'PI 6 ~ 7 d ID2 6 ~ 8 d 'Dl

6~5d ID2 +

65' IS,, 6s6p 'PI 5d6p IF3

- 2.240 - 1.098 - 0.828

0.620

0.725 1.003 1.259 1.508 1.794 2.400 2.539

- 1.412 0.828 1.912

I 91.3 3 1.4 4 4.6

1 1 43.0

12 11.0 14 I .4 17 44.7 20 69.3 23 4.6 28 21.9 29 -

2 1.6 13 2.7 24 50.6

11.8 1.4 0.70 36.0 0.43 0.12 - 0.70 0.08 0.12 1.7 0.08

0.29 - -

5.5 0.77 0.57 12.7 0.57

0.93 0.14 0.06 2.9 - 0.24 - - 0.25 0.07 5.9 - 0.28 12.0 -

31.0 - - - -

- - - - - 2.5 0.37 0.37 9.0 - 2.0 0.57 0.40 12.0 0.10

0.46 0.06 0.08 0.30 0.08 0.42 0.05 0.07 1.00 0.05 1.9 0.07 0.06 13.3 -

elastic scattering processes could not be determined in these experiments. Below and above this diagonal, the squares correspond to superelastic and inelastic scattering processes, respectively. Observed processes are indicated by designation numbers in the appropriate squares. The measured DCS results for excited species are given in Table XIV. The DCS measurements were carried out on the 138 isotopes selected by the laser pumping from the naturally occurring isotopic mixture of Ba atoms. However, the DCS are the same for all isotopes and hyperfine levels if the effect of nuclear structure and nuclear spin (Percival-Seaton nuclear hypothesis; Percival and Seaton, 1958) can be disregarded. Even if some effects due to nuclear structure or spin occur, these are completely negligible compared to the experimental errors (-50%). It was found that the excited-state to excited-state cross sections were similar in magnitude to those associated with ground to excited-state processes and the dominant ones were those associated with A J = k 1 transitions.

Muller-Fiedler et af. (1984) reported DCSs for electron impact excitation of He (23S) metastable atoms to the 23P, 33S, 33P, 3'0 and to the sum of the n = 4 triplet states at 15, 20 and 30 eV residual energies between 10" and 40" scattering angles. They utilized a discharge at the region where the target He beam was


formed to prepare a mixture of ground and metastable atoms (population ration los to 1) and used conventional differential electron scattering techniques for generating energy-loss spectra. Intensities measured in these spectra were converted to cross sections by normalization to elastic scattering by ground state atoms. (The small perturbation from elastic scattering by excited atoms was disregarded.) The results are summarized in Table XV and shown in comparison with ground state cross sections in Fig. 11. The method of preparation in these measurements precluded orientation or alignment in the 23S level, therefore, the DCS results correspond to isotropic and unspecified distributions in the initial and final magnetic sublevels, respectively. At 15, 20 and 30 eV residual energies the cross section values obtained for exciting the 23P, 33S, 33P and 33D levels from the 2’s level were found to be strongly forward peaking and much larger than those associated with the excitation of these same levels from the ground state.

With suitable kinematic analysis, the atomic beam recoil technique can also

TABLE XV

DATA OF MUELLER-FIEDLER Er A L . , (1984) FROM K. JUNG (1991) DIFFERENTIAL CROSS SECTIONS FOR ELECTRON IMPACT OF HE(z3S) (UNITS ARE 7T At SR-I ) ;

~~~ ~~ ~

Angle 33s 33P 3)D n = 4

Detection Energy = 15 eV: 10 594 10.2 4.65 32.3 17.8 15 134 2.40 2.49 15.1 6.69 20 39.3 1 . 1 1 2.42 6.67 4.05 25 11.7 0.79 1.94 2.10 1.51 30 4.07 0194 0.54 I .35 1.62 35 2.28 40 1.60 Detection Energy = 20 eV: 10 314 5.00 5.26 24.7 9.74 15 86.1 I .32 1.99 10.9 5.46 20 26.5 1.02 2.27 4.18 1.67 25 7.98 0.62 0.79 1.69 0.77 30 2.86 0.40 0.60 0.36 0.49 35 1.48 40 0.95 Detection Energy = 30 eV: 10 318 5.35 2.68 27.2 9.72 15 64.9 0.81 3.47 8.51 4.88 20 11.0 0.76 2.03 2.05 1.56 25 3.99 1.01 0.55 0.92 1.13 30 1.87 1.33 0.31 0.48 0.56 35 1.20 40 0.77

96

1 o2

10’

2 1 oo 0,

n

v v) u

lo-’

1 o-:

S . Trajmar and J.C. Nickel

I

2%

3’D

3%

3%

2lP

0 10 20 30 40

Angle(Degrees)

FIG. 1 1 . Inelastic DCS for metastable (2’s) He (from Muller-Fiedler et al., 1984). The final level for the excitation processes are indicated. For the purpose of comparison, DCSs for the ground (IS) --* 2IP excitation are also shown.

be used to obtain elastic and inelastic differential scattering cross sections (including spin-exchange and spin-flip cross sections). This approach for ground state atoms was described Rubin et al. (1969), Collins et al. (1971) and more recently by Jaduszliwer et al. (1984), Vuskovic et al. (1989a) and Jiang et al. (1990a). Application of the recoil method to Na (3*&, F = 3 ) excited atoms for the purpose of differential cross section measurements was made by the New York University (NYU) group. The principles and foundations for analyzing atomic recoil measurements involving both ground and excited initial atomic targets in terms of differential cross sections was described in detail by Vuskovic et al. (1989a). We briefly summarize these works now.

Zuo et al. (1990) determined elastic DSC for laser-excited sodium. In these measurements, the magnetic field in the interaction region was kept below


G . Therefore, the nuclear and electron magnetic momenta were coupled, and the magnetic sublevels were nearly degenerate. A hexapole magnet eliminated the 3 3 , , , ( F = 1) and 32S,,2 ( F = 2, M, = - 2) species from the original sodium beam. For the excitation, a traveling-wave laser field (with laser beam perpendicular to the atomic beam) was utilized with both cr and 7~ pumping. The laser frequency was tuned to the 32S,,, ( F = 2) + 32P,,, ( F = 3) transition. The composition of the atomic beam for the various pumping cases is listed in Table XVI. The electron beam was perpendicular to both the atom and laser beams and was square-wave modulated. The excited atoms were displaced from the original atom beam direction by the laser excitations (first recoil) and then by electron collisions (second recoil). The doubly recoiled beam consisted of concentric rings in a plane perpendicular to its propagation direction corresponding to elastic and various inelastic scattering processes. The atom detector could be moved in this plane, and measurements on the recoiled beam components corresponding to elastic and inelastic electron scattering associated with various polar (0) and aximuthal (4) scattering angles could be made. The measurements were, however, restricted to a line parallel to the incoming electron momentum, displaced from that by the photon recoil and only to the section along this line that corre- sponded to in-plane elastic scattering. Restriction of the detector to this line segment yielded elastic differential scattering cross sections with azimuthal scattering angle fixed at 0" and 180", undistinguished (the reported data, obtained by this technique, are the average of these two DCSs). Absolute elastic DCS for oriented ((T+ or (T- pumping) and aligned (r-pumping) excited Na (3*P,,,, F = 3) atoms were obtained at 3 eV impact energy in the 0 = 25" to 40" angular range. To obtain these absolute cross sections, the knowledge of the relative density of the excited species (with respect to ground species) was required. This fraction was obtained from the magnitude of the photon recoil. In addition, the velocity distribution of the atoms, the ground state differential elastic and (for minor correction purposes) the total electron scattering cross sections for the ground state species and for the excited species were needed. The velocity distribution was determined from measuring the position for foreward inelastic electron scattering corresponding to the 32P (32p3,2 and 32P; j2 undistinguished) exci-

TABLE XVI

EXPERIMENTS OF Zuo Er AL., (1990) COMPOSITION OF THE NA ATOMIC BEAM FOR VARIOUS PUMPING SCHEMES IN THE

Pumping Ground Species Excited Species

"The hexapole magnet eliminated the MF = - 2 species and the small population in the MF = 2 sublevel was neglected.


tation, ground state elastic DCS was obtained in the same paper and the total cross sections were available from earlier measurements (Kasdan et al . , 1973; Jaduszliwer ef al . , 1980). The elastic DCS for the excited atom were found to be about a factor of four and ten larger for cr- and 7~ pumping, respectively, and more foreward peaked than the ground level cross sections. It should be noted that the elastic scattering DCS associated with the 32P,,2 aligned or oriented Na atoms also include the experimentally undistinguished small contribution of high-angle superelastic scattering to the (32q,2, F = 1 and 2) levels. Elastic DCS measurements were also reported by Jiang et al. (1991a, 1991b) at E, = 2 eV, 8 = 36" to 44" and 4 = 0" and 180" (undistinguishable) for fully oriented Na 32Q,2 (F = 3, M, = 3) atoms. In these cases the excitation of Na was achieved with a standing-wave laser beam (utilizing a mirror to reflect the laser beam) and the single (electron collision) recoil approach was utilized. The detector was moved along a line parallel with the electron momentum (no photon recoil). These DCS values are about 2.5 times larger than those obtained by Zuo et al. (1990) at E , = 3 eV.

The NYU group also reported superelastic DCS and partial integral cross sections on laser-excited Na. Vuskovic er al. (1989b) carried out superelastic scattering cross section measurements on oriented Na 32€$2 (F = 3) atoms at E, = 3 eV to 20 eV in the 8 = 0" to 30" range, utilizing a standing-wave laser field excitation. Part of these superelastic results has also been reported by Jiang et al. (1990b) at Eo = 3 eV in the 8 = 1" to 30" range. In these measurements, the detector was moved along a line parallel to the electron momentum. These cross sections show a forward peaking character that becomes more pronounced with increasing impact energy. The DCS have a value in the range of 10-l6 to 10 - l 4 cm2/sr.

Jiang et al. (1991b) obtained inelastic DCS and partial integral cross sections for the Na 32P3,, ( F = 3, M, = 3) + 42P;,2 process at E, = 2 eV, in the 8 = 3" to 30" angular range (averaged over 4 = 0" and 180"). The value of 6.2 x cm2 was obtained by them for the partial integral cross section.

Acknowledgments

The authors wish to express their gratitutde to D. C. Cartwright, G. Csanak, T. Gay, P. Hammond, H. Hotop, A. D. Khakaev, K. Lam, A. C. H. Smith and L. Vuskovic for valuable discussions and to K. Jung for supplying numerical data and their thanks to Yvette De Freece-Gibson for preparing the manuscript. Support by a joint UCR-Los Alamos CALCOR grant, by NSF and NATO is greatfully acknowledged.


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tributed Papers,” post-deadline paper. Stumpf, B.. and Gallagher, A. (1985). Phys. Rev. A 32, 3344. Svanberg, S . (1972). Phvsica Scripra 5, 73.

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Tiffanny, W. B. (1975). Modern Utilization of Infared Technology Civilian and Military 62, 153. Trajmar, S., and Register, D. F., (1984). In “Electron Molecule Collisions” (I. Shimamura and

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Physics).

Phys. 69,2978.


ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS. VOL. 30

THE DISSOCIATNE IONIZATION OF SIMPLE MOLECULES BY FAST IONS COLIN J . LATIMER School of Maihemaiics and Physics The Queen's University of Belfast Belfast, Northern Ireland

1. Introduction . . . . . . . . . . . . . . . . . . . . . . 11. The Dissociative Ionization Process . . . . . . . . . . . .

A. Potential Energy Curves and Collision Energetics . . . . . B. The Angular Distribution of Fragments . . . . . . . . .

111. Energy Distributions of Fragment Ions . . . . . . . . . . . A. Hydrogen and Deuterium . . . . . . . . . . . . . . . B. Oxygen. . . . . . . . . . . . . . . . . . . . . . C. Nitrogen. . . . . . . . . . . . . . . . . . . . . .

IV. Energy Distributions of Fragment Ion Pairs: Coulomb Explosions A. Hydrogen and Deuterium . . . . . . . . . . . . . . . B. Nitrogen Target: Quasibound States of Nj' . . . . . . . . C. Multicharged Fragment Pairs . . . . . . . . . . . . .

V. Angular Distributions of Fragment Ions: Orientated Molecules. . A. Hydrogen and Deuterium . . . . . . . . . . . . . . . B. Other Molecules. . . . . . . . . . . . . . . . . . .

VI. Partial Dissociative Ionization Cross Sections . . . . . . . . A. Charge and Mass Analysis of Fragment Ions . . . . . . . B. Cross Sections with State Identification . . . . . . . . . C. Double Capture in Hydrogen. . . . . . . . . . . . . . D. Energy Loss Spectrometry . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. . . . . . 105 , . . . . . 107 . . . . . . 107 . . . . . . 110 . . . . , , 112 . . . . . . 113 . . . . . . 118 , . . . . . 120 . . . . . . 121 . . . . . . 121 . . . . . . 125 . . . . . . 126 . _ . . _ . 129 . . . . . . 129 . _ . . . . 132 _ _ . _ _ _ 132 . . . . . . 133 . . . . . . 133 . . . . . . 134 . . . . . . 136 . . . . . . 136

I. Introduction

The process of dissociative ionization in heavy particle collisions has until recently been the subject of relatively little study despite its fundamental nature and its significance in determining the ionization, thermal balance and chemistry in interstellar clouds, planetary atmospheres and comet tails (Huntress, 1974; Clarke et al . , 1982; Neugenbauer, 1988; Ajello et al . , 1991). Therefore many important features of even the most basic processes involving hydrogen molecules are still not completely understood.

In contrast the intimately related process, dissociative ionization by electrons, has been extensively studied over the years (Lozier, 1930; Hagstrum and Tate,

Cvpyrighl 0 1993 by Academic Press. Inc. All righa of reprvduclivn in any form reserved.

ISBN 0-12-003830-7 I05

106 Colin J . Larimer

1941; Dunn and Kieffer, 1963; Crowe and McConkey, 1973; Kollman, 1978; Burrows el a f . , 1980; Landau et af., 1981; Cho et a f . , 1986). This field has been the subject of several relatively recent reviews (Teubner, 1985; Mark, 1985; Compton and Bardsley, 1984). In addition dissociative photoionization experiments have been performed by Doolittle et a f . , 1968; Fryar and Browning, 1979; Gardner and Sampson, 1975; Strathdee and Browning, 1979; Dujardin et a f . , 1987; Kossmann et a f . , 1989 and Wu et a f . , 1990. Theoretical studies of the dissociative photoionization process have been carried out mainly by Dalgarno and his colleagues (Ford et a f . , 1975; Kirby et a f . , 1979, 1981). Much recent work has been stimulated by the fact that, in both the electron and photon impact experiments, mass analysis of the product ions coupled with studies of their energy and angular distribution have shown that the fragment protons produced in hydrogen exhibit many features that cannot be accounted for by simple direct one-electron excitation to repulsive states of Hf . Rather they arise via-two- electron excitation processes to autoionizing states of H 2 , although it is not easy to identify all the states involved or the mechanism by which the H+ fragment ions are produced (Kirby et af., 1981; Guberman, 1983). Studies of another two- electron excitation process, double ionization, are also receiving a great deal of attention currently, largely because it is a sensitive probe for electron correlation effects and subtle collision mechanisms. Correlation effects are of greatest importance for target electrons moving in a field of low effective nuclear charge, since in this case the mutual electronic interaction is largest. Thus the most important systems for study are H - , He and also H2 where double ionization to form H i + will necessarily lead to fragmentation (Andersen, 1988; Kossmann et a f . , 1989; Edwards er a f . , 1990).

In recent years three main groups have commenced detailed experimental investigations of different facets of the dissociative ionization of simple molecules in heavy particle collisions. Latimer and coworkers in Belfast have concentrated mainly on studying the energy and angular distributions of fragment ions and ion pairs produced in simple diatomic gases (especially H,) by 3-30 keV ion beams. The only previous experimental work on these processes, which has been limited simply to a charge and mass analysis of the product ions to provide partial cross sections, indicates that within this energy range fragment ions and ion pairs arise predominantly through the electron capture processes (Browning et al., 1969; Afrosimov et al . , 1969).

x+ + B C + A + B + + C + x + B + + C + + e

A series of similar but complimentary experiments has been performed at much higher energies 0.2-4.0 MeV by Edwards, Wood and coworkers in Geor- gia. In this energy region pure ionization processes normally dominate

A+ + B C + A + + B + + C + e + A + + B + + C + + 2e

THE DISSOCIATIVE IONIZATION OF SIMPLE MOLECULES BY FAST IONS 107

The group in Kansas (led by Cocke) have concentrated on studying the dissociative ionization of deuterium by various multicharged ions Aq+, where 1 6

q 6 9 over a wide range of energies. At high energies (8-20 MeV) pure ionization processes again dominate, capture to form A(4- ’ )+ is two or three orders of magnitude smaller and double capture to form A‘4-2)+ is immeasurably small. However at low energies, 0.5- 1 .O keV, both single and double capture processes are important.

It is perhaps worth noting that virtually all investigations to date have, mainly for experimental convenience, studied the dissociative ionization of homonuclear diatomic molecules by beams of positive ions. Clearly future experiments will have to address the special problems of heteronuclear and polyatomic molecules and other heavy particle beams.

II. The Dissociative Ionization Process

A. POTENTIAL ENERGY CURVES AND COLLISION ENERGETICS

It has been known for a long time (Condon, 1930) that the kinetic energies of fragment ions formed through dissociative ionization depend upon the detailed nature of the potential energy curves of the states involved in the collision. The dissociative ionization of simple homonuclear diatomic molecules, in particular the hydrogen molecule, has dominated this field of study since the early days, irrespective of the type of incident beam. Indeed it is interesting to note that over 60 years ago Bleakney (1930), in the introduction to a paper on the dissociative ionization of hydrogen by electrons, said, “The ions produced by electron impact in hydrogen has been studied by the method of positive ray analysis so many times that it might, at first sight, seem useless to try to make much more progress in this direction.” Such studies are still in progress today! A diagram showing some potential energy curves for some selected bound and dissociating states of H2, H’iand H+; is shown in Fig. 1.

Now, as has already been pointed out in recent volumes in this series, comprehensive theoretical investigations of ion molecule collisions are extremely rare (Pollack and Hahn, 1986; Kimura and Lane, 1989), owing to difficulties in (a) obtaining accurate electronic wave functions for the appropriate ion- molecule system and (b) the complexity of performing calculations of all the collision observables. Therefore discussions of dissociative ionization processes involving ion beams are normally conducted in simple terms involving Franck- Condon excitations and the reflection approximation, even in situations where their validity is somewhat marginal. Recently however Sidis, Gauyacq and coworkers (Sidis and Courbin, 1987; Gauyacq and Sidis, 1989; Aguillon er al., 1991) have made significant attempts to address the problem of dissociative

108 Colin J . Latimer

4 <I 7 I

V

t r

t [ I-

I . I

I I I 1

1 2 3 4 5 6 7 8 9 10

INTERNUCLEAR SEPARATION ( 8 1 FIG. 1. Potential energy curves for selected states of H, and H; (Sharp, 1971). Also shown are

the reflection approximation predictions of fragment proton energy spectra produced by Franck- Condon excitation (Wood er al . , 1977).


charge exchange at relatively low energies using a coupled wave packet method. Interesting phenomena have been revealed in as yet idealized systems.

Figure 1 shows a schematic representation of several dissociative ionization processes in hydrogen. The projectile collides with the H2 molecule in the XlC: ground state and excites the molecule vertically at fixed internuclear separation within the Franck-Condon region. When the transition is to a point above the asymptotic limit of the final state potential, the fragments separate, gaining the energy difference between the excitation energy and the asymptotic limit. The assumption that the nuclei of the molecule remain fixed stems from the comparison of characteristic molecular rotation and vibrational times (trot = lo -" secs, fv lb = lO-I4 secs) with typical collision times (tColr = 10-I5- lo-" secs, at keV energies for light projectiles). Moore and Doering (1969) and Dhuicq et al. (1985, 1986) have, in experimental studies of vibrational excitation in ion-molecule collisions, shown that the transition to non-Franck- Condon behaviour occurs around a collision velocity v = lo8 cm s-l (corresponding to a proton energy of 5 keV). Fournier et al. (1972) has shown that the Franck-Condon principle is obeyed in charge transfer collisions involving fast protons with energies down to 4 keV. A detailed theoretical description of all the necessary requirements, which can be quite stringent, for the valid use of the Franck-Condon principle can be found in a recent volume in this series (Sidis, 1989).

The transition probability R,c in a bound-free transition from a vibrational state v, to a continuum state of energy E = k 2 / p , where k is the wave number of the separating particles and p is the reduced mass, is given by

P , , E = l(v,lPtk(r)Ik)l2 (1)

where p J r ) is the dipole strength of the transition. In the Franck-Condon approximation pulk(r) is assumed to be slowly varying; and it is replaced by an average transition strength ~ , ~ ( r ) , giving

-

P " , C = [ a l 2 l ( v , l k ) l 2 (2)

a transition probability that is simply proportional to the Franck-Condon factor. An alternative approximation, the r centroid (Fraser, 1954; Nicholls and Stew- art, 1962) allows account to be taken of the variation of the dipole strength with internuclear distance when this is (unusually) known. Here ~ , ~ ( r ) is simply replaced by P,~ (T ) , where r is the r centroid given by f = (v,lrlk)/(v,lk). This simple approximation is rigorous when plk( r ) is a linear function of r. However, in the simple dissociation of molecules in their ground vibrational state, any such variation in p(r ) can usually be neglected over the small range of r covering the Franck-Condon region in this case (Le Rouzo, 1988).

A further simplification can be made by using another approximation, the reflection method, first introduced by Winans and Stueckelburg (1928) and later

-


investigated by Coolidge et al. (1936), Hagstrum and Tate (1941), Kieffer and Dunn (1967), McCulloh (1968) and Dunn (1968). In this approximation the un- bound wave function is replaced by an appropriately normalized &function (Coolidge et al., 1936; Buckingham, 1961; Tellinghuisen, 1985)

at the classical turning point r,, where E = V(r,)-V(W) is the energy above the dissociation asymptote. Therefore the probability densities in the initial state are "reflected" from the upper repulsive potential. The reflection method has been shown to be most appropriate and accurate (Tellinghuisen, 1985; Child, 1980; Dunn, 1968) when the final repulsive potential is sufficiently steep and when only small vibrational quantum numbers are involved in the initial state. This is equivalent to requiring that the de Broglie wavelength of the bound state wave function be much longer than the oscillation period of the continuum wave func- don near its turning point. These criteria normally are satisfied in simple direct dissociative ionization processes involving ground state molecules. The results of such calculations, which include a small contribution to account for thermal broadening, performed by Wood et al. (1977), in the case of hydrogen are shown in Figure 1. Conservation of momentum in this case requires that each fragment takes away half the available kinetic energy. Furthermore since the dissociation asymptotes can vary from state to state the resultant fragment proton energy distribution will consist of broad composite features.

B. THE ANGULAR DISTRIBUTION OF FRAGMENTS

When dissociation occurs in a time that is short compared with the period of molecular rotation, the fragment ion trajectory from a dissociating diatomic molecule will indicate the orientation of the target molecule in space. Thus the study of several accompanying processes such as electron capture or photoelectron emission with aligned or orientated molecules in a coincidence experiment is now feasible (Cheng et al., 1991; Eland, 1984). If the molecule is aligned parallel to the incident ion beam, the angular distribution of fragments will have maxima at 0" and 180", while a perpendicular orientation will give maxima at 90" and 270".

In addition yet more information on the potential energy states involved in the collision can be obtained from the angular distribution of the fragments, since the differential cross sections depend on the symmetries of the states involved. In the absence of a complete theoretical description of collisions involving dissociation, approximate selection rules for diatomic molecules, first proposed by Dunn (1962), are often used to provide a qualitative description of dissociative

THE DISSOCIATIVE IONIZATION OF SIMPLE MOLECULES BY FAST IONS 1 1 1

attachment excitation and ionization processes. He considered the two limiting cases described previously; namely when the target molecule has its internuclear axis either parallel or perpendicular to k,,, the momentum vector of the incident beam. In the case of attachment the symmetry axis is in the direction of k,. In the case of dissociative excitation the symmetry axis becomes the momentum exchange vector K = ko - k', where k' is the momentum transferred in the collision. For pure dissociative ionization, where the final state comprises both a molecular ion and an ejected electron, there are clearly additional complications, and the axis of symmetry is even less well defined. However, at high collision energies the ejected electrons are emitted preferentially in the momentum transfer direction (Ehrhardt et al . , 1980), and the symmetry axis can again be taken along K. Different selection rules apply to homonuclear and heteronuclear molecules and are tabulated by Dunn (1962). As an example, in the former case transitions from 2; to excited states 2$, Il,, or Ag would lead to fragments normal to the symmetry axis while transitions to 2; would give fragments parallel to the symmetry axis.

In general it can be shown (Dehmer and Dill, 1978; Zare, 1967) that the fragment angular distribution in the axial recoil approximation, which applies when the fragments have kinetic energies much larger than the rotational spacings, is given by the familiar expression

(4)

where (+ is the total cross section, p is an anisotropy parameter and in the dipole approximation ( e = 2), P,(cos 6 ) is given by 2(3 cos*O - 1). The parameter p, which depends on the nature of the molecular orbitals, can range from - 1 (giving a sin26 distribution with maxima at 0" and 180") and can provide information on the symmetry of the final states formed in the collision.

However, it should be noted that there are several situations where these ideal angular distributions will not be observed. They arise due to the following:

a. Rotational efect and predissociation. Clearly rotation of an excited molecule after excitation but before dissociation can destroy the initial orientations induced in the collision process, with the result that /3 is significantly reduced. Such a situation is most likely to arise when the final electronic state is a bound state that fragments through a radiationless transition to the repulsive portion of a lower-lying state. This predissociation process is considerably slower than direct dissociation, and the molecule may undergo several rotations before break- ing apart. It is possible however for the angular distribution of fragments to retain some anisotropy, as discussed in detail by Jonah (1971) and van Brunt (1974) in the case of photodissociation. An initial anisotropy cannot be completely abo- lished easily, because rotation cannot supply or remove angular momentum in any axis. It should be noted that, for predissociation, the angular distribution is characterized by the symmetries of the initially excited states and not the final

i (e) = (c /4r ) [ i + ppe(cOs e)]


dissociating state. However, these states are expected to interact most strongly with dissociating states of the same symmetry (van Brunt and Kieffer, 1974).

b. Momentum transfer effects. Peek (1964), Zare (1967) and van Brunt (1974) have shown that when small amounts of momentum are transferred in the collision process, the angular distributions can become quite eccentric although still possessing symmetry about 90". A model for parameterizing non-dipolar angular distributions has been proposed by van Brunt (1974) and van Brunt and Kieffer ( 1974) that allows for higher-order multipole corrections to the simple dipole-Born approximation. These terms dominate electron impact dissociation and presumably dissociative ionization, near thresholds involving large momentum transfer.

c. Two-electron processes. Edwards et al. (1991) have generalized the simple dipole-Born approximation to produce a function of the form

m(e) = a,( i + A cos2e + B C O S ~ ~ ) ( 5 )

where m,, is the value of the cross section at 90", which can be used to describe fragmentation through two electron processes such as double ionization, ionization plus excitation and double excitation of hydrogen by fast (>0.35 MeV/amu) electrons and protons. The coefficient B is zero for a single dipole projectile- electron interaction. Double excitation can then arise through a subsequent electron-electron interaction. However the c0s4e term is necessary if the projectile makes two dipole interactions one with each electron in separate events. In this way, assuming the final state can be identified by other means such as kinetic energy spectra, the intimate collision mechanism can be studied (Ezell et a l . , 1991).

III. Energy Distributions of Fragment Ions

The energy analysis of charged particles is readily achieved using magnetic, electrostatic or time of flight methods. These well-established techniques are not unique to the study of dissociative ionization and detailed descriptions of these methods can be found, for example, in the reviews of Berkowitz (1979) and Browning (1979) or the recent teatise of Scoles et al. (1988). However, several features unique to the dissociative ionization process are worthy of mention.

Thermal motion of the target gas will produce an unavoidable spread in fragment ion energies. The thermal broadening AE of the fragment energy distribution produced by dissociation taking place in a collision chamber has been given by Chantry and Schulz (1964, 1967) for a homonuclear molecule as

(6) AE = [ 1 l(mf/m,)kTE]


where m, and m, are the fragment and molecular masses, respectively, and E is the dissociation energy. At room temperature this width can, in the case of hydrogen for example, be as large as 0.7 eV for a fragment proton energy of 3.5 eV. This effect has been observed in the experiments of Landau et al. (1981) and Tronc et al. (1977). Substantial reductions in the broadening to insignificant levels can be achieved however by replacing the collision chamber with a molecular gas jet beam (Lindsay et d., 1987; Landau et al., 198 1) and by observing fragment ions at right angles to the direction of motion of the gas. A small fragment ion energy shift due to recoil of the target molecule is also possible (Schermann, 1979; McDaniel, 1989). In most cases such shifts are not large compared to apparatus resolution, and the fact that frequently the structures to be observed (see Figure 1) have widths greater than 1 eV mask this effect.

The simultaneous measurement of fragment ion energy and mass distributions is not trivial (Wood et al., 1976). The most common approach to this problem has been to omit a mass filter and to limit investigations to situations where this is not a problem. For example, in studies of the dissociative ionization of hydrogen, the H: ions produced all have energies below -1.0 eV (Lindsay et al., 1987), and therefore all ions observed with energies above this can safely be assumed to be fragment protons.

A. HYDROGEN AND DEUTERIUM

The energy spectra of fragment protons produced in the dissociative ionization of hydrogen has been studied with a large variety of projectiles over a wide energy range by Crooks and Rudd (1975) with 50-200 kV H + ions; Wood et al. (1977) and Edwards et al. (1977) with 0.5-4 MeV H +, He + , 0 + and H,' ions; Huber and Kahlert (1980) with 1-10 keV Krq+, 1 S q S 7; Lindsay er al. (1987) with 5-25 keV H' ions; Savage et al. (1990) with 15 keV He+ ions and Edwards et al. (1990) with 0.3-3.5 MeV H + ions. The corresponding process in deuterium have been investigated by Wood et al. (1977) with 0.5-4.0 MeV He+ ions and Cheng et al. (1989, 1990) with 8-20 MeV 08+ ions.

A schematic diagram of the apparatus used by Lindsay et al. (1987) and Sav- age et al. (1990) is shown in Fig. 2. A 3-30 keV ion beam was crossed at 90" by a low-pressure gas jet of target molecules in the ground vibrational state at the centre of a ramp voltage labelled region defined by two parallel circular plates. A weak radial draw-out field of a few volts per cm allowed the observation of all secondary ions produced without significantly perturbing their angular distribution. Fragment ions, appearing at a prescribed energy and angle 8, at right angles to the gas jet and in a plane parallel to, and midway between, the circular plates, were selected using two identical parallel plate analysers that viewed the interaction region from opposite directions to allow the detection of


FIG. 2. A schematic diagram of the apparatus used by Lindsay et al. (1987) and Savage el al. (1990) to measure the energy spectra of fragment ions and fragment ion pairs produced in 3-30 keV ion-molecule collisions.

fragment ion pairs produced with equal and opposite momenta in a Coulomb explosion. Fragment ions could be identified by time of flight. Coincident spectra (corresponding to proton-pair production plus random coincidences) and non- coincident spectra (corresponding to random coincidences and therefore total proton production) were accumulated simultaneously using a coincidence/mixer router/MCA system. The technique used ensured that neither sin 8 nor analyser transmission factors were necessary and that the thermal motion of the H, gas had a negligible effect.

An example of an energy spectrum taken at 75" and 15 keV incident proton energy is shown in Fig. 3 and can be seen to be diffuse, consisting of two broad composite features at around 6 and 9 eV. As can be seen, this can be successfully interpreted as arising from Franck-Condon transitions in accord with the reflection approximation to the four repulsive states 2p7ru, 2suU and H + H + as shown in Figure 1. This deconvolution process allows relative differential cross sections for the production of fragment protons from these states to be determined to within 10%.

It is apparent that, although the fit is not perfect in the region 2.5-5.0 eV, where an additional group or groups of protons arising from autoionizing states of H, (Gubermann, 1983) have been clearly observed with electrons (see, for example, Burrows et al . , 1980) and photons (Strathdee and Browning, 1979), it would appear that these states are relatively unimportant in keV H + charge transfer collisions. The results of Crooks and Rudd (1975) lead to the same conclusion.

However, as can be seen from Fig. 4, this conclusion is not valid in the case

THE DISSOCIATIVE IONIZATION OF SIMPLE MOLECULES BY FAST IONS 1 15

. ...

2 4 I I 1II (2 1 4 I I

FIG. 3. An energy spectrum of fragment protons produced at 75" in 15 keV H+-Hz collisions (Lindsay er al., 1987). The full curve is a theoretical fit to the data based on the reflection approximation (see Figure 1).

mrm w n a v I ~ V I AWLEX 75 KO9 L)(CIIO*= 15 KIV

t - 4 r

e 3 -2

c

x

c

0

c - i 2 M e u 1

FIG. 4. An energy spectrum of fragment protons produced in 15 keV He+-Hz collisions (Savage et al., 1990). Full curve: equivalent results for 15 keV H+-Hz collisions (Lindsay et al., 1987) that correspond to transitions to the 2pu, , 2p?r,, 2su, and H + H + states. Broken curve: proton kinetic energy distribution arising from dissociative autoionization of the '2: state of H1 (Kanfer and Sha- piro, 1983).


of 15 keV He+-H, collisions. The higher-energy group around 9 eV is similar to that obtained using 15 keV H+ projectiles and can readily be described in a similar manner as arising from excitation of 2pvU and the H + H + state. However, the lower-energy group differs considerably from that observed with keV H +

projectiles, having a broad maximum around 4 eV rather than 6 eV. It cannot therefore be simply described in terms of excitation to the 2pr , and 2sa, states. The only satisfactory explanation is that there is now a significant contribution arising from dissociative autoionization of the doubly excited states, which cross the Franck-Condon excitation region between 26-33 eV (Gubermann, 1983), giving rise to 2.5-5.0 eV fragments. An example (Kanfer and Shapiro, 1983) of a calculated distribution from the lowest ' 2 ; state of HT* that agrees well with the photoionization experiment of Strathdee and Browning (1979) is also shown in Figure 4.

At higher energies, where ionization processes dominate, Wood et al. (1977) and Edwards et al. (1977) in their studies with 0.5-4.0 MeV H +, He + , 0 + and H,' ions were also unable to fit the fragment proton energy spectra to the four states shown in Figure 1. They concluded that a group of ions at 3.27 eV arising from autoionizing states, contributing up to 20% to the total fragmentation, was needed to fit the data properly.

In more recent studies these workers (Edwards et al . , 1990) have reinvesti- gated the fragment proton spectra produced by electrons and protons in the range 350-3500 keV/amu. The positive ion fragments entered a hemispherical analyser positioned at a chosen angle relative to the beam direction. Time of flight and energy were measured simultaneously for each ion allowing fragments of different charge-to-mass ratio to be separately identified and analysed. An example of this work is shown in Fig. 5. The H + spectrum is a smoothly varying overlap of ions from several dissociation channels. Significant yield is again observed in the interesting 3 eV region. Kinetic energy distributions were calculated using the reflection approximation and slightly broadened to take account of the transmission function of the hemispherical analyser before fitting to the data. The expected contribution from autoionizing states was calculated semiclassically used the potential energy curves of Guberman (1983). The autoionizing lifetimes, as a function of the internuclear separation r, were obtained from the energy widths calculated by Tennyson and Noble (1 985) that have been confirmed by the recent R-matrix calculations of Shimamura et al. (1990). The doubly excited states In, '2 ; and '2: were considered as possible contributors. Although all gave distributions that were similar in shape, the best fit (curve A) was obtained for an equal mixture of the 'II, and l2; distributions. It is immediately apparent that these calculated distributions bear little resemblance to that of Kanfer and Shapiro (1983) as shown in Fig. 4. The semiclassical model


0 5 10 15

KINETIC ENERGY (eV) FIG. 5 . An energy spectrum of fragment protons produced at 90" in 0.5 MeV H+-H2 collisions

(Edwards et al . , 1990). The smooth line through the data is the result of a least squares fit of the predicted kinetic energy distributions from the states indicated. The A state distribution is a mean of several doubly excited, autoionizing states (see text).

predicts an H+ energy distribution with a maximum at zero energy while the more complete quantum mechanical theory of Kanfer and Shapiro gives rise to interference effects between the direct and discrete processes resulting in discrete energy spectra. To date no experiment has been able to distinguish unambigu- ously between these two models. Clearly further work is required.

Cheng et al. (1989, 1990) have measured the energies of fragment deuterons produced in collisions of fully stripped oxygen Os+ projectiles with deuterium at energies between 8-20 MeV. In addition they were also able to separate simultaneously the single capture (to form 0") and ionization channels by extracting the secondary target ions and detecting them with a two-dimensional position-sensitive detector incorporating time-of-flight analysis coupled with the coincidence detection of the charge analysed projectile product ions. The energy spectra of fragment deuterons were analysed in the manner of Edwards et al. (1990), as already described. The 2pu, and double ionization D + D + channels are the most important, with the latter dominating at the lowest energies. How- ever, no contribution from autoionizing states was observed. The projectile analysis shows that ionization dominates at all energies although the probability of capture processes increases rapidly with decreasing energy.

All these features are in accord with the only other data involving multicharged ions, that of Huber and Kahlert (1980) involving Krq+ ions ( 1 G q G 7) at much lower energies (<lo keV). Both the fast projectiles and slow secondary ions were energy analysed. Electron capture processes are now observed to be dominant. For low q values the H + ions are formed by all the processes shown


in Figure 1 . However, as q is increased, the H + yield results from the two electron capture process, necessarily producing H + H + fragments. Their energy analysis of the fragment ions produced a surprising result. In addition to the normal H + energy group -8-9 eV a high-energy contribution whose energy depended on the collision energy and detection angle was also observed. For the Kr+ + H2 system at a collision energy of 900 eV the energy distribution of these extra-fast H + fragments was centered around 35 eV. When the collision energy was increased to 100 eV the peak shifted to about 90 eV. Huber and Kahlert suggest that these ions arise from close collisions with significant momentum transfer in which a large amount of energy is likely to be transferred to the nuclei, causing rotational and vibrational excitation of the Hf ion formed. Ac- cording to Russek (1970) and Lange et af. (1977) this may lead to the vibrational-rotational induced dissociation of the Hf ion. Further observations of these high-energy fragments coupled with a more detailed analysis is clearly highly desirable.

B. OXYGEN

The energy spectra of fragment oxygen ions produced in the dissociative ionization of oxygen molecules has been studied by Steuer et af. (1977) with 1 MeV H +, He+ and 0 + ions, Bischof and Linder (1986) with 0.5-200 eV He + ions and Yousif et af. (1987) with 3-25 keV ions.

For many years it has been assumed, following a suggestion of Stebbings et al. (1963, that the dominantly large cross section for the dissociative ionization in He +-02 collisions from thermal energy to many tens of keV (Stebbings et af., 1963, 1965; Browning et af., 1969) arose from the near resonant process

(7)

followed by predissociation to one of a number of lower-lying dissociation asymptotes to form energetic 0' fragment ions. In recent years however the pre- ceeding investigations of the fragment ion energy spectra have shown that this explanation is too simple. Other processes are just as important.

The experimental set-up of Bishof and Linder (1986) is shown in Fig. 6. A mass and energy selected ion beam was crossed with a supersonic nozzle target beam. The detector, which could be rotated in the plane of the crossed beams, incorporated both an electrostatic energy analyser and a 90" magnetic analyser for mass analysis of the reaction products. The well-defined kinematical conditions of the crossed beam geometry allowed a detailed analysis of the data using Newton diagrams, which played an important role in the conclusions that could be drawn from the data. An example of their measured energy spectra is shown

He+(*SI12) + 02(X3Cg) + He(lS,) + Of(c4C;, v = 0) + 0.02 eV


Detector \

v-----r--- Ion beam

ton gun

- 0 1 2 3cm

FIG. 6. A schematic diagram of the crossed beam scattering apparatus of Bischof and Linder (1986) used to study the 0' and 0; ions produced in 0.5-200 eV He+-O? charge transfer collisions.

in Fig. 7 and can be seen to consist of four discrete, rather than diffuse, energy groups, which are a result of transitions to bound states that rapidly predissociate. With the aid of photoelectron-photoion coincidence spectroscopy data (Hayashi et al., 1986; Frasinski et af., 1985; Richard-Viard et af., 1985) it is possible to identify the two reaction mechanisms that lead to four groups of 0 +

products, (1) process (7) with predissociation of the "2; state of 0: to form O(3P) + O+(4S) and O('0) + O+(4S) fragments with 0' energies of 2.97 eV and 1.94 eV respectively, and (2) a slightly exothermic charge transfer process via the 111211, state of O:, which predissociates to form the two remaining groups. The work of Yousif et al. ( 1 987) at higher energies ( 2 3 keV) also shows that nonresonant channels are important. However, in this work the B22,;, v state predissociates to give 0.79 eV ( v S 4) and 0.05 eV ( v > 4) O+(4S")


He++ 02- He+O + 0' Elob (He*) = 6.5 eV

.'. y

' . . , 60'

30°

0 1 2 3 4 5

Kinetic energy of the O* product ions I r V l FIG. 7. Energy spectra of 0' fragment ions produced in 6.5 eV He+-Ol collisions at different

scattering angles (Bischof and Linder, 1986). Channels I and I1 arise from the predissociation of OT(c4X,;) to form O+(") + O('P) and O + ( 4 S ) + OOD) fragments. Channels 111 and V arise from dissociation of O:(IIIlrI,,) to from O+('D) + O('P) and O+(?P) + O('P) fragments.

fragments that, along with a broad feature at -4.6 eV due to direct dissociation via the repulsive lII, state of O;, are observed in addition to the resonant process (7). Further discussion of these reactions can be found in Section VI.

C. NITROGEN

The energy spectra of fragment nitrogen ions produced in the dissociative ionization of nitrogen molecules has been studied by Crooks and Rudd (1975) with 50-200 keV H+ ions, Edwards et al. (1988) with 1 MeV H' and He+ ions, and Yousif et al. (1990) with 5-25 keV H + ions.

All these experiments have produced similar energy spectra that, in addition, bear a remarkable resemblance to electron impact (Delanu and Stockdale, 1975)


and photoionization data taken with 80- 150 A photons (Sampson et al., 1987). Discrete features due to predissociating states are observed at 1 .O eV and 3.6 eV. The 1 eV group is thought to arise through predissociation of the N: (c%.:, v a 3) state (Govers et af., 1973), but the 3.6 eV group has not been unam- biguously identified, although it is possible that N: (D'II,) may be the state involved (Edwards et af., 1977).

IV. Energy Distributions of Fragment Ion Pairs: Coulomb Explosions

Studies in which both ionic fragments produced in the dissociative ionization of simple diatomic molecules are detected in coincidence can provide definitive information about the charge state and potential energy curves of the multicharged molecular ion produced in the collision. The two fragments explode apart under Coulomb repulsion and have equal and opposite momentum vectors in the laboratory frame of reference (neglecting thermal effects). Typically two identical energy analysers view the interaction region from opposite directions (see Fig. 2).

A. HYDROGEN A N D DEUTERIUM

The energy spectra of fragment protons produced in the dissociative double ionization of hydrogen have been studied by Yousif et al., 1987 with 5-30 keV H +

ions and Savage et af. (1990) with 15 keV He + projectiles. The corresponding process in deuterium has been investigated by Giese et af. (1988) with 500 eV and 1000 eV Ar5+ ions, and Latimer (1991) with 15 keV He+ ions.

These simple molecules provide a special situation, since only in hydrogenic cases is there a single exactly known (pure Coulomb) doubly charged repulsive curve for the molecular ion. Therefore the fragment proton energy distribution (see curve 4 in Fig. 1) is directly related to the ground state wave function of H,. An example of an energy spectrum taken at 90" with 15 keV.H+ ions (Yousif et a!., 1987) is shown as Fig. 8(a) and can be seen to consist of a single broad peak centered at 9.8 eV, which is clearly consistent with the Franck-Condon transitions from the ground state of H,. The transformation of this energy distribution using the reflection approximation to give the square of the ground state wave function is shown in Fig. 8(b). Theory and experiment are clearly in good accord although there is a small, as yet unexplained, displacement of about 0.08 a, towards a smaller equilibrium internuclear separation.


8 r I 1 1 1 I 1 I 1 8 . I

5 6 7 8 9 10 11 12 13 14 15 ENERGY l e v )


FIG. 9. An energy spectrum of proton pairs produced in 15 keV He+-H, collisions (Savage ef al., 1990). The full curve is the sum of direct transitions to the H + H + potential (dash-dot curve) and two-step transitions via the HT( Isu,) state, which is populated through the two-electron excited state H2(’Z,)(2pu.)* (dash curve).

The energy spectrum of fragment proton pairs, also taken at 90” but with 15 keV He+ ions (Fig. 9), is not a single broad peak at 9.8 eV in accordance with the preceding picture (Savage et al . , 1990). An additional group of ion pairs is observed, centered around an energy of 5.0 eV. Since this lower-energy group is discrete and found to be insensitive to projectile velocity, a simple breakdown of the Franck-Condon principle cannot provide an explanation of this interesting observation.

Now the study of double ionization in helium (isoelectronic with H2) is currently receiving a great deal of interest, largely because it is a sensitive probe of two electron correlation effects. At high velocities where the first Born approximation should be very accurate and predict cross sections independent of charge sign, cross sections for double ionization by equivelocity electrons and antipro- tons are much larger than for protons. Both classical (Olson, 1987) and quantum mechanical (Reading and Ford, 1987) theories can reproduce these observations and indicate that “there is apparently a rather subtle interplay during the collision between projectile-electron and electron-electron interactions. If electron correlations are ignored . . . , the effect is completely missed. But equally important, if the projectile-target interaction is treated in lowest order (as in the first Born approximation), the effect also disappears, even though electron-electron correlation is fully included” (Ford 1989). However, even more recently Peder- sen and Hvelplund (1989) have, in the same velocity regime, surprisingly ob-


served much smaller, nearly negligible, differences between electrons and protons in cross sections for another two-electron process: the double excitation of helium. Product angular distributions are however different. These effects are currently inexplicable but again, according to Ford (1989), “Undoubtedly electron correlation plays a role but quite what this is has yet to be uncovered.”

Several workers (Andersen et al. , 1987; McGuire, 1987) have explained the observed charge effects in double ionization as arising from interference between different mechanisms. In the first Born approximation, double ionization normally arises through a shake-off mechanism that involves a single-step (SS) interaction between the projectile and a target electron; the second electron is then ejected during the subsequent rearrangement of the target electrons. However, double ionization can also arise through the second Born or two-step (TS) processes. When the projectile interacts sequentially with each of the two target electrons the process is known as TS-2, and such a mechanism is expected to dominate at lower projectile velocities. When the projectile interacts with just one target electron, which then recoils and collides with a second electron, the process is called TS-1 and is important even at high velocities.

The work of Savage et al. (1990) is in the lower-velocity regime where TS-2 processes should be important and indeed provides a possible explanation for the -5 eV energy group. The H2+ (1 su,) state has the correct internuclear separation for an intermediate step. However, given the collision time scale these workers show that a satisfactory description of events can be obtained only if this state is populated indirectly via autoionization of one of the double excited states of H2, e.g., ‘Z8 (2pu2) (Hazi, 1974; Kanfer and Shapiro, 1983; Gubermann, 1983), which are known to be significantly excited in such collisions. Simple calculations based on this model, shown in Fig. 9, are in accord with the experimental data.

In order to explore these phenomena further, the double ionization of deuterium has also been investigated (Savage and Latimer, 1991; Latimer 1991). The heavier deuterium molecule will change the picture in two ways. First the ground state potential well is some 16% narrower than in H, and in the simple reflection approximation this should be reproduced as a similar narrowing of the fragment ion pair distribution. Second the increased mass of the nuclei means that nuclei on a repulsive curve, such as the doubly excited intermediate, will separate more slowly. This in turn implies that in a two-step process of the type described earlier in the case of He+-H, collisions there could be an enhancement of the intermediate step that requires the repulsive doubly excited state to autoionize into D,’ rather than dissociate.

Figure 10 shows an energy spectrum of fragment D + ion pairs produced at 90” in 15 keV H+-D, collisions (Latimer, 1991). The previous expectations are clearly apparent. The direct double ionization process, centered as usual at 9.8 eV is indeed proportionally narrower than in H2. Furthermore, the TS-2 process giving -5 eV fragments, which is unobservable in H +-H, collisions (Figure 8(a)), is now clearly seen.


, I

3 4 5 6 7 6 9

E n t r g y (ev) FIG. 10. An energy spectrum of fragment D + D + ion pairs produced at 90" in 15 keV H+-D,

collisions.

Giese et al. (1988) have indirectly studied the energies of deuteron pairs produced in Ar5+-D2 collisions at 500 and 1000 eV (v = 0.022 and 0.032 au). Single deuterons emitted in the Coulomb explosion of the target were energy analysed by time of flight and counted in coincidence with the charge changed projectiles Ar4+ and Ar3+ to observe the double capture process. Using a computer simulation of the kinematics of these reactions they were able to deduce the effective two-electron bonding energies for D, as a function of collision energy. These authors conclude that at these low velocities the Franck-Condon principle is not applicable.

B . NITROGEN TARGET: QUASIBOUND STATES OF N:'

In the case of collisions with nonhydrogenic molecules we have a situation opposite to that discussed previously in that the ground state wave function of the target molecule is much better known than the many potential energy curves of the doubly ionized states. The reflection approximation procedure can now be reversed to provide information about the potential energy curves of the doubly charged molecular ions produced in the collision. Such studies have been made by Edwards and Wood (1982) in the case of 1 MeV He+-N, collisions and Yousif et al. (1990) for 5-25 keV H +-N, collisions. An energy distribution of N + N + fragment ion pairs produced in the dissociation of N:+ ions obtained by the latter workers is shown in Fig. 1 1 . Considerable structure can be seen due to the production of quasibound predissociating states of N i + . Six dissociation channels are observed and have been tentatively identified from the theoretical predictions of Whetmore and Boyd (1986), which predict that all the states in-


12 1

Energy teVI

FIG. 1 1. An energy spectrum of N + N + ion pairs produced at 90" in 15 keV H +-N2 collisions. The features are identified in Table I (Yousif er al . , 1990). The full curve is the expected distribution arising from the purely repulsive Ni' ( * A v ) state.

volved predissociate to the N+(3P) + N+('P) limit at 38.84 eV. In addition a purely repulsive state 2Au, giving a broad distribution around 7.4 eV, is also observed. All the states observed are listed in Table I along with comparable data from other ion and photon impact experiments.

C. MULTICHARGED FRAGMENT PAIRS

The multiple ionization of an atom or molecule generally requires small impact parameter collisions, which can be achieved either in violent encounters with projectiles of low charge that will transfer large recoil energies to the target or in more gentle multicharged ion encounters (Cocke and Olson, 1991). This latter approach has the advantage in the atomic case of allowing (a) the spectroscopy of the multicharged target atoms with little Doppler broadening and (b) the development of multicharged recoil ion sources since collection of the low-energy recoil ions is not a problem. However, in the case of molecules the problem of recoil energy is replaced by the dissociation energy of the fragments caused by their mutual Coulomb repulsion.

These effects have been investigated indirectly by Mann et al. (1978) in 56 MeV collisions with a wide variety of molecular targets. They observed a kinematic line broadening of Auger electron spectra in the lithiumlike ions

4

0 tj

TABLE I TOTAL KINETIC ENERGIES (EV) OF N'N' FRAGMENT IN PAIRS (SUM OF BOTH IONS) FORMED IN THE 3

DISSOCIATION OF N;'. FIGURES IN PARENTHESES INDICATE WEAK FEATURES. THE DlSSoClATION LIMIT IS N' ('P) + N' ('P) At 38.84 EV.

v,

Ion Impact Electron Impact Photon Impact Theory B 5

(5.5) In. 5.3 3 5 2

9.1 'C ; 8.9 4 !2 10.0 9.7 10 ? ?

;r! 12.2 (12.4) 12.4 'Z ; 12.6 m 14.8 14.8 14 14.2 14.2 f

? 15 ? 0.6 ? ? F E

'Edwards and Wood (1982). 5 bStockdale (1977). !.2

< S i t 0 and Suzuki (1987). %

Whetmore and Boyd ( 1986). 3

< d ? m

B N: + N+N+

15 keV H+ 1 MeV Heia - 300 eVb - 400 e V -400eVc -55eV' State Energy

6.6 (6.8) 7 6.4 6.8 In" 6.8 7.4 7.8 7.8 8 7.3 7.3 3z ; 7.4

8.1 8.1-8.6 'A, 8.7

(10.8) 10.6 10.6 10.8 In, 10.7

W -e

cFeldmeier er al. (1983). dBrehm and de Frenes (1978).

'Besnard et al. (1988).

n

4 BCrosby era/. (1983). B

- 14 4


produced due to the fact that the electrons were ejected from moving fragment ions. An example of their data is shown in Fig. 12, which shows the ls2p2p4P line for C ions originally bound in CH, and CO (Stolterfoht, 1987). The width may be readily understood from the steriometric structure of these molecules. Because the C in CH, is located at the centre of mass of the exploding molecule it is left essentially stationary after the collision. However, the C in CO is not in the middle and so receives significant energy.

Coulomb explosion energies have been determined more directly in diatomic molecules by Watson and Maurer ( 1987) using time-of-flight spectroscopy with the aid of an extraction field. Double peaks are then observed corresponding to fragments whose explosion velocities are directed towards and initially away from the detector. The fragment energies can be obtained from the sepa-

0- 226 229 232

-Electron Energy lev]

1200 =

800 -

LOO -

3 = rzo” AE = Q6 eV

0- 229 232 226

-Electron Energy lev1

-- -)c -- . \

c o H * CHL

FIG. 12. Carbon-K-Auger electron spectra produced in 56 MeV ArI3+-CHp, CO collisions (Mann er al., 1978) and a pictorial representation of the resulting Coulomb explosions (Stolterfoht, 1987).


ration of the peaks in the spectrum. Tawara et a f . (1986) have used a magnetic analyser and compared the shift in the energies of the fragment ions relative to the ions from an atomic target. Both these experiments show that charge is distributed nearly symmetrically between the fragments, as might be expected since the fast electronic relaxation time (- 10 - l 6 secs) relative to the dissociation time (- 10 - l 5 secs) should allow relaxation of the electronic charge cloud before dissociation. An important implication of this result is that, in order to produce a highly charged ion from a diatomic target, the molecule must be roughly twice as highly ionized as the required fragment ion (Cocke and Olson, 1991).

V. Angular Distributions of Fragment Ions: Orientated Molecules

It has already been seen (see Section 11.B) that the observation of fragments from diatomic molecules at a fixed angle define the orientation of the target molecule in space and hence can provide information on the symmetries and lifetimes of the states involved in the collision. However, only a few relatively simple studies of this kind have been performed to date. More work is expected and clearly desirable in this area. Details of the experimental problems and techniques can be found in Berkowitz (1979), Scoles et a f . (1988) and the papers cited later.

A. HYDROGEN AND DEUTERIUM

The angular distribution of fragment ions and fragment ion pairs in hydrogen have been measured by Edwards e t a f . (1985a, 1985b) in 0.4-3.5 MeV H + , D + and He+ collisions, Lindsay er a f . (1987) and Yousif er a f . (1988) in the case of 5-25 keV H + collisions; and Ezell et a f . (1991) in 1.0 and 2.0 MeV H + collisions.

Examples of angular distributions from Lindsay et a f . (1987) are shown in Figs. 13(a) and 13(b) for the 2 p , state and 2pm, states. In the former case, where three degenerate channels are available, the p value ( - 0.97) indicates a near sin2B distribution with over 90% of the transition being 2 + II with both (T and 6 electrons being ejected (Dehmer and Dill, 1978). The transition to the 2pa , state has a /3 = 0.5, implying that the transition cross section to both 2 and II are comparable and both (T and T electrons are ejected. These results show a remarkable similarity to the equivalent photoionization data. The ion projectile apparently suffers no momentum transfer in the collision process (note the symmetry about 90") and simply provides an electrical impulse along the direction of motion that can be compared with the transverse E field of a polarized photon. The H + H + ion pair distributions in which two electrons are ejected have an isotropic distribution both in charge transfer (Yousif et a f . , 1987) and ionization

130

- - ta

c - 3

w. L ,

c .-

z c 2 - 0

* - c m t, u c

-

c

I

Colin J . Latimer

0 0

- -

- I I I I I I I I I I I I

0 30 60 90 120 150 180 Angle (deql

(a)

- - ta

c - 3

w. L ,

c .-

z c, 2 - 0

* - c m t, u c

-

c

I

- -

- I I I I I I I I I I I I

Angle (degl (b)

FIG. 13. The angular distribution of fragment protons from (a) the 2p7r. state of H,’ and (b) the 2pu . state of H; , produced in 15 keV H +-H2 collisions. The full curves are a fit of Eq. (4) to the data. (Lindsay er al., 1987).


(Edwards et al., 1985a) processes involving H + projectiles. However, Edwards et al. (1985b) have observed a small orientation effect in 0.025-0.875 MeV/ amu He+ collisions and conclude that at low velocities there is a contribution from double collision or two-step processes.

In their most recent work Edwards, Wood and coworkers (Ezell et al . , 1991) have studied the angular distribution of fragment ions from the two electron excitation processes producing the 2pvu, 2p1r,, 2sv,, H + H + and doubly excited autoionizing Hz** states with equivelocity electrons and protons (1 .O and 2.0 MeV/amu). For both projectiles the 2su, and 2pu, states have angular distributions that can be fitted to a 1 + A cos20 expression corresponding to simple dipole transitions. Similarly however, fragments from the 2p7r. and H + H + states can be fitted only to a 1 + A cos*O + B cos40 expression corresponding to nondipole or double collision interactions. The exception is the doubly excited “state” that exhibits dipole behaviour for electron impact and nondipole for proton bombardment. A full interpretation of these interesting results and their implications in unfortunately not yet available.

In another recent experiment Cheng et al. (1991) have examined the angular distributions of deuterons produced in electron capture collisions of 08+ ions bombarding deuterium. The experiment was performed by applying an extraction electric field perpendicular to the beam to project the velocity distribution of the ions on to a two-dimensional position sensitive detector. The time of flight of the recoil ions was also determined, and thus the dependence of the cross sections on the molecular orientation could be obtained. These results show (see Fig. 14) that the molecules prefer to be orientated perpendicular to the incident beam. A simple explanation of this result has been provided by Wang and

c

0- 0 45 90 135

Angle e 10

FIG. 14. The electron capture cross section for 10 MeV On+ ions in DI as a function of 0, the molecular orientation relative to the beam axis (Cheng e? al . , 1991). The full curve is the result of an OBK calculation incorporating two scattering centre interference effects (Wang and McGuire, 1991).


McGuire (1991) in terms of two centre interference effects within the Oppen- heimer-Brinkman-Kramers approximation. When the molecular orientation is perpendicular to the beam, the phase difference between wave amplitudes originating on the two scattering centres of the molecule is a minimum, resulting in constructive interference.

B. OTHER MOLECULES

Angular distribution studies have also been performed with nitrogen and oxygen targets. Ezell et a f . (1984) measured the coincidence yield of pairs of N + ions produced in 0.2-3.0 MeV He+-N, collisions. They showed that (a) any molecular recoil effects were much smaller than those due to the thermal motion of the N, target, and (b) the angular distributions of the groups of ion pairs (see Section IVB) were neither isotropic nor of a simple sin28 or cos’8 dipolar form. Similarly eccentric distributions, still possessing symmetry about 90°, have been observed in 5-25 keV H +-N, collisions by Yousif et al. (1990) and accounted for by including higher-order multipole corrections to the simple dipole-Born approximation (van Brunt, 1974).

Varghese et a f . (1989) have investigated the recoil N4+ ions ( q = 1-5) produced in Coulomb explosions in 19 MeV F9+-N2 collisions using the spatial distribution of the fragments projected onto a two-dimensional position sensitive channel plate detector located at right angles to an extraction electric field coupled with time-of-flight analysis. They found that the highly charged fragments (e.g., N 5 + ) are preferentially produced when the molecule is aligned along the beam axis while the production of fragments of low charge (e.g., N 2 + ) requires that that molecule be aligned perpendicular to the beam axis.

The energy distribution of fragment 0 + ions produced in the He +-Oz charge transfer process have already been discussed in Section Ill (Bishof and Linder, 1986; Yousif et al., 1987). At low energies (<60 eV) the angular distributions are anisotropic (see Figure 7), indicating that the total charge transfer probability is generally higher for the parallel orientation of the 0, molecule with respect to the relative translational motion. Above 60 eV most of the orientational dependence of the charge transfer process is lost.

VI. Partial Dissociative Ionization Cross Sections

Dissociative ionization involves the collision of a projectile with a target molecule and the subsequent production of a wide range of secondary ions. A measure of the probability of this reaction is the total dissociative ionization cross section for all the secondary ionic products. Such measurements can be readily made using the traditional condenser plate method (Gilbody, 1968; McClure and


Peek, 1972). However a more detailed and valuable measure is the partial dissociative ionization cross section that separately identifies the probability for each ionization channel, including the charge-to-mass ratio of each fragment and the electronic states involved in the collision. Such measurements clearly require a detailed analysis of all the reaction products summed over all angles.

A. CHARGE AND MASS ANALYSIS OF FRAGMENT IONS

The complete collection and identification of all fragment ions is difficult since they are emitted with considerable energies anisotropically into 47r solid angle. Electric field ion collectors with contiguous mass spectrometers tend to discriminate against ions that are emitted perpendicular to the collecting field. In the case of fast heavy particle collisions these problems were first successfully overcome by Browning and Gilbody (1968) and Afrosimov er al. ( 1969).

In a series of experiments Browning, Gilbody and coworkers studied the dissociative ionization of a wide range of simple molecular gases (H,, N,, O,, CO, CO,, CH,) by protons (Browning and Gilbody, 1968), He+ ions (Browning et a!., 1969), He atoms (Browning et al., 1970), Ne+, Na+ ions and atoms (Gra- ham er d . , 1973) and 3He + + ions (Graham et al., 1974) within the energy range 5-85 keV. It was found that dissociative processes are generally dominant in the ionization of the target molecules and that the fragmentation pattern is substantially independent of the projectile species. In the experiments of Afrosimov and coworkers (Afrosimov et al . , 1969, 1972, 1974, 1980) a delayed coincidence method was employed to provide a simultaneous analysis of the final charge states of both collision partners and thus separate the electron capture and ionization channels in 5-50 keV collisions of H + and H" with H, and CO. In proton collisions electron capture processes dominate while Ho projectiles remain un- changed in the collision process (pure ionization).

The most recent work in this field of investigation has been performed by Shah and Gilbody (1982, 1989, 1990), using a crossed beam technique employing time-of-flight analysis and coincidence counting of all the collision products to identify all the main reaction channels. H + , HeZ+, Liq+, Cq+, N4+ with 1 < q < 5 have been employed within the energy range 6.7-550 keV/amu. In all cases electron transfer processes are found to be very important. Unfortunately no theoretical studies of any of these processes are available at this time.

B. CROSS SECTIONS WITH STATE IDENTIFICATION

Most recent experiments have incorporated energy and angular analysis of fragment ions in addition to charge to mass identification. This allows, in situations where the potential energy curves are well known, final state identification by


deconvolution, including integration over all angles. Thus Lindsay et al. (1987) were able to determine elementary cross sections for each dissociation process in 5-25 keV H+-H2 collisions. Within this energy range u(H,’ , 2pu,) is the dominant channel followed by u(H,’, 2pr , ) , u(H2+, 2sa,) and u(H+H+) .

The H + H + channel of course can be directly observed by detecting the pairs of H + ions in coincidence, as was done by Yousif et al. (1988) for 5-25 keV H+ projectiles and Edwards et al. (1986) for 80-3500 keV H+-H2 collisions. This latter experiment at high velocities is especially interesting. In the equivalent double ionization process in helium equivelocity, electrons and protons behave differently due to electron correlation effects (see Sections I and IV). The ratio R of double to single ionization cross sections is about a factor of two greater for electrons than protons, contrary to the predictions of the Born approximation. In H2, which is isoelectronic with He but clearly lacks the same spherical symmetry, the same factor of two is observed and no effects due to molecular orientation are apparent (Edwards er al., 1988).

Partial state analysed cross sections for the production of secondary 0’ and 0,’ ions in He +-O2 collisions obtained by Bischof and Linder (1986) and Yousif et al. (1987) are shown in Fig. 15. At low energies the near resonant channel 0,’ (c4Z;) and the slightly exothermic channel 0,’ 111211, dominate with an unidentified channel 0, which produces near zero energy 0’ ions, becoming significant at higher energies. At energies above about 2 keV the 0,’ ( B 2 C ; ) state is also important although ions arising unidentifiable processes (labeled R) dominate.

c. DOUBLE CAPTURE IN HYDROGEN

The dissociation of a molecule that has n bound electrons can be inferred from the conversion of projectile atomic ions of charge m into ions of charge m - n under single-collision conditions. The complete stripping of the electrons from the target molecule implies a Coulomb explosion. This type of measurement has been used primarily where the target molecule is hydrogen. Therefore cross sections c + , ~ in proton collisions (Fogel et a l . , 1959; Williams, 1967) where H- is detected necessarily implies the process

H + + H 2 + H - + H + + H +

Recently Kusakabe et al. (1990) have investigated the process

He2+ + H 2 + Heo + H + + H +

by observing the neutralization of 1 - 10 kV 3He + ions and confirmed the existence of a cross-section minimum at 10 keV (Afrosimov et al., 1980). At very low energies (<1 keV) the cross section for this process, which may be important in astrophysics as a source of energetic protons, rises to over 10 - l4 cm2.


D. ENERGY Loss SPECTROMETRY

Fournier et al. (1972, 1986) have combined the double capture technique in hydrogen with energy loss measurements on the product H- ions at near-zero scattering angles. At energies between 3 and 9 keV a mean energy loss of 49.6 eV was observed confirming that the Franck-Condon principle was applicable in such collisions (see Figure 1). Furthermore, the detailed distribution of energy loss is well described by the reflection approximation.

The successful extension of this technique to other molecules requires a detailed knowledge of the molecular potential energy curves. In an exciting series of experiments Dowek et al. (1981, 1982, 1983) have used energy loss spectrometry to study electron capture by 200 eV-3 keV He + ions from H,, N,, O,, CO and NO. These results, which have been interpreted in terms of cubic correlation diagrams, have provided a great insight into ion-molecule collision processes, including dissociative ionization. This work has been reviewed in a recent volume of this series (Pollack and Hahn, 1986).

Roncin et al. (1986) have studied two electron capture by 8 keV N 7 + , 07+, 08+, Ne7+, Ne*+ ions in hydrogen. These differential cross-section measurements show that, for highly charged ions q = 7, 8, the two electrons are captured successively rather than simultaneously during the collision.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS. VOL. 30

THEORY OF COLLISIONS BETWEEN LASER COOLED ATOMS P. S . JULIENNE Molecular Physics Division, National Institute of Standards and Technology, Gaithersburg. MD

A. M. SMlTH and K . BURNETT Clarendon Laboratory, Department of Physics, University of Oxford, United Kingdom.

I . Introduction . . . . . . . . . . . . . . . . . . 11. Cold Collisions in the Absence of Light. . . . . . .

A. Theory of Cold Collisions. . . . . . . . . . . B . Examples of Cold Collisions . . . . . . . . . .

111. Cold Collisions in a Light Field . . . . . . . . . . A. Formal Theory . . . . . . . . . . . . . . . B. Trap Loss Processes . . . . . . . . . . . . .

D. Effects of Long-Range Collisions on Laser Cooling Acknowledgments . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

C. Optical Manipulation of Collisions . . . . . . .

. . . . . . . . . . 141

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. . . . . . . . . . 157

. . . . . . . . . . 157

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I. Introduction

Rapid advances in laboratory techniques for laser cooling and trapping of neutral atoms have enabled ultracold temperatures, below 1 mK, to be obtained experimentally. The reader is referred to the special, November 1989, issue of J . Opr. SOC. Am. B for standard references on this subject. We will include specific references where required in this chapter. Laser cooling techniques offer many new opportunities for science and technology, including greatly improved time and frequency standards (Rolston and Phillips, 1991) and the ability to study the very novel physics that characterize the collisions of such cold atoms.

It is important to understand ultracold collisions for two reasons. First, collisions in a trap can limit the trap lifetime or otherwise degrade the performance of a desired application. Second, ultracold collisions exhibit new and unusual effects that are interesting in themselves. Collisions of cooled and trapped neutral atoms have been studied experimentally (Gould et al . , 1988; Prentiss et al., 1988; Sesko et al., 1989; Lett er al . , 1991) and theoretically (ViguC, 1986;

Copyright Q 1993 by Academic Press. Inc. All rights of reproduction in any form reserved.

ISBN 0-12-003830-7 141

142 P.S. Julienne, A . M . Smith and K . Burnett

Pritchard, 1986; Thorsheim et a/ . , 1987; Julienne, 1988; Julienne et al., 1988, 1990, 1991; Julienne and Mies, 1989; Gallagher and Pritchard, 1989; Julienne and ViguC, 1991; Julienne and Heather, 1991; Smith and Burnett, 1991; Gal- lagher, 1991; Tiesinga et al., 1991, 1992; Trippenbach et al. , 1992). A special case of low temperature, T, collisions is the study of spin-polarized H, using nonlaser methods of cooling, which has been studied extensively experimentally (Silvera and Walraven, 1986; Mashura et af . , 1988; van Roeijn et al., 1988; Doyle et al., 1991) and theoretically (Stwalley, 1976; Berlinsky and Shizgal, 1980; Uang and Stwalley, 1980; Ahn et al., 1983; Lagendijk etal. , 1986; Ver- haar et a f . , 1987; Koelman et a/. , 1987, 1988a, 1988b; Stoof et al., 1988; Agosta et al., 1989). We concentrate in this chapter on collisions of laser cooled atoms instead of hydrogen.

Although a variety of magnetic and optical trapping methods can be used, trap depths are generally much less than 1 K and the density is typically in the range lo9 to lo’* atoms/cm3. Temperatures of laser cooled and trapped atoms can be in the range from near 1 p K to 1 mK, and the kinds of species subject to trapping are alkali, alkaline earth, and metastable rare gas species. The presence of a fine or hyperfine structure in the cooling transition can lead to extra cooling to temperatures well below the Doppler cooling temperature & (Lett et al., 1988; Dalibard and Cohen-Tannoudji, 1989; Ungar et al., 1989; Salomon et al., 1990; Monroe et al., 1990). At & the thermal energy is K ~ T = hy/2, where y is the linewidth of the cooling transition and K~ is Boltzmann’s constant. Experimental temperatures for laser cooling will remain above the recoil temperature, T R , where the atomic momentum k is equal to the photon momentum k,, or equivalently, the atomic de Broglie wavelength is equal to the wavelength of the cooling transition.

Two different methods of observing collisions of trapped atoms have been used so far. One is to stop loading the trap and watch the trap density decay due to processes that eject atoms from the trap. This has been done for Cs (Sesko et a/. , 1989) and Na (F’rentiss et al., 1988) traps. The other method is to observe the appearance rate of products of a collision, as done for Naf ions produced by associative ionization (AI) of two excited Na atoms (Gould et af . , 1988; Lett et al., 1991). The experimental field awaits development. One exciting prospect would be collision studies with velocity control using laser cooled beams (Nel- lessen et af., 1989; Riis et al., 1990; Zhu et a/. , 1991; Ketterle et a f . , 1991). The Na A1 cross section has been measured at 60 mK collision energy using laser velocity group selection techniques in a fast beam (Weiner et al . , 1989; Thorsheim et al., 1990), and such techniques could be extended to lower velocities. One-dimensional velocity selection to produce effective one-dimensional temperatures around the pK level has been demonstrated (Kasevich et al. , 1991). Such techniques are bound to produce a profound effect on the study of atomic processes.

THEORY OF COLLISIONS BETWEEN LASER COOLED ATOMS 143

It is important to distinguish between two fundamentally different kinds of collisions at low temperature. The first, which we call Type I collisions, occur in the absence of a light field. These collisions are described by the normal, well- developed wave-function methods of scattering theory using a conservative Hamiltonian and exhibit the familiar Wigner threshold law quantum effects associated with the long de Broglie wavelength as T + 0 (Wigner, 1948).

The second kind, which we call Type I1 collisions, are collisions in the presence of a light field. If the light frequency is sufficiently near to resonance with the cooling transition, the “preparation” of the atoms for the collision can not be isolated from the collision itself, because of the very long time and distance scale of the collision (Julienne and Mies, 1989). Such collisions should be treated by quantum mechanical methods that explicitly treat the dissipation due to excited state spontaneous emission during the “collision.” It is this dissipative aspect of ultracold collisions in a light field that gives ultracold collisions of laser cooled atoms novel and distinctive features that are not yet very well understood. If the light field is sufficiently far off resonance, or the collision velocity is high enough, these Type I1 collisions may be treated to a good approximation by Type I methods that use a conservative Hamiltonian.

This chapter will be divided into two distinct parts. The first will treat the issues associated with a Type I collision, using examples of collisions of ground states and metastable rare gas species. The second part will treat the issues re- lating to Type I1 collisions and give several examples of excited state collisions that lead, for example, to loss of atoms from an optical trap. In this category we should also distinguish between collisions that may be thought of as disturb- ing the cooling process (i.e., collisions affecting cooling) and those collisions where the cooling is critical for the mechanism of the Type I1 collision (cooling affecting collisions). By this we mean that one gets the special nature of the Type I1 collision only because the atoms have been cooled prior to the collision.

II. Cold Collisions in the Absence of Light

Since these collisions occur in the absence of a light field, they could be observed in cold beams or fountains in free space, in magnetic traps, or in optical traps during brief intervals during which the light is off. All of these represent possible ways of actually doing collision experiments with currently developed methods for optical manipulation of neutral atoms. Although we will discuss collisions of like atoms, there is no fundamental reason why collisions of two different species could not be studied, especially if cold beam technology could be developed.

Cold collisions can be elastic, in which only the momentum of the colliding partners changes, or inelastic, in which a change of atomic state also occurs for


one or both atoms. Alkali species all have a hyperfine structure in the ground state and can experience inelastic collisions that change hyperfine quantum numbers:

As mentioned in Section I, such collisions have been extensively studied for collisions of cold hydrogen atoms in a magnetic field, and similar calculations for Na and Cs atom hyperfine changing collisions have recently been reported (Tiesinga el a l . , 1991, 1992). Rare gas metastable atoms also are excellent candidates for cold collision experiments (Julienne and Mies, 1989). The He* ) S , level can be laser cooled on the 1.08 pm 3S + 3P transition (Aspect et a l . , 1988), and the 3pZ level of the other rare gas species can be cooled (Shimizu et a l . , 1987; Katori and Shimizu, 1990). These species are very energy rich and during a collision can easily eject an electron by Penning ionization (PI):

(2) A* + A * + A + A + + e -

or associative ionization (AI):

A* + A * + A : + e - . (3)

The ionization of two He 3Si metastables has recently been studied by Miiller et al. (1987, 1991), who resolved the electron spectrum that resulted from metastable collisions within a single beam at 20 K relative collision energy. This is a very fast process at normal collision energies, for which a larger fraction of the cross section leads to PI rather than AI.

It is important to make the distinction between the collisional rate coefficients for polarized and unpolarized samples of atoms. An unpolarized sample is one in which all Zeeman sublevels are populated (but we will not specify with what distribution), whereas we define a polarized sample to have all the atoms in the “stretched” state with maximum total angular momentum (Fmax) and maximum Zeeman quantum number, M = F,,,; for rare gas species without nuclear spin, F = j = electronic angular momentum only. Unpolarized samples of either alkali or metastable rare gases will decay rapidly by virtue of collisional processes that have high intrinsic probability. In both cases polarized samples will decay by much slower processes, having smaller intrinsic probabilities. Alkalis will behave similarly to hydrogen, for which there are two mechanisms for hyperfine structure change. One is a strongly allowed spin-exchange process, which occurs by virtue of strong exchange interactions at small internuclear separation R and affects unpolarized samples. The other is the dipolar process by which polarized samples decay by virtue of very weak spin-dipole-spin- dipole interactions occurring at relatively long range. The ionization processes, Eqs. (2) and (3), for rare gases will be very rapid for unpolarized samples, but are expected to be much slower for polarized samples. This is because, when all


spins are lined up in the molecular state formed during a collision, the ejection of an electron to the ground state of the ionic system (a spin doublet system + electron) is spin forbidden and occurs only because of spin-orbit mixing. Noth- ing is known about the ionization rates for collisions of spin-polarized rare gas metastables. We would guess that the ratio of rates for polarized versus unpolarized samples would be largest for the low Z species He and smallest for the high 2 species Xe. The ratio could easily be several orders of magnitude for He.

A. THEORY OF COLD COLLISIONS

1 . Quantum Scattering Theory

The fundamental question to ask is how the collision cross section U ( E ) for the process of interest varies as the collision kinetic energy E + 0. An equivalent question can be asked of the rate coefficient K ( E ) = (T (E)v (E) , where v is the relative collision velocity of the colliding species; the experimental observable is the average of K ( E ) over the energy distribution, K = ( K ( E ) ) , where the ( ) average is usually taken to be Maxwell-Boltzmann. We prefer to work with the rate coefficient, since it is more closely related to what is observed in an experiment. The number of events per unit time in a uniform region of volume V is KNINzV, and an atom in state 2 experiences a mean time between events of IIKN,, where N i is the density of atoms in state i = 1 or 2. As an order of magnitude example, a rate coefficient of lo -” cm3s-’ and an atomic density of loio cm -3 generates 109V events per second and a mean time between events of 10 s. Thus, collisions between atoms are relatively rare, but if enough atoms are confined, there should be enough events to detect the consequences of the collisions (e.g., appearance of ions or fluorescence, disappearance of atoms).

Since Type I collisions can be treated by the normal methods of scattering theory, let us briefly review these methods. Numerical scattering calculations are usually based on the time-independent wave-function formulation, as applied to cold H collisions by Stoof et al. (1988) and discussed in the context of ultracold collisions by Julienne and Mies (1989).

This formulation is based on a time-dependent viewpoint in which the colliding species a and b are prepared in the distant past in some state I?) = Iyoyb) and, as a consequence of the collision, evolve in the distant future to a set of states I?’). Usually it is convenient to expand the relative motion in an angular momentum basis and define the asymptotically separated atoms by the set of channel states Ip) = Ilmy), where 1 is the relative angular momentum quantum number and m is its projection on some space fixed axis. The transition amplitudes between channel states for total energy E , T ( E , p + p’) = 1 3 ~ ~ ~ - S ( E , p + p’), are expressed in terms of the unitary S-matrix. The cross section for


cell experiments, averaged over all collision directions relative to a laboratory space quantization axis, is defined by

(4) V ( E ~ , y + y ' ) = 7 c (21 + l ) P ( E , 1, y + yf),

where k, is the channel momentum wave vector. The channel kinetic energy is

7T

k ,

In this equation the channel energy E is the energy of the separated fragments, and the opacity function is

Elastic collisions are those for which a state change does not occur, y = y ' , whereas y # yf for inelastic collisions. Care must be taken in the case of collisions of like atoms to build in correctly the effects of homonuclear symmetry into the calculation of the 7'-matrix elements and the rate coefficients, a problem that has been throughly discussed by the Eindhoven group (Stoof et al., 1988; Manders et a l . , 1989).

The 7'-matrix elements can be readily calculated by solving the matrix Schro- dinger equation

dZ 2 1 -F(E, R ) + 7 [ E l - U ( R ) ] F ( E , R ) = 0, dR R (7)

generated by the expansion of the wave function of the colliding system in a basis Ip) that spans the space of all coordinates except the magnitude of the internuclear separation R:

(8)

Since the terms in the potential matrix U ( R ) are made up of the electronic Born-Oppenheimer potentials and various coupling terms that mix these states, the basis IP(R)) depends parametrically on R . These coupling terms can be due to electronic, Coriolis, fine structure, or hyperfine structure interactions, depending on the specific problem. The effect of external static or oscillating electromagnetic fields can also be incorporated into U . Standard numerical methods are available for solving Eq. (7) for F, from which the T-matrix can be calculated. In this way calculations have been done for ground state collisions of cold trapped H (Stoof et al., 1988), D (Koelman et a l . , 1987), and Na (Tiesinga et a f . , 1991) in a magnetic field.

A variety of approximations are available for interpreting the cross section and

Q ~ ( E , R ) = c I P ' ( R ) ) F ~ , ~ ( E , R Y R . P'


giving physical insight into the nature of the collision. One particularily useful approach at normal temperatures is the semiclassical picture, in which the cross section is written as

(T(E,, y + y ’ ) = Jr 2rbP(E, b, y + y’)db, (9)

where P in Eq. (9) is the semiclassical probability of the y + y’ transition. The impact parameter b, the distance of closest approach for a straight-line trajectory in the absence of an interaction potential, is related to the angular momentum by

Using Eq. (lo), the quantum expression Eq. (4) and the semiclassical expression Eq. (9) are easily seen to be equivalent, where the discrete summation over 1 in Eq. (4) is replaced by the continuous integration over b in Eq. (9).

2. Quantum Threshold Laws as T + 0

The de Broglie wavelength A,, for the entrance channel y is related to k , by

Whenever A, is large compared to some scale ‘‘size’’ of the interatomic interaction potential, the T-matrix elements exhibit typical quantum effects (Wigner, 1948; Delves, 1958), and the semiclassical picture must be modified. These’effects have been known since the earliest days of quantum mechanics and were first studied in the context of scattering of cold neutrons by atomic nuclei, where the neutron de Broglie wavelength was large compared to the size of the nucleus (Bethe, 1935). These effects are a consequence of the analytic properties of the quantum wave function in the vicinity of a channel threshold where E + E , and the channel kinetic energy E? + 0. For collisions of neutral atoms, the contribution to the cross section and rate coefficients all vanish for all partial waves 1 except for the s-wave, for which 1 = 0. The s-wave T-matrix element for an elastic collision cross sections (y‘ = y ) is proportional to k , , giving a finite cross section, but vanishing rate coefficient, as B , + 0. Clearly no endothermic processes contribute in this limit, but exothermic inelastic processes (y’ # y , E,, > E , ) have s-wave T-matrix elements proportional to kli2, giving a cross section that diverges as l l k , (or l /v) , but a rate coefficient that remains finite as E , + 0. Elastic collisions thermalize the velocity distribution in a magnetic trap, whereas inelastic collisions typically have the undesirable effect of producing states that are no longer trapped. The quantum threshold laws have the consequence that, at sufficiently low T in the quantum regime, inelastic exothermic


processes will always dominate elastic processes if s- wave inelastic processes are possible.

Although several methods have been used for obtaining the threshold laws (Wigner, 1948; Delves, 1958), we have used the generalized form of the multichannel quantum defect theory (GMQDT) to gain some physical insight into the nature of these quantum threshold effects (Mies, 1984; Julienne and Mies, 1984a, 1989). Although the GMQDT gives a rigorous quantum mechanical analysis of the analytic properties of the radial wave function F(E, R ) as a function of E and R , the theory is readily stated in a WKB form that makes good use of semiclassical insights. The first step of the theory is to define a set of reference potentials U p ( R ) for each asymptotic channel p. These would normally be chosen so the full U ( R ) matrix is diagonal at large R and is diagonal or nearly diagonal over much of its range. The reference potentials define single-channel reference solutions to the uncoupled Schrodinger equation:

There are two linearly independent solutions to Eq. (1 2), but we will work here with only the solution that is regular as R + 0. This solution can be written in phase-amplitude form:

f p ( E , R ) = a p ( E , R ) sin b p ( E , R ) ; (13)

and the Schrodinger equation can be transformed into the Milne equation for a and b (Greene et a l . , 1982; Pan and Mies, 1988). However much insight is found by writing Eq. (13) in its WKB form, where

(14) upWKB(E, R ) = K,(E, R ) - ' / 2 = [2p(E - U p ( R ) ) / h * ]

bpWKB(E, R ) = IR; K,(E, R')dR' + 7~14,

and R , is the inner classical turning point of the reference potential. The WKB normalization in Eq. (14) is a classical time normalization; that is, the probability of finding the colliding particles in element dR is proportional to the classical time spent there, dRlv, since the square of apWKB is proportional to the inverse of the local velocity, namely l / v , ( R ) .

If the channel kinetic energy E , is large enough, the criterion for the validity of the WKB approximation is satisfied for all R > R , ; namely,

where A,(E, R ) is the local de Broglie wavelength, and the WKB form in Eqs. (14) and (15) will apply as a good approximation at all R > R , , How-


ever, as E , + 0, there will always be some range of E and R where Eq. (16) is not satisfied, the WKB form will no longer apply, and GMQDT predicts a simple form of the departure. If we assume a potential that varies at long range as -C, /R", then it is simple to obtain the point ( R Q , e n ) at which the function in Eq. (16) reaches some maximum specified value as a function of R . Taking dAp(.sp, R,) /dR = 1/2 as a criterion of WKB breakdown defines (Julienne and Mies, 1989)

2"

EQ = E [ ( 2 q) . (-)". n - 2 (- 2n + 2 . T ) ~ ] " ~ - ~ (17) El. 6n n - 2 pC,

It is convenient to imagine the range of R divided into two zones, an inner zone with R << R , and an outer zone with R >> R Q . In the outer asymptotic region the wave function approaches

(19)

According to Levinson's theorem, the phase shift qo approaches n n as E , + 0, where n is the number of bound states in the potential. An s-wave (1 = 0) has the asymptotic behavior in the outer region:

(20)

(21)

as E , + 0, where is a parameter called the scattering length, and the phase shift vanishes as A,,,k, ; the elastic scattering cross section associated with reference channel p is just .rrAa,,. GMQDT shows that f ,(E, R ) in the inner zone has the form (for all E )

f p ( E , outer) = k ; i / 2 sin (k ,R + d / 2 + 7,).

a p ( E , outer) = k ; i / 2 = apWKB

b,(E, outer) = k,(R + Ae, , ) # bpWKB,

a,(E, inner) = C,(E)-'apWKB (22)

b,(E, inner) = bpWKB, (23)

that is, the phase is that of the WKB function, but the amplitude may be modified in a way that is independent of R but depends only on the incident energy. For high collision energy, E , >> E ~ , C,(E) - I = 1 and the WKB form applies at all R . However, for low collision energy, E , << E ~ , and so C,(E)- ' is different from 1 . At sufficiently low energy, GMQDT shows for s-waves,

Cp(E) - ' = (kyA, .p) ' / ' , (24)

where we call the proportionality constant A which should not be confused with A e , p .

the inelastic scattering length,


We see that when E , << ep a semiclassical connection cannot be made between the reference wave functions f in the inner and outer zones. The consequence is that the WKB amplitude off is modified in the inner zone and the WKB phase is modified in the outer zone. These modifications lead to the quantum threshold properties of the rate coefficient that were mentioned previously. A simple picture can be worked out for estimating the order of magnitude of rate coefficients for exothermic inelastic collisions. Let us assume that the p + p' process in question is due to channel mixings that are due to interactions occuring in the inner zone. If the interaction potential has accelerated the atoms to have much larger local kinetic energies than initially, i.e., cp(inner) >> E , , the probability of the inner zone process, P(inner, p -+ p'), will be very insensitive to the initial E , , depending instead on local inner zone kinetic energies, curve crossings, etc. But as the initial energy E , + 0, the probability is modified by the amplitude changes that occur in the inner zone:

P(E? << cp, p +. p') = C,(E) - T p . ( E ) -2P(inner, p + 0'). (25)

This is obvious for weak inner zone coupling. If strong coupling occurs, then Julienne and Mies (1989) showed a correction factor may apply, which can effectively be accounted for by using a modified inelastic scattering length in the entrance channel. If the channels p and p' are degenerate, both C - * factors in Eq. (25) are proportional to k, the probability approaches zero as k2, and the T + 0 limit of the rate coefficient is zero, just as for elastic scattering. Since, for exothermic processes, the exit channel factor C,.(E)-' is just a constant (= 1 if the exothermicity is >> c Q ) , the $-wave probability in Eq. (4) can be written as

(26)

where A:.p has units of length and includes possible modifications of A,,@ by strong coupling and C,' # 1 for the exit channel. The T + 0 limit of the rate coefficient for exothermic processes becomes

P(E, 0, p-+ p' ) = A ' ,.pk,P(inner, P -+ p ' ) ,

x P(inner, p -+ p')cm3s-l.

An upper bound to K is set by the unitarity of the S-matrix. This bound is also found from Eq. (27) by setting the probability P equal to unity and replacing A' by the thermal expectation value of (hJ27r). If the inner zone process is strongly allowed, then P is near unity. Use of typical values for A' and p in Eq. (27) shows that the T + 0 limit of ultracold inelastic exothermic rate coefficients is


not very different from the range of typical room temperature coefficients, say, 10-I2 to 1O-Io cm3/s.

3. Onset of Threshold as T + 0

The GMQDT picture not only gives a rigorous analytic representation of quantum threshold properties of the T-matrix (Julienne and Mies, 1989) but also enables us to see where to expect the onset of such properties as E , is lowered towards zero. The condition E , < E , is a necessary, but not a suficient, condition that the threshold law behavior in Eqs. (23) and (24) applies. The onset of this behavior depends on the proximity to threshold of the last bound state in the U , ( R ) potential, and E , may have to be one or more orders of magnitude smaller than E , for these forms to apply. As the binding energy of the last bound state, eb.@ = E, - Eb,,, approaches zero, the A,,@ parameter becomes arbitrarily large. We find that as E , decreases from E , towards zero the inner amplitude function C,(E) - will decrease monotonically from unity if E ~ , ~ > E ~ , but first rise above unity before decreasing if & b , p < E , . This latter behavior gives rise to s-wave resonances in inelastic processes when E , < E ~ . The position of the last bound state in a potential cannot be predicted from a knowledge of the long-range potential; instead, the position is a property of the whole potential. However, the spacing of the last levels, and the “bins” in which the levels must lie, can be calculated from a knowledge of the long-range potential alone (Stwalley, 1970; LeRoy and Bernstein, 1970). Thus the last level will have a binding energy less than the permissible maximum value &b.ma* and an outer turning point R b larger than the permissible minimum value Rh.min.

Another set of characteristic parameters that influence the threshold properties are the positions R , ( l ) and heights ~ ~ ( l ) of the centrifugal barriers in the long- range potential. Given the long-range potential,

c,, ti241 + 1) K ( R , 1 ) = - - +

R” 2 p R 2 ’

these are readily calculated. If E , < E , ( / ) , penetration inside R = R , ( l ) is classically forbidden. This is one reason why only s-waves contribute to the cross section at threshold. Since we are interested in ground state collisions for which the n = 6 van der Waals potential is the lead term, we will illustrate the theory for the n = 6 case (although very weak spin-spin interactions generate long-range interactions varying as l / R 3 for two *S atoms [Meath, 19661, these are too small to distort the motion significantly for the energies considered here).

We find the remarkable result that for n 2 3 the parameters R , , R c ( l ) , and Rb.min all scale with mass and Cmi, in exactly the same way, being proportional to s, = (pC,)l/n-2. In addition, E,, ~ ( l ) , and &b.max have the same scaling factor, proportional to p - ’ s i 2 = p-I(pC,J - 2 i n - 2 . Other than this scaling, these

152 P.S. Julienne, A . M . Smith and K. Burnett

2 -

0-

cp * -2- L3

-4 -

-6 -

-8

quantities depend on factors that depend on n or 1 alone. Therefore, it is possible to represent these quantities on a plot of E versus R that is universal for a given n. Figure 1 shows such a plot for the van der Waals potential, n = 6 . For this case, using Eqs. (17) and ( I @ ,

(29)

E Q I K B = 28.6K/p,(amu)si, (30)

R, = 3.83s6 = 3.83[j~(arnu)C,(au)]”~

where s6 is evaluated with p, in atomic mass units and C6 in atomic units ( e 2 a i ) and RQ is given in Bohr atomic units. The ordinate and abcissa in Figure 1 are given in units of E~ and RQ, respectively. We find for n = 6,

Rb,min = 0.920RQ (31)

(RQ,rQ) ‘b..! XQ/2T . ..*.,. f , .......................

’*- ...... ........- p -

...................

I I

2.670RQ = [ 1 ( 1 + 1) ] l /4

Eb,max = 5.754EQ (33)

E c ( f ) = 0.0193[1(1 + l ) ] 3 / 2 ~ p . (34)

The long-range s-wave potential becomes, in these reduced units,

Figure 1 also indicates hQ/2n = 0.857RQ, where AQ is the asymptotic de Broglie wavelength for a collision kinetic energy of E ~ . We see from the figure that the (RQ, E ~ ) point occurs where the potential makes a “sharp” bend from asymptotic


TABLE I . CHARACTERISTIC MAGNITUDES FOR THE BREAKMWN OF WKB CONNECTIONS

Li 32 120 He * 34 I80 Na 44 19 Ne * 40 26 K 64 5 . 3 Ar* 60 5.7 Rb 82 1.5 Kr* 19 1.6 c s 101 0.6 Xe* 96 0.6

flatness to short-range attraction. The centrifugal barriers are outside of RQ for low partial waves.

Actual values for real species can be used to put real units on Figure 1 once the long-range C , is known. Table I shows parameters for alkali and metastable rare gas species, using the known alkali pair interactions (Tang ef a l . , 1976). The C6 coefficients were estimated for the metastable rare gas pairs from the known value for He* + He* (Krauss and Neumann, 1979), scaled using the measured metastable polarizabilities (Molof et a l . , 1974). Although the rare gas j = 2 metastables have small quadrupole moments, we neglect them here for the purpose of making these order of magnitude estimates of the smaller contributions near RQ of the long-range quadrupole-quadrupole potential, which varies as 1/R5; the full potentials should be used for more accurate estimates. Figure 1 is not useful for H atom collisions, since RQ is small enough for H2 that exchange interactions are dominant in determining R,, not the C , potential. For the heavier and much more polarizable systems in the table exchange interactions are expected to be negligible at RQ.

For the lighter species in Table I, laser cooling has achieved, or could in the future achieve, temperatures that are well below eQ. Cs traps have operated in the range from a few hundred p K to near 1 pK. Therefore, it is likely that collisions of many of the species in the table could be studied under conditions where quantum threshold effects are significant. Let us now turn our attention to examining the low T limit for particular types of collisions.

B. EXAMPLES OF COLD COLLISIONS

1 . Hyperfine Changing Collisions

We have already noted the extensive literature on collisions of ground state H atoms with hyperfine structure, the process in Eq. (1). These studies include the effect of a magnetic field and show that an unpolarized gas decays by fast spin- exchange collisions, and a polarized gas decays by the much slower spin-dipolar mechanism. Preliminary results of similar calculations have recently been reported for Na (Tiesinga et a l . , 1991) and Cs (Tiesinga ef a l . , 1992). Although


alkali systems are qualitatively similar to hydrogen, there are important differences that have yet to be explored experimentally and theoretically. There is evidence that fast hyperfine changing collisions in a Cs trap contribute to loss of atoms from the trap under some conditions (Sesko et al., 1989). Hydrogen collisions reach the s-wave T + 0 limit at experimentally realizable temperatures. Although the lighter alkalis may well be in this limit at TD (depending on the details of interactions), the heavier alkalis will begin to approach this limit only at the lower end of the temperature range obtainable by sub-Doppler cooling.

Recent calculations by Williams and Julienne (1991) have shown that the rate coefficients for spin-exchange collisions for hydrogen are sensitive to nonadiabatic corrections to the interatomic interaction. This is significant, since it indicates the sensitivity of some rate coefficients in the Wigner law limit to small terms in the Hamiltonian and subtle details of the calculation. The hydrogen system is unique in that the interaction potentials are known better than will probably ever be the case for the alkali systems, so that hydrogen provides a fundamental system for comparing theory and experiment. Williams and Juli- enne set up a close coupling formalism identical to that used by Stoof et a / . (1988) and have checked it on the hydrogen system using the very accurate new hydrogen potentials of Schwartz and LeRoy (1987), based on the calculations of Kolos et a/. (1986). These adiabatic potentials used fully converged variational Born-Oppenheimer potentials corrected by relativistic, QED, and diagonal mass polarization matrix elements. Williams and Julienne used the recommendation of Schwartz and LeRoy to use the bare nuclear masses to calculate the solutions to the Schrodinger equations, but the calculation neglected the nonadiabatic corrections due to mixing of the ground 'Zg state with the distant E , F ' 2 , double minimum state.

Although Williams and Julienne calculate identical rate coefficients to those of Stoof et a / . (1988) for the transitions that go by the spin-dipole mechanism, they find large differences, about 30%, for the transitions that go by the spin exchange mechanism. These differences are due to the neglect of nonadiabatic corrections and can be understood by using the approximate analysis provided by Stoof et al. (1988), based on their degenerate internal states (DIS) approximation (Parenthetically, this approximation has many of the features of the GMQDT analysis.) In this approximation the rate coefficient is proportional to the square of the difference of the elastic scattering lengths for the '2, and 3Z,, ground state molecular potentials. The difference between rate coefficients is just due to the difference in the '2, elastic scattering length, where Williams and Julienne find 0.45a" versus 0 . 3 2 ~ " reported by Stoof et al. By introducing approximations to include the nonadiabatic corrections to the scattering length, Williams and Julienne calculate spin exchange rate coefficients close to the results of Stoof et al . , who include the effect of these corrections following the method of Bunker and Moss (1977), by which the calculation with the adiabatic potentials is done using the atomic mass (nucleus + electron) rather than the


bare mass. This procedure at least approximately brings in the effect of the nonadiabatic corrections, although Wolniewicz ( 1983) does not believe that this procedure is as accurate for the eigenvalues as his method of using the bare mass and nonadiabatic corrections. We believe that the calculation of Stoof et al. is reasonably accurate in treating the nonadiabatic effects on the rate coefficients. In any case, it is interesting that even so thoroughly studied and fundamental a system as hydrogen still has questions that need to be carefully resolved. The conclusion is that great care must be taken in calculating the low temperature rate coefficients for alkali atom collisions, in order to be sure that the dependence on small uncertainties in the Hamiltonian parameters is understood.

Alkali collisions may be very different from hydrogen ones. Figure 1 and Table I show that the distance at which the potential becomes important is very much larger for alkalis than for H. The collision times will be many times longer because of the longer range and the smaller velocity. The DIS approximation works well for H exchange collisions because the short-range exchange collision occurs on a fast time scale compared to the precession time due to the hyperfine interaction. This condition may not apply to the heavier alkalis at trap temperatures, and new approximations may be needed for interpreting the results. Pre- liminary calculations by Williams and Julienne (199 1) for hyperfine changing collisions between Cs atoms show that the dipolar collisions of fully polarized atoms occur at a rate similar to that for H collisions, but exchange collisions occur at a much faster rate than in H. However, the results for exchange collisions are very sensitive to the potentials, as for the H case, and should not be taken too seriously until a very careful analysis of the potentials is done. Much work remains to be done in order to understand these ground state collisions in alkali atoms. It will be much more difficult to understand these collisions in an optical field, as will be evident after the discussion on Type 11 collisions in Section 111.

2 . Rare Gas Metastable Ionization

The collisions of rare gas metastable states can be studied using laser cooling methods. This should be possible not only in atom traps, but also with metastable atomic beams. Optical methods of beam manipulation should make possible the brightening and slowing of beams, and we will have to wait to see what kind of velocity control and experiments can be done. According to Table I, He* 3,Sl at its Doppler cooling temperature of 30 pK can be expected to be well into the quantum threshold range, where only the s-wave will make any appreciable contribution to the rate. Julienne and Mies (1989) used an estimated lower bound on the inelastic scattering length A , for the known long-range He* + He* potential to estimate a lower bound to the rate coefficient of ionizing collisions of unpolarized atoms. This gave results of about 5 x 10 ~ l o cm3s - I in the T + 0 limit and an upper bound of 10 -9 cm3s - I for the unitarity limit at TD. The rate coeffi-


cient for ionization of the spin-polarized 4He (j = 1, m = 1) isotope should be much smaller, possibly by several orders of magnitude. The rate coefficient for collisions of the spin-polarized 3He isotope should be even smaller, since this Fermion can collide only in p-waves (and higher odd waves) when spin polarized, and the p-wave rate coefficient vanishes in the T - 0 limit. This same result applies also to the two isotopes of lithum, 6Li and 'Li, which are respectively a Fermion and a Boson; the spin-polarized Fermion system should have a much lower rate of hyperfine changing collisions. In any case, magnetically trapped spin-polarized )He* metastables may be exceptionally stable relative to collisional decay. We also note that, if an optical field is on, excited state Type I1 collisions, to be discussed in Section III.B.3, will give rise to loss processes, the rates of which can be much larger than the T + 0 rate for the Type I collision.

The other rare gas '4 metastables will behave in a way similar to the He 3S,. Unpolarized systems should have very large decay rates relative to Penning or associative ionization, whereas the spin-polarized species will have reduced rates. The Xe* species is especially interesting, in that it has possibilities for use as a very precise optical clock (Rolston and Phillips, 1991). Table I shows that Xe* will be barely in the quantum threshold regime at its Doppler cooling temperature of 0.1 mK. We would hope that ionizing collisions of rare gas metastables will be a fruitful subject of experimental and theoretical studies in the future.

3. Pressure Shifrs

One other aspect of cold collisions that might be important for applications to time and frequency standards is the magnitude of pressure shifts in the transition frequency. Although such shifts should be extremely small at the very low densities that might be used in new atomic clocks, even small shifts could be significant given the very high level of precision desired; i.e., transition frequency measured to one part in lOI5 or better. A careful study of the hyperfine interactions during spin-exchange collisions of ground state hydrogen atoms has shown that pressure shifts are important in limiting the frequency stability of the cryo- genic hydrogen maser (Verhaar et a l . , 1987; Koelman et a l . , 1988b). A very recent study by Tiesinga et al. (1992) has used quantum scattering calculations to calculate the line shift and line broadening coefficients for the Cs clock transition due to spin-exchange collisions of two cold Cs atoms. The predicted shifts are large enough to be an important factor in limiting the anticipated accuracy of Cs fountain clocks. No calculations have yet been reported for pressure shifts that might affect a Xe metastable atom clock (Rolston and Phillips, 1991). This subject of pressure shifts is one that will require careful attention in the future.

4 . Surface Scattering

We will conclude this section by mentioning the subject of collisions of ultracold atoms with surfaces. The interesting question is whether the atoms will stick to


the surface as T + 0 or undergo perfect quantum reflection. This question can be addressed by using the GMQDT analysis in Section II.A.2 to examine the wave function in the one-dimensional potential that characterizes the interaction of an atom in normal incidence to the surface. The short-range amplitude of the inner zone wave function near the surface is proportional to C ( E ) - ’ . Since C(E)-* decreases as k when E is sufficiently smaller than E , , the probability of surface interactions that lead to sticking must also decrease towards zero as k, thereby leading to a reflection probability that increases towards unity as E 4 0.

The experiments of the Amsterdam group (Berkhout et al . , 1986, 1989) demonstrated a strong increase in reflection as T decreased to about 100 mK. But very recent experiments by Doyle et al. (1991) show an increase of sticking as T is lowered by three orders of magnitude to about 100 p K . The experimental behavior between 1 K and 100 p K is consistent with one of several model calculations by Goldman (1987), who demonstrated the strong sensitivity of the sticking probability to the potential parameters. Since E , / K ~ is above 1 K for the H + He surface interaction, this confirms our warning that the actual approach to the T + 0 limit depends on model details in such a way that the threshold law behavior may not be observed until E is possibly orders of magnitude below E,.

The atom-surface interaction varies as -C31R3 at moderate range, and as - C4/R4 at longer range, where retardation corrections are important. Equation (17) can be used to estimate E , for a 1/R3 potential:

where mass p is in amu and C , is in atomic units (e2a%). This is only a very crude estimate, since R, may be in the retarded region where the potential varies as 1/R4. The main point is that, except for light atoms with weak surface interactions, E , will be in the p K range or less, and extremely low T, possibly a nK or less, would be required to observe quantum reflection of heavy atoms incident on typical surfaces.

III. Cold Collisions in a Light Field

A. FORMAL THEORY

1 . Nature of Problem

Collisions in a light field are radically different from the conventional Type I collisions we have been considering up to now. This is because the dissipation due to excited state spontaneous emission can dramatically affect the way in which we describe these collisions theoretically (Julienne and Mies, 1989) and also strongly modify the effective collision rate coefficients. Such effects have


been widely discussed in the context of collisional losses from alkali atom traps (Julienne et al., 1988, 1990; Gallagher and Pritchard, 1989; Julienne and Vigut, 1991); associative ionization in Na atom traps (Julienne, 1988; Julienne and Heather, 1991; Gallagher, 1991); and interruption of laser cooling (Smith and Burnett, 1991). The long time scale of ultracold collisions is a consequence of the very long distance scale associated with the normal resonant dipole-dipole interaction, which is of the form (Meath, 1968)

V(R) = +aF~y()i/R)~, (37)

if retardation corrections are neglected. Here X = A/27r, where A is the wavelength of the atomic transition, y is the natural decay rate of the excited state, and a is a constant on the order of unity. As laser cooled atoms have kT on the order of Ay or less, the natural distance scale of the excited state potential is R = X. A near-resonance cooling laser excites the atom near R = X, which is much larger than R, for the ground state potential, so that amplitude changes in the inner zone ground state wave function do not affect the collision. In addition, since typical temperatures from laser cooling are much larger than E ~ / K ~ for the excited state potential (see Eq. (36)) and since the excited state l/R3 potential is effective in capturing many partial waves with 1 >> 1, the motion in both the ground and excited state potentials is essentially semiclassical and can be well described by WKB wave functions. As long as the temperature remains above the recoil temperature, the atomic de Broglie wavelength will be smaller than X. These characteristics will allow us to develop classical path approximations to the collision dynamics. The most significant new feature of these Type I1 collisions is the dissipation due to the very long time scale of the collision relative to the natural lifetime 7A = I/y of the atom.

To discuss the novel phenomena that may occur in Type I1 collisions at sufficiently low temperatures, it is useful to define the quantity

X

7.4 v,v = -.

Thus, v, corresponds to an atomic velocity where one optical wavelength is traveled in an atomic lifetime. Normal collisional physics is the regime where v >> v,; in this regime dissipation from the excited state is unimportant and there is little distinction between Type I and Type I1 collisions. However, if we are in a temperature regime where v << v s then this has considerable ramifications on the collisional process.

For alkali atoms, such as cesium and sodium, the atomic temperature at the Doppler limit corresponds to a velocity significantly below v,, and so we would expect novel phenomena related to dissipation from the excited state during the collision. In other words, the distance traveled in one atomic lifetime, R, = V T ~ ,

is now much less than X. For the collisions to be considered in Sections 1II.B and IILC, this means that as the two atoms collide they may, under the right


conditions, be excited and reemit many times. The production and survival of the excited states is central to the mechanisms of trap loss and associative ionization to be presented. A general theory of this production and survival can be based on the formal approach we give later. We can also give explicit results in certain limits. The first of these corresponds to the case where the laser is sufficiently detuned from resonance for us to use a linear response approach (Section 1II.B). When the laser is close to resonance a good deal of optical pumping can take place as the collision complex forms. In that case the method of Band and Julienne ( 1992) can be used to consider saturation of the long range pumping.

In Section III.D, where we analyze the effect of very long-range collisions on the laser cooling process, the fact that R, << X means that the collision can be regarded as quasi-static; i.e., we can assume that the interatomic separation, R , may be considered as fixed over several atomic lifetimes. Therefore, the very long-range collisions in an atomic beam or trap can be treated (at least in a binary collision model) as a series of quasi-static nearest neighbor interactions, with appropriate angular and density averages.

This model breaks down as the atoms are cooled towards the recoil limit, where the de Broglie wavelength is of the order of X. In this regime, which is now experimentally realizable (using polarization gradient forces, Dalibard and Cohen-Tannoudji, 1989, and others) where low intensity lasers are employed, it will be important to consider the pumping time r,, where r, >> rA , so that the velocity limit

X v, = -

7, (39)

could signal some interesting physics. We therefore require a formalism that allows dissipation from the excited state

at the same time as treating the atomic motion, internal atomic structure, and collisional interaction. To do this in the next section, following the theory developed in Smith and Burnett (1991), we shall introduce a formal Hamiltonian approach for considering the collision between two like atoms in a light field. The form of the Hamiltonian assumes that the collision is sufficiently long ranged that it is dominated by the dipole-dipole interaction, rather than the higher-order (in R ) van der Waals interactions (on at least order 1/R6) , which were considered in Section 11.

2 . Hamiltonian Description of Atoms in a Light Field

We shall begin with a Hamiltonian description of two atoms colliding in a standing-wave laser field. In this approach, both the internal state of the atoms and their translational degrees of freedom are treated completely quantum mechanically. The spontaneous emission is given by coupling the atoms to a bath of

160 P.S. Julienne, A.M. Smith and K . Burnett

reservoir field modes, with the laser fields being coherent state field modes. This is also the approach adopted by Trippenbach et al. (1992) in their formalism. The internal structure of each atom may consist of an arbitrary system of energy levels. However, for simplicity in what follows, we shall write out the exact form for a two-level system only:

H = (Hi + H?)TRANS + (Hi + WINTERN (40)

Introducing the Pauli matrices to represent the internal atomic variables, which satisfy the equal-time commutation relations,

+ HFIELD + (H1.t + H ~ . ~ ) I N T E R .

[(T;, q+] = - [(T;, v;'] = 2 ~ ~ 6 , ~ [a,+, a;] = - 2 ( ~ , + 6 , ~ , (41)

then it may be shown that the terms in the Hamiltonian are of the form

P: P: (Hi + &)TRANS = - + -

2 M 2M'

In these equations arA and a& are the photon annihilation and creation operators, respectively, for a plane wave of wave vector k and polarization EA(k). They must satisfy the commutation relation

V is the volume of quantization and w k is the transition frequency associated with the plane wave. Since the atoms are identical we give them the same dipole frequency oo, the same mass M, and magnitude of dipole moment [dl (however their respective directions of electric dipole moment d, may be different). The translational degrees of freedom for the atoms are described by operators representing their momenta 2 and position Xi.


The Heisenberg equation of motion, for any system operator Y, may be written as

. I = -lHTRANS + HINTERN + H F + H I N T E R , Y l . (45)

Operator equations for the internal atomic variables and the field modes are then derived using Eq. (45). The field modes solutions may be written as integrals and substituted in the internal atomic equations. A set of complicated equations is obtained, but these may be simplified and all integrals calculated by making the secular approximation. This essentially corresponds to making a Markov assumption for the interaction between the atoms and the reservoir modes. This approximation is used in nearly all of quantum optics and is certainly expected to be valid until the temperature approaches the recoil limit.

h

3. Operator Optical-Bloch Equations

After some extensive, but straightforward, algebra we obtain the equation for s,+ (the other equations may be obtained by an identical procedure; for full details, see Smith and Burnett, 1991):

s: = -[V$, - ih - v$Js: - - is, + [:(L + i K ( R , ) ) + 2iwo] 2M

(46) 1 2

+ iz(t)si - - s ; s ~ [ a ( R ) + a(R‘)

1 - i(P(R) - P(R’))I + 5 G ~ i [ a ( R ) + a@’) + i(P(R) - P(R’))I.

This is still an operator equation, but to obtain it we have taken a trace over the initial state of the radiation field and taken nondiagonal matrix elements of the position eigenstates IX,). The internal operators are now defined by

s;, = (v;,) (47)

where

) = I$, [OI)~X,)lX,), (= ($,loll(xlI(x;l

and

(48)

The choice of Eq. (48) corresponds to an initial field state where a standing wave is directed along the k, axis, with polarization A , , and that all the other radiation modes are in the vacuum state.

1 akA(o)/$, lo]) = ,$I$? [0 ] ) [6k ,k , + 6 k , - k ~ 1 6 k , A ~ ~

162 P.S. Julienne, A.M. Smith and K. Burnetr

The spread in the quantum uncertainty in the wave packet of each atom (as opposed to classical diffusion from spontaneous emission) is characterized by the vector quantities R , = X I - Xi and R, = X2 - X;. The vectors R = X I - X; and R' = X, - Xi give the separation between the two atomic wave packets.

The remainder of the parameters in Eq. (46) are as follows. First, y is the standard one-atom spontaneous emission term, given by

The variable TI ( t ) represents the interaction of the standing-wave laser field with atom 1, where

T,(t) = ~d l*EA, (k l ) [Eoe + E$eiwl'](cos(k ,.X ,) + cos(k ,ex',)) (50)

= T ; e-iwlr + T : elwl'.

The constant Eo has dimension s - I and is given by

where + is determined by the intensity of the laser field and is normally given a phase to make Eo real. It is also assumed that it is possible for this two-level atom to renormalize the atomic energy levels to allow for the Lamb shifts.

The effect of the spread of the wave packet can be seen quite clearly in the equation for s:. First it creates a gradient dependent on position, due to the translational terms in the Hamiltonian. However, the quantum uncertainty in the atom's position also causes a change in the damping rate of the atomic levels, given by the term

sin koR , K ( R , ) = ( 1 - (dl*Rl)2)- -k ( 1 - 3(d,.kl)2)

koR 1 ( 52)

In a classical path description we have R , + 0, which gives K ( R , ) + 2/3, so that the overall damping rate is given by Eq. (49), as in semiclassical analyses.

The remaining variables, a ( R ) and P(R)(a(R') and P(R')), represent the retarded interaction between the two atoms. The a(R) term describes the process of two-atom spontaneous emission, i.e., where atom 1 emits a proton that is absorbed by atom 2 and is then subsequently spontaneously emitted, and is given by (Meath, 1968)


A A sin k,R (I - RR)- + (I - 3RR)

(53) koR

The P(R) term corresponds to a shift in the energy levels of one atom because of the presence of the other atom. It is therefore equivalent to a two-atom interatomic potential and can be written as

The potential Eq. (54) may be obtained using perturbation theory (see McLone and Power, 1965; Meath, 1968) and is relativistically correct for the dipole- dipole interaction. For R < X, ( k , = I l k ) the 1/R3 term dominates, and we obtain the form of the potential in Eq. (37). However, as mentioned earlier, Eq. (54) does not include the van der Waals potential and other higher-order interactions.

So far we have made the restriction of two-level atomic systems. The evolution of the colliding pair is described fully quantum mechanically in three dimen- sions. In order to solve these equations approximations have to be made. We should emphasize that a full three-dimensional treatment is out of the question on computational grounds. In the rest of the article we shall deal with simplifying approximations.

The first of these is considered in Sections III.A.4 and III.B, where the effect of the cooling process on the atomic velocity distribution can be considered independent of the collision processes. The collisions can then be treated as individual events. It can be further assumed that any excitation during an individual collision can be dealt with in a rate equation fashion. This will be valid if reemission and double reexcitation during a single collision is negligible. We shall also show how it is possible to use optical-Bloch equation to treat local path corrections.

In Section 111. B we will give simple physical models for describing collisional processes that cause loss of atoms from an optical trap. In Section 1II.C we will show how light can be used to manipulate the dynamics of ultracold collisions and to do photo-association spectroscopy. Finally, in Section III.D, we shall make a local classical path approximation and assume that the collisions are sufficiently weak that a single collision is insignificant in comparison to the cooling process. It is then the cumulative effect of many collisions that may cancel out the cooling at sufficiently low temperature.


4 . Application to Trap Loss Processes

By using the equation of motion for the atomic operators in the presence of the radiation field we have found an effective equation for the pair of atoms driven by the laser field and exchanging excitation. We would now like to show how the formal two-particle density matrix can be used to treat individual ultracold collisions. We shall specifically address the calculation of radiative escape (RE) due to an excited atom acquiring kinetic energy during a collision. This process will then be described in more detail in Section 1II.B.

A general theory of this process is, of course, extremely complex. We shall first address the case of linear excitation of the collision pair. We shall speak of excitation during a collision since the process is a significant source of trap loss only when the laser is detuned to the red and excites atoms entering a collision. We shall assume that the excitation may be described using perturbation theory: the population excited will then depend linearly on the intensity of the driving laser. We start with the equations of motion for the pair in the form obtained in the last section.

If we suppose that the effect of the driving field can be handled perturbatively we can start with the zeroth order approximation to these operator equations. To get this we simply put si = - 1 in Eq. (46). We can then see that the equations of motion can be decomposed into two equations: one for the symmetric and one for the antisymmetric excitation of the pair (singlet and triplet states). The symmetric equation takes the general form

where

s: = s: + s:. (56)

In writing Eq. ( 5 9 , we have assumed that the spatial extent of each atomic wave packet is less important than the uncertainty in the distance between the two wave packets. This means that we ignore R , and R 2 , but retain R and R’. The decom- position in Eq. (55) effectively diagonalizes the two-atom problem for the case of two state atoms. For real atoms, i.e., atoms with degeneracy, we need to assume that we are deep enough into the collision to use a single potential, i.e., an adiabatic approximation for the molecular problem.

The potential and damping terms in this expression are what one would expect from a simple Dicke style analysis (see Sargent et a l . , 1974). Following Eq. (55) we can make the identification T,(R) = y + 2a(R) . In addition, for R < )i, the form of the potential, V,(R) = P(R), is exactly that of the usual long-range potential written in Eq. (37).

Assuming that we can make this single potential approximation we proceed to project the equation of motion onto the relevant wave functions; i.e., singlet or


triplet states. By this we mean, of course, the symmetric and antisymmetric combinations along with the explicit translation states (at this point we shall keep to the position representation). We then obtain the following general equations of motion:

- V,(R’))p,,k + driving terms, j , k = e, g.

Here, e refers to the excited state of the pair and g to the state with both atoms unexcited. The ground state potential is assumed to be zero so we have V,(R) = 0 as well as T,(R) = 0. The form of the driving terms can be easily obtained from Eq. (50) and give a nonzero contribution for p , and pne.

To proceed we now have to solve these inhomogeneous equations. This solution can be analyzed in various ways. We shall first describe the distribution of particles on the excited surface. We shall also discuss the result for the rate of transitions to states of specific final relative momentum of the pair of atoms. These occur when the pair reemits a photon. We shall see in this way that one has to be quite careful in using the excited state distribution in a calculation of other collision processes that depend on the excitation that is present. In Smith et af. (1992b) we give a fully quantum-mechanical analysis based on Eq. (57).

In the semiclassical limit this gives a rigorous basis for the distributions used by Gallagher and Pritchard (1989) and Julienne and Vigut (1991), which are discussed in Section 111. B. We assume that the dependence of the imaginary part of the potential on distance may be ignored (T,(R) = re). This gives a quite accurate and useful result rather easily.

We use WKB wave functions and the method of stationary phase. In this way we obtain the following result for the distribution on the excited state:

where A is the detuning, ci is the initial kinetic energy, and S 5 ( R 2 , R , ) is the survival probability from R , to R 2 , given by

S14R2, R , ) = exp[ & I R 2 -1, T,dR’ R I &(R‘) (59)

166 P.S. Julienne, A . M . Smith and K . Burnert

The positive form of the exponential is required when R 2 < R , and the negative exponential when R z > R , . We now use the reduced mass p of the pair of colliding atoms, as this allows us to connect more closely with the analysis in Section 1II.B. The result, Eq. (58 ) , for the excited state distribution is the same as that used by Gallagher and Pritchard (1989) and Julienne and ViguC (1991). It shows that one should use the Franck-Condon principle to determine the excited state distribution, but that the point of excitation is blurred by the presence of spontaneous emission. We do, however, have to be careful in interpreting this distribution. This is relevant to the calculation of processes that go on after excitation to the upper surface.

As a very important example of a two-step process we shall now consider the transition to states of different final kinetic energy of the pair that takes place due to reemission during the collision. Full details of this calculation are given in Smith et al . (1992b). In the same semiclassical limit the result is as follows:

- I - ”’* dV, Sb.(R,, R , ) , (61)

2

( V , ( R , ) - hA)* + r2 4

[ d R I R i

where C! is the Rabi frequency, and the combined effect of the delta functions in the original expression is to fix R , so that

E/ - E , = V,(Ri) - U R 2 ) . (62)

We can see that this result follows closely the form of the excited state distribution, as one would expect. It may also be observed that the smearing of the excitation process does not violate conservation of overall energy (as one would hope): the excited distribution has energy conservation “hidden” in it!

These results for the excitation during a collision will be valid only in the region where one can use the ordinary quasi-static theory. We should emphasize that the theory given so far cannot handle the case of strong coupling between the states. A recent study of Band and Julienne (1992) has shown how optical- Bloch equations can be used to handle near-resonance excitation. This analysis demonstrates that the wing excitation does in fact give a reasonable description of the excitation rate.

B. TRAP Loss PROCESSES

I . Rate Equation Theory

A good example to illustrate the novel physics of ultracold excited state collisions is the trap loss process for alkali atom traps, by which atoms are heated and ejected from a trap by either of two processes:


A(S) + A(S) + Aw + AT(e, 2Pj/2 + *S) + A(S) + A ( S ) + fiw‘ + AE + A(’Pl/2) + A(S) + AEFS.

(63) (64)

The colliding ground state atoms are excited to an upper molecular state e , which connects adiabatically to 2&2 + 2S separated atoms. For small detunings A ( = o - w o , where wo is the resonance frequency), Eq. (37) shows that the excitation occurs at extremely long distance, R = X. The atoms are slowly drawn together on the attractive potential curve of the state e , during which time the scattering flux on the excited state may decay via spontaneous emission. Once the atoms are close enough together, hot ground state atoms can be produced by either of the preceding two mechanisms, and the hot atoms are ejected from the shallow trap.

In the first radiative escape (RE) mechanism, kinetic energy AE picked up on the excited state is transferred to the ground state atoms when a red shifted photon o’ is emitted during the collision: ho = Aw’ + A E . In the second fine structure (FS) changing mechanism, an amount of kinetic energy equal to the 291i2 - 2512 splitting, is picked up due to molecular interactions at small internuclear separation R. Julienne et al. (1988) showed that large loss rate coefficients, di- minished by excited state decay, were possible for the RE mechanism in Na atom traps. Gallagher and Pritchard (GP) (1989) showed that the escape rates due to the FS mechanism were dominant over those due to RE for most alkalis. They also proposed a simple model for calculating the loss rate based on weak field perturbation theory for exciting a quasi-static distribution of ground state atoms. Although this model contains in an elegantly simple way the essential physics of ultracold collisions, it contains numerous oversimplifications of detail. Julienne and ViguC (JV) (1991) generalized the GP model by introducing the specific molecular mechanisms of FS and RE loss for the various alkali species and showing that it is necessary to incorporate the role of the relative angular momentum 1 of the colliding atoms.

Although the formulation of GP and JV has the appearance of a quasi-static theory, it can actually be derived as a limiting case of the more fundamental dynamical theory in III.A.4 and thereby be connected more closely to the conventional scattering viewpoint. The rate of FS or RE transitions per unit volume per unit time in the JV formulation of the theory is written in the following rate coefficient form:

where N is the ground state density, v = hk/p is the asymptotic velocity for reduced mass p and kinetic energy E , , hw+ is the intensity of the light, d, is the ground state degeneracy (= 2 when hyperfine structure is ignored), and the symmetry factor of 1 /2 accounts for homonuclear symmetry. The summation


extends over all contributing attractive upper states e . The probability fiL that the FS or RE transition occurs at small R = RTL once the atoms are drawn together is PTL(e, I), which was calculated for the various mechanisms by JV and was shown to be nearly independent of incident collision energy over a range comparable to room temperature k,T.

The novel aspects of ultracold collisions are contained in the excitation- survival probability PE,(R, e , 1, A, +) in Eq. (6% which gives the total probability that the upper state e excited at rate G , ( R ’ ) at some R’ > R by light with detuning A and intensity hw+ will survive without decay during motion from R’ to R:

The differential element dR‘lv , which gives the amount of classical time spent absorbing light near R ’ , depends on the ground state trajectory for E , and 1. The survival factor, S’,<(R, R ‘ ) , for moving from R’ to R on the excited trajectory with initial excited state kinetic energy E , , is exactly that in Eq. (59). Therefore, PEs in Eq. (66) depends on both the ground and excited state trajectories.

Julienne and Vigue (1991) showed that an ambiguity in the choice of initial energy E , for calculating the excited state trajectory can lead to uncertainties on the order of a factor of 2 in the calculated rate coefficients for T near the Doppler cooling temperature. One obvious choice is to pick E , so that the excited state has a total energy equal to E , + hw. This is in accordance with the usual conservative Hamiltonian description of a collision in a radiation field (Julienne, 1982; Julienne and Mies, 1984b), where the radiative distorted wave approximation gives a transition probability proportional to the Franck-Condon factor ~(@&&,)[@~(&, + hw))l?. Such a choice satisfies the classical Franck-Condon principle (FCP) at the Condon point R c . The classical FCP states that the local kinetic energy and momentum in the semiclassical picture do not change when the transition “occurs” at some point R . However the classical FCP is violated by the energy conserving trajectory when off-resonant excitation is permitted with R # R c .

Julienne and ViguC proposed an alternative choice of initial excited state energy, E , = E ~ ( R ’ ) , which satisfies the local FCP at each R at which off-resonant excitation occurs. Such a choice prepares an excited state wave packet with a spread of energy associated with the lifetime of the upper state. Although this choice appears to violate the conventional energy conservation, this is not the case. The finite lifetime and broadening due to excited state decay permit a spread in the excited state distribution, and the problem can be resolved within the framework of the general quantum theory of motion, as was shown previously in Section III.A.4. The form of Eq. (66) is immediately obtained from Eq. (58) in Section III.A.4 by making the integration over intermediate energy E ’ , where the delta function in Eq. (58) specifies E , exactly in accordance with


the local FCP while total energy conservation is maintained. A more detailed discussion of this point is made in Smith, et al. (1992b).

The rate expression, Eq. (65), looks like an ordinary rate expression for a process that involves two sequential steps: first, the probability PEs of the excited state being formed and surviving to R T L , and second, the probability of the trap loss transition itself. Both GP and JV write the excitation rate in terms of the weak field photo-absorption cross section,

(67)

where the R’-dependent Lorentzian line shape function xe is normalized to unity at the peak, 6(R,) = 0:

GAR’, A, 4) = a,(R’, A M = a$ak(R’ , AIxAR’, A M ,

Here 6(R’) is the detuning from the molecular resonance frequency w,(R’) , given by the difference between the upper and lower potentials:

hw,(R’) = hwo + V,(R’) - V,(R‘) = hwo + 6,”(R’), (69)

or

6(R’) = fiw,,,(R’) - hw = 6,(R’) - A . (70)

In the usual quasi-static picture, absorption only occurs at the Condon point Rc where the molecule is in resonance with the light, 6(Rc) = 0. But in the ultracold collision, off-resonant excitation, when R is much smaller than the Condon point, is also important, since the survival factor is more favorable in the integral in Eq. (66).

Band and Julienne (1992) have recently given a classical path optical-Bloch equation treatment of the trap loss process. In such a formulation, the excitation rate of GP and JV in Eq. (66) is shown to be equivalent to using the local steady- state, weak-field solution to the optical-Bloch equations at each R‘:

(71) G:s(R’, A , 4) = 2RIm[p,,(R’)] = T,p.,(R’)

where the Rabi frequency R is given by I12

= (EQ) (73)

Band and Julienne (1992) show that the optical-Bloch equations can be formulated in such a way as to include the motion on both the ground and excited state trajectories. Numerical solutions of these equations give per(RTL), the density


matrix at RTL, which corresponds to PES(RTJ in the JV theory. Good agreement between the rate coefficients calculated from the JV formula and the optical- Bloch equations was found for Cs trap loss. The semiclassical optical-Bloch equation method has the advantage that saturation can be treated when the field is no longer weak. The method could be generalized to include additional states and hyperfine structure. The disadvantage to the semiclassical optical-Bloch equation method is that a single excited state trajectory must be chosen, which cannot represent the spread of energies within the wave packet, as in Eq. (58). Ultimately, the full quantum density matrix treatment is needed to properly describe the effect of dissipation in strong and weak fields. The hope is that simple models such as Eq. (65) will prove to be practical approximations.

The probability PEs can be reduced to a simple expression in two limits. First, if the detuning is large compared to rc, the line shape function xe is very sharply peaked near the Condon point. In this case the slowly varying S and l / v factors in Eq. (66) can be pulled out and the integral over the line shape can be done using the form Eq. (72). After transforming the integration over R’ to a unit integral over a normalized Lorentzian by using dR’ = (dR’ /dhS)dhS , we find the pleasing result

The factor multiplying the survival factor is just the semiclassical stationary phase evaluation of the radiative distorted wave approximation mentioned earlier, equivalent to the Landau-Zener curve crossing probability for a field- induced transition at R c between the ground and excited field-dressed potentials. It is gratifying that Eq. (66) gives this conventional result for a “sharp” crossing.

The other simple limit occurs when the velocity is much larger than v, defined in Eq. (38). The integral defining PEs is then almost completely determined by its long-range part R >> X, where the parameters take on their atomic values. It is a simple exercise to show that in this weak-field high-temperature limit

N * PES = P e r = - ” (75)

where pee is the excited state density matrix for a single free atom and N * is the excited state density. Then the rate of trap loss events is just

Rate = K N 2 = K * N N * , (76)

where K* = K/PEs is the conventional excited state rate coefficient; that is, Eq. (65) without the PEs factor. This corresponds to the conventional view that the excited state is prepared as an independent atom, and there is no excited state decay during the fast collision inside R = X.


2 . Alkali Metal Trap Loss

The specific molecular mechanisms of FS and RE trap loss for the alkali metal species Li, Na, K, Rb, and Cs were determined by Julienne and ViguC (1991). The potential curves of all the alkali metal dimer molecules are qualitatively similar. The long-range molecular potentials and transition dipoles for the states that diagonalize the electronic plus spin-orbit Hamiltonian are also known (Dash- evskaya et al . , 1969; Movre and Pichler, 1977, 1980; Bussery and Aubert- Frecon, 1985). There are five attractive potentials correlating with *& + *S,,, separated atoms. These states, in order of increasing attraction, have I , , 0; , 2, , l,, and 0: symmetry labels, where the integer gives the projection of (electronic + spin) angular momentum on the internuclear axis. In the dipole approximation, these states have decay rates re that are respectively 0.54, 2 , 0, 1.21, and 1.33 times the atomic decay rate y. Because of this variation, the survival factor in Eq. (59) is very sensitive to the state e. The 2 , state plays a special role, since retardation corrections to the forbidden dipole transition rate of this state allow it to decay to the ground state with a decay rate r ( 2 , ) = 0.2y(R/X)* , which decreases rapidly when R < X. Thus, the 2 , state can be excited near its Condon point R,, but with excellent survival probability between R , and RTL.

Julienne and ViguC (1991) verified by numerical quantum scattering calculations the mechanisms of the FS transitions proposed by Dasheveskya (1979). Only the 0: and 2 , entrance channel states give a nonnegligible contribution to the FS transitions. Both of these states can lead to an FS transition through a Coriolis mechanism, in which the spin projection becomes uncoupled from the rotating molecular axis during the short-range part of the collision. This mechanism is dominant at room temperature for the light species Na and K. The 0: entrance channel can also lead to an FS transition by spin-orbit mixing at a short- range curve crossing. This mechanism is dominant at room temperature for the heavier species Rb and Cs. The probability PTL of the FS transition is nearly independent of 1 for the spin-orbit mechanism but decreases approximately as 1' for the Coriolis mechanisms. Since the range of partial waves 1 that contribute to the sum in Eq. (65) decreases with increasing T, the average probability of FS at low T by the Coriolis mechanism cannot be extrapolated from room temperature experiments. Such an extrapolation caused the probability estimate of Gal- lagher and Pritchard (1989) for FS transitions in Na traps to be an order of magnitude too large.

Julienne and ViguC calculated the FS probabilities at low T of the spin-orbit and Coriolis mechanisms for the species Na, K, Rb, and Cs. They also calculated the RE probability for all species, including Li. Because the fine structure splitting is so small in Li, FS transitions will not lead to loss from traps deeper than about 250 mK. The RE probability is also greatly reduced for Li, since the emission occurs at small enough RTL that the Hund's case (a) molecular coupling


scheme must be used instead of the asymptotic Hund’s case (c) coupling scheme discussed earlier. The Li RE transitions are very weak in the Hund’s case (a) coupling scheme.

Figure 2 shows the overall trap loss rate coefficients versus T calculated by JV for all the alkali metal species. These are twice K in Eq. (65), since two atoms are lost per FS or RE event. The results are for a detuning of one linewidth to the red of resonance and for a total power of 10 mW/cm*. The RE contribution to the rate was calculated for a trap depth of I K . The survival factor, Eq. (59), was calculated using the classical FCP to choose the excited state trajectory. The figure also shows the good agreement with the points taken from the experimental trap loss rate measurement of Sesko et a / . (1989) for a Cs trap at 300 pK. The calculations find that the spin-orbit mechanism for the 0: entrance channel gives the dominant trap loss process for Cs, but the Coriolis mechanism for the 2, entrance channel is favored for Na, K, and Rb. The latter is true because of the excellent survival factor of the 2, state for the reasons discussed previously. Radiative escape is smaller than FS in all cases except Li. For Cs the FS rate is only twice the RE rate. Since RE scales as D -516 for the species other than Li, and as D - for Li, where D represents trap depth, the RE contribution to the rate increases as the trap depth decreases. For small enough trap depths, ground state hyperfine changing collisions may become the dominant loss rate (Sesko e t a / . , 1989).

Figure 3 shows the effective excited state rate coefficient, K* = K ( N / N * ) , for Cs FS transitions over seven orders of magnitude of T. The experimental data at 300 K and 300 pK are also shown. The dashed line shows the conventional K * ,

Rap Loss (FS + RE)

FIG. 2. Trap loss rate coefficients due to S + P collisions for all alkali species calculated by JV for a laser power of 10 mW/cm2 and a trap depth of I K . The measured loss rate (Sesko et al . , 1989) for a Cs trap is indicated by the points.


FIG. 3. Effective excited state rate coefficient for FS transitions during S + P collisions of two Cs atoms. The dashed line labeled C is the conventional rate coefficient calculated by ignoring excited state decay. The curves labeled CEC and CFCP calculate low T survival factors using conventional energy conservation and the classical Franck-Condon principle, respectively. The arrow indicates the velocity v , . The curve labeled FS + RE adds the RE rate to the FS rate calculated using the CFCP. Measured rate coefficients are indicated by the points for 300 pK and 300 K .

Eq. (65) with PEs = 1, which is just the Langevin capture rate coefficient for the 0: entrance channel multiplied by the FS probability for this channel due to spin- orbit mixing. The figure shows the obvious departure of the actual rate coefficient from its high-temperature form when v becomes less than v, and pEs is influenced by molecular excitation-survival. The small probability of survival on the excited state during the collision causes a dramatic drop of FS rate below about 100 mK. The two curves show the consequence of the two choices of initial kinetic energy for calculating the survival factor. The upper curve corresponds to the choice based on conventional energy conservation, whereas the lower one is based on the classical FCP.

Since a predictive theory is available that shows there should be a wide variation in magnitude and mechanism for trap loss rate coefficient in alkali species, it is very desirable that new experiments be carried out to test the theory. It would be especially useful to measure the variation of loss rates on laser detuning and intensity to see how realistic the predictions of the theory are. In particular, the theory neglects hyperfine structure, which could be important. Band and Juli- enne ( 1992) use the semiclassical optical-Bloch equation method to verify the near linearity of trap loss rate with laser power observed by Sesko er al. (1989) for a Cs trap, although the on-resonance Rabi frequency Q of the experiment spanned a range where it was larger than the atomic saturation value. The lack of saturation for Cs arises because most of the excitation is off-resonant excitation occurring well inside R c , as this process is favored by improved survival. Experiments on a Na trap (Prentiss er al., 1988) found a nonlinear power depen-


dence. Although this experiment has large error bars and has been criticized by Sesko et al. (1989), the existence of two very different mechanisms of FS loss in Cs and Na traps could lead to different saturation properties. Additional theoretical and experimental investigation of the power and detuning dependence of trap loss rates for all alkali species is certainly desirable.

3. Rare Gas Metastable Trap Loss

In Section II.B.2 we examined the Penning and associative ionization of trapped rare gas metastable atoms in the absence of light. We showed the possibility of experimentally reaching the regime where the collisional rates exhibited Wigner threshold law behavior with only s-wave collisions. This regime should be obtainable for the lighter rare gases, especially He. We also showed that fully spin- polarized gases should have much lower ionization rate coefficients than for an unpolarized gas.

We wish to call attention here to the fact that the rate coefficients for collisional processes may be dramatically modified if near resonant light is on, such as would be provided by a cooling laser. The reason is that the ability to excite an atom during the collision can greatly increase the available phase space that can contribute to the collision. Excitation to the upper state, with the resonant dipole-dipole interaction varying as 1/R3, permits many more partial waves I to contribute, instead of just s-waves in the ground state potential. For the unpolarized gas, the probability for a short-range ionization event, PTL(RTL), is essentially unity. If the intensity of the light is large enough that the cooling transition is saturated, very large rate coefficients are possible. Enhanced loss rates are even likely for a polarized gas, since loss by RE transitions is always possible, and, especially for the heavier rare gases, FS transitions also. So the presence of light could strongly modify the stability of a polarized gas relative to collisional destruction.

The case of He 3S metastable ionization offers an instructive example. The upper 3P state of the cooling transition has a long lifetime of 100 ns. If an unpolarized 3S gas at 100 pK is assumed, excited state decay during the collision is unlikely since the rms velocity is comparable to v,. A rough estimate of the trap loss rate coefficient is

where a unit ionization probability is assumed for PTL. The mean factor now should include not only the effect of excitation and survival but also the role of molecular degeneracy; that is, not all paths lead to ionization (e.g., the quintet states). Taking I,,, = 5 from the Langevin capture range of the excited state potential gives


Thus, a loss rate coefficient in the range lo-’ to cm3s-’ is possible if the intensity is high enough to saturate the excited state population, depending on the details that determine g . Such a value is over an order of magnitude larger than the s-wave unitarity limit for the ground state collision in the absence of light. This illustrates how Type I1 collisions with light on can be very different from Type I collisions with light off.

c . OPTICAL MANIPULATION OF COLLISIONS

An important new feature of ultracold collisions is the ability to manipulate the rate coefficients for various processes by the light used to produce the excited state. This is different from an ordinary collision, where the light is used to prepare the excited states of the independent, separated atom(s), which then go on to collide. In the ultracold regime, the “preparation” is an intrinsic part of the collision dynamics and gives extra leverage in influencing the ultimate out- come of a collision. This offers the prospects of new kinds of collisional spectroscopy and new ways to manipulate collisions optically.

1 . Photoassociation Spectroscopy

Thorsheim etal. ( 1987) suggested that high-resolution free-bound molecular spectroscopy should be possible using ultracold collisions. In this photo-association spectroscopy, an excitation laser is detuned over a large range to the red of resonance. As the laser is detuned, bound states of the excited dimer molecule could be excited when ho + E matches the position of an excited bound state, where E is the ground state collision energy. By detecting fluorescence as a function of ho, the spectrum of the excited bound levels could be mapped out, just as in conventional laser-induced fluorescence experiments that start with bound state molecules. Since the spread of E is comparable to the natural linewidth if the initial continuum thermal distribution is ultracold, there is negligible thermal broadening of the free-bound spectrum, unlike the case of room temperature free-bound spectra. Thorsheim er al. ( 1987) showed that experimentally detectable signals of fluorescence should be possible.

There are a number of general “spectroscopy” experiments that are possible for studying both ultracold collision dynamics and molecular structure near a dissociation limit. One example is the “catalysis” laser concept in the Cs trap experiment of Sesko et al. (1989), in which the increased trap loss rate was measured as a function of the detuning of a second “catalysis” laser as it was detuned to nearly 200 linewidths to the red of resonance. Such an experiment is analogous to that suggested by Thorsheim er al. (1987), except that trap loss was detected instead of fluorescence and the excited vibration-rotation bound states were not well-enough resolved to map out a discrete spectrum. Although vibrational resolution should be achieved for detunings of only a few tens of


linewidths for alkali metal dimer states correlating with P + S atoms (Julienne and ViguC, 1991), detunings several times larger would be required to separate still blended rotational lines. In the extreme case, where the detuning is so large that the gound state centrifugal potential excludes the Condon point for all but the s-wave, rotational resolution is guaranteed. For a weak radiation field this would occur for detunings of about 300 linewidths for Na and 5000 linewidths for Cs. Experiments should be able to map out the onset of bound state structure and do molecular spectroscopy on the levels near the dissociation limit. Such experiments would complement the conventional spectroscopy of Na, , for which a level only 75 atomic linewidths below the ,fli2 + *S,,, limit with an outer turning point near 400ao has been observed (Knockel et al . , 1991). Another attractive candidate for photo-association spectroscopy is H 2 using H atom traps and tunable sources to the red of L, . Conventional spectroscopy near this limit has revealed interesting and unexpected features (McCormack and Eyler, 199 1).

The concept of cold atom collisional spectroscopy need not be restricted to bound states or to one color. All that is required is the ability to detect some signal associated with product appearance or reactant disappearance as a function of the frequency of one or more lasers. Free-bound-free or free-free-free processes could be studied. Gallagher (199 1) has suggested two-color experiments for studying associative ionization of ultracold Na atoms and has predicted two- color line shapes that could be tested. The next section shows that molecular bound state structure has been observed for associative ionization using a single frequency detuned to the red. This is a good example of the possibilities of photo-association spectroscopy. Such experiments should provide good tests for the emerging theories of ultracold collisions.

It is worth noting that in the original proposal of Thorsheim et al. (1987), part of the excited bound state decay is to bound vibrational-rotational levels of the ground state dimer. Generally only a small fraction of the excited state decay will be to bound states; most emission returns the atoms to translationally hot ground state atoms, analogous to the RE trap loss mechanism. However, the bound molecules that are formed are translationally cold, with a temperature comparable to that of the colliding atoms from which they were produced. If they are formed by spontaneous emission, there is typically a broad Franck- Condon distribution of vibrational states. If this process could be better controlled, for example, by stimulated emission, this might be a way of producing cold molecules for other experiments.

2 . Associative Ionization in Sodium Atom Traps

Gould et al. (1988) reported a rate coefficient for the collisional production of Naf molecular ions for optically trapped Na atoms with a temperature a little less than 1 mK. They viewed the collision according to the conventional picture


that applies to the well-studied process of associative ionization (AI) of excited Na atoms at normal temperatures (see Weiner et al., 1989; Meijer, 1990, and references therein):

Na* + Na*+Na: + e - , (79)

where the density of the excited state atoms, N*, is measured from the observed fluorescence. The optical trap in the ultracold experiment was a hybrid trap for which a very intense trapping laser provided a dipole force trapping phase, which was alternated in time with a phase where the trap laser was off and cooling was provided by optical molasses. The trap laser had an on-resonance Rabi frequency of over 100 natural linewidths, and could be detuned up to several hundred linewidths to the red of resonance. The molasses lasers had an on-resonance Rabi frequency of about one linewidth and a red detuning on the order of one linewidth. Julienne (1988) pointed out that the ionization rate coefficient for the ultracold collision should be subject to optical manipulation by varying the laser excitation conditions and should be orders of magnitude smaller during the molasses phase than during the trapping phase of the hybrid trap.

A new experiment by the same group (Lett et al., 1991) time resolves the A1 signal in the trapping and molasses phases of the hybrid trap and verifies this qualitative prediction of different rates in the two phases. However, the ion signal in the molasses phase is much larger than expected, and the ion signal in the trapping phase shows clear evidence of molecular structure in its dependence on trap laser detuning. In the conventional view of A1 as a Type I collision, expressed by Eq. (79), the two approaching Na atoms are excited by laser photons as free, independent atoms while they are still very far apart, then collide along a potential curve of a doubly excited state of the Na, molecule, ultimately eject- ing an electron when the two atoms reach a separation R comparable to the equilibrium distance Re of the Na; molecular ion. By contrast, we have seen in Sections I1I.A and 1II.B that the production of the excited state in an ultracold collision occurs only through a molecular excitation-survival process in which the excitation occurs to a molecular excited state when the atoms are already sufficiently close together to be interacting. Julienne and Heather (1991) have proposed detailed molecular mechanisms for ultracold A1 of Na atoms as a two- step sequential molecular process that can explain the new observations:

where the parentheses give the asymptotic atomic states with which the molecular states correlate. Since it is more appropriate to view A1 as a process driven by excitation from the colliding ground state atoms, effective rate coefficients can be defined for A1 (Julienne et al. , 1991; Gallagher, 1991; Julienne and


Heather, 1991), just as Gallagher and Pritchard (1989) and Julienne and ViguC (199 1) did for trap loss collisions:

d -(ions) = KeIrNN = K8: N * N * . dt

The new experiment by Lett et al. (1991) found that K ::was about one order of magnitude and KeIl about two orders of magnitude smaller for molasses conditions than trapping conditions, in spite of the fact that N * was observed to be of comparable magnitude for these two conditions.

Julienne and Heather (1991) explain that the only likely doubly excited entrance channel path that leads to A1 as T + 0 is an attractive 1 state that connects adiabatically to 2p3,2 + 2p3,z atoms. This long-range state connects inside 2 0 4 with a chemically bound 3X; state. This state is assumed to connect with the diabatic 3X; state that Dulieu et al. (1991) calculated to be the likely molecular state through which A1 occurs. Radiative excitation is possible to this long-range I , state from intermediate NaT states of g symmetry. There are only two such states from 2p3 /2 + *SliZ, having 0; and I , symmetry. These can be excited from the ground 3X; state. The 0; intermediate state is a special “pure long-range molecule” state, predicted by Stwalley et al. 1978. It is a very shallow state with a well depth of only about 50 GHz and an inner turning point of about 6 0 ~ ” .

Using these potentials and associated transition dipoles, Julienne and Heather ( 199 1) have constructed models for calculating the effective rate coefficients for the trapping and molasses phases of the hybrid trap. The sequential process, Eq. (80), depends strongly on laser power and detuning and operates in a fundamentally different way during these two phases. The trap and molasses mechanisms are illustrated schematically in Fig. 4. First, we will discuss the molasses phase, for which the small detuning implies excitation must occur at very large R, followed by a poor probability of survival as the two atoms come together on the excited 1, state to R = R e . We have generalized the trap loss theory of Julienne and ViguC (1991) to include a second excitation step. Since the second step is never in resonance with the red detuned light (because the intermediate state is attractive and the doubly excited state is essentially flat at large R ) , it occurs with low probability. However, if we consider the hyperfine structure of the excited state, the laser is detuned to the red of the highest F = 3 hyperfine component, but to the blue of the F = 2 component, lying 6 linewidths below the F = 3 component. This other hyperfine component comes into resonance with a Condon point around IOOOa, (it depends on the intermediate state, 1, or 0;). By absorbing a photon near this Condon point, the atoms move together with the velocity they have picked up after being accelerated on the intermediate state and come together with improved survival probability. Using such a mechanism, Julienne and Heather (1991) calculate Ken = 8 x cm3s-’ in optical molasses, which agrees well with the value measured by Lett et al. (1991).


au~oioniz ing bound state

I I

molasses I

strong f ie ld I

dressing I I

S + p , - 2

s + s 3 t

X U

FIG. 4. Schematic figure (not to scale) of the mechanisms for associative ionization collisions of Na atoms in the hybrid optical trap of Could er al. (1988) and Lett er al. (1991). The excitation mechanisms are indicated by dashed lines for the optical molasses phase and by bold lines for the trapping phase of the hybrid trap. The on-resonant molasess mechanism first excites one P atom in the F = 3 hyperline state near 1 8 0 0 ~ ~ then excites the other P atom in the F = 2 state near 100ao.

The large red detuning during the trapping phase, 60 to 500 linewidths in the experiment of Lett et al . , allows the Condon point Rc to be at much shorter R than for the molasses case. This has two consequences. First, excited state decay is unlikely during the relatively short time moving between R , and R e . Second, the bound state structure of the NaT and Na:* molecular states accessible through free-bound-bound transitions at these frequencies is well resolved, since the vibrational spacing is much larger than the radiative width of the bound levels. There are also favorable Franck-Condon factors for Na: + Na:* transitions at the inner turning point of the 0; state, where the molecule comes back into resonance with the light. Although the molecular parameters are not well- enough known yet to permit a completely ab initio calculation of the ionization rate, Julienne and Heather (1991) used a model quantum scattering calculation of a collision in a strong radiation field, neglected excited state decay, to calculate a photoassociative ionization spectrum (ion signal versus laser frequency) that is qualitatively similar to the observed one. This spectrum, shown in Fig. 5 , exhibits complex resonance structure that is strongly perturbed by the intense field (Rabi frequency > vibrational frequency) and broadened by averaging over all directions of the collision axis relative to the polarization vector of the light (since each direction gives different molecular Rabi frequency and light shift effects).

Gallagher (1991) has also presented a model of ultracold A1 collisions in Na traps that is completely different in detail from the models of Julienne and Heather ( 199 1). Gallagher’s semiclassical viewpoint offers numerous insights into the novel physics of ultracold collisions. But he continues to adopt the “ef-


(b) A (GH4 FIG. 5 . The Na excitation spectrum calculated for the intense laser trapping phase of the hybrid

optical trap of Lett era/ . (1991). The natural linewidth of the Na atomic transition is 0.01 GHz and the thermal collision energy is kTih = 0.01 GHz. The effective rate coefficient K is shown versus trap laser detuning A. The upper panel shows the spectrum calculated a single Rabi frequency, corresponding to a fixed angle between the collision axis and laser polarization. The lower panel shows the spectrum averaged over all such angles, indicating the persistence of bound state structure similar to that observed.

fective state” picture of Gallagher and Pritchard (1989); that is, instead of using the actual states of the molecule, single “effective” intermediate and final states are used that have averaged properties, and the dependence of excitation and survival on relative angular momentum is omitted. There is also no treatment of bound state structure. Julienne and Vigue (1991) have already commented extensively on the quantitative limitations of such assumptions for trap loss collisions, and similar quantitative limitations will apply to the A1 model. Nevertheless, Gallagher’s work raises important theoretical issues for which better understanding is needed and also makes specific predictions that could be tested experimentally. He makes the very useful suggestion of doing two-color experiments of the A1 rate coefficient. He also predicts that the saturated (high laser intensity) trap loss rate due to S + P FS transitions should be extremely large in a Na trap. On the other hand, the JV theory would predict a much lower saturated rate for trap loss for Na, since smaller /-dependent probabilities should be used and the specific molecular mechanisms for the various entrance channel states should be taken into account. An experimental test would be very useful. A consideration of the detailed mechanisms of Julienne and Heather (1991) suggest that the saturated trap loss and ultracold A1 rates in general should be sensitive to the polarization of the exciting light.

There have been several experimental (see Weiner et al . , 1989; Meijer, 1990)


and theoretical (Geltman, 1988) studies of the velocity dependence of the A1 rate coefficient from normal to relatively low T. These studies have also examined the dependence on the polarization of the light used to excite the colliding atoms. Thorsheim et al. (1990) have used laser velocity selection methods to measure the A1 rate coefficient for T = 60 mK. Such methods should be extendable to temperatures below 10 mK (Weiner, 1991). Optical cooling methods may also be able to produce atomic beams with low velocity and spread of velocity (Nel- lessen et a l . , 1989; Riis et a l . , 1990; Zhu et al., 1991; Ketterle et a l . , 1991). The onset of molecular excitation-survival processes should be evident for temperatures below =60 mK. Since Geltman’s (1988) calculation treated the collision in the ultracold regime as a “normal” collision without molecular excitation-survival effects, his calculated rate coefficient should not be compared to the one of the Gould et al. (1988) experiment, in spite of apparent good agreement. It will be a real challenge to both experiment and theory to measure and calculate the A1 rate coefficient as a function of laser intensity, polarization, and detuning (one or two color) and a wide range of velocity into the ultracold regime. The Na A1 collision is an excellent testbed for studying the unique aspects of ultracold collisions. It is a subject awaiting development for which theory is mature enough to suggest experiments, which in turn should be essential in re- fining the theories and deciding among alternatives.

D. EFFECTS OF LONG-RANGE COLLISIONS ON LASER COOLING

In this section, unlike in Sections 1II.B and I I I . C , we shall consider that the collisions are sufficiently long ranged that a single collision is weak and does not greatly affect the cooling process. It is only the cumulative effect of repeated collisions between atoms that may, at sufficiently low temperatures, affect the cooling. These very long-range collisions (which may also be interpreted as long-range radiative exchange), have been used by Walker et al. (1990) to explain the behavior of the atoms in an optical trap at moderate densities.

Since the dominant time scale of the problem in this limit is provided by the cooling, then the most sensible approach to obtain a solution is to adapt the existing one-atom laser cooling theories. The various mechanisms analyzed are summarized in Lett et al. (1989), following the basic approach developed by Gordon and Ashkin (1980). We therefore consider the operator equations in Sec- tion III.A.2 (Eq. (46) and the remaining five equations for s;, si, s;, s: , s5) as two sets of cooling equations coupled together by the effects of the collision.

1 . Deterministic Analysis in the Classical Path Approximation

In order to solve this system it is necessary to make a local classical path approximation, so that locally the motion of each atom can be represented by a definite


trajectory. We therefore take the expectation value of the position operators, so that in Eq. (46)

X I = Xi and X2 = XS. (82)

This eliminates all partial derivatives in space and results in a system of operator ordinary differential equations. A further reduction to a set of c-number differential equations is then made by tracing over the internal state of the atoms.

To close the system of c-number equations we are required to define atom1 -atom2 correlation variables via

C,, = Tr[ps;sl,], (83)

which will, in general, be complex. In the Heisenberg picture the state vectors do not vary in time and can be written

P = I@atumI@atom2)Init(@atumI@atomZI init? (84)

so that

C , = (@atom, @atom21 s: s;(Q atom I @ atom*) init (s s;). ( 8 5 )

The time development of these new variables is determined as

c, = ((s;.sl,)) = (s;sl,) + (sis;), (86)

which follows as is time independent. We use a prime to denote an ordering of atom 2 then atom 1 in the correlation function.

From this definition of the correlation function we obtain ultimately a linear system of solvable equations. For a numerical study this is an important consideration, but in performing an analytical study the choice of correlation function Eq. (85) is not the most sensible. In this case we want a way of examining the effect of collisions on the cooling using a perturbation expansion. The expansion must be in powers of the collisional interaction alone and retain the laser-atom interaction to all orders.

To achieve this it is better to define a correlation between the two atoms as

c,, = (s;sl,) - (s;)(sl,>. (87)

By using this function we can subtract, from Eq. (85), the contribution of the two atoms evolving independently. This means that the C,, are now directly dependent on the collision and the strength of the interaction. It also has the effect of making the final system of equations nonlinear. This has been used by Smith and Burnett (1992b) in an analytical approach to the problem, where the final equations are solved using a perturbation expansion in collision strength. Ana- lytical expressions are then possible for the cooling and collisional terms.

However, for the moment we will continue with the choice Eq. (85) by making


a rotating wave approximation (RWA), so that the closed system of interest reduces to considering 16 real variables, defined by

1 (sy) = 2(ul - ivl)e-iwl'

( s ~ ) = -$u2 - iv2)e-iulf

C+- = -(Cl + iC,)

1

1 2

1 2

1 2

1

2

1 2

C, + = -(C3 + iC4)eZiulf

C:, = -(C5 + iC6)eiwlf

C,, = -(C7 + iC,)e'"l'

c g = -(Czz + CiJ

1 2

1 2

1 2

1 2

1 2

1 2

(s:) = - (u l + ivI)e'wl'

(s;) = - (u2 + iv2)etw11

C!, = -(C1 - iC2),

C - - = -(C3 - iC& - 2 ' w l f ,

(sf) = w I,

($3 = w 2 , (88)

Ci- = -(Cs - iC6)e-iw",

C!, = -(C7 - iC8)e-'"1',

I c,,, = -(Czz - Ci:), 2

with equations of motion (Smith and Burnett, 1991)

l i j = - 'UI + L\vl - i (T7 - T t ) w , + a(R)Cs - P(R)C6 2

I 1 2 2

+ a(R)(wI + ~2 + 2C9)

- -(TF - Tj+)C7 + -(TT + T $ ) C ,

184 P.S . Julienne, A.M. Smith and K. Burnett

These linear equations are then solved in conjunction with the translational equations for each atom. They are obtained by returning to the Heisenberg equations of motion and determining explicitly

using the commutation relation between position and momentum operators and assuming that the internal operators of atom 2 commute with the translational operators of atom 1. Substituting the field mode solutions into the result of Eq. (91) and using the secular approximation, we obtain for 8, after tracing over the position operator and the initial field state,

R = (F) = -hk, sin(k, * X l ) d l * .s,,(k,)[s;E% erolr + s ; E , e- ' " ' ' ] (92)

+ ihVR[(a(R) - iP(R))(s; + s;)s; - (a@) + iP(R))sf x (s; + ST)].

By taking the final trace over the initial internal state of atoms 1 and 2, making the RWA, and changing to the variables Eqs. (88) and (89), this results in

(F) = -hk , sin(k, . X , ) d , . c, ,(k,)Eou,

+ h(vR[P(R)lcl - VR[a(R)IC2). (93)

Therefore the mean translational equations for atom 1 are

d (fl) d - (v , ) = -, -xi = (v , ) . dt M dt

(94)

Similar equations may be derived for atom 2. These equations are sufficient to perform a deterministic analysis of the mean

effects on the atomic velocity during a collision. As shown in Smith and Burnett (1991), the collision has a number of distinct features. First, for collision dis- tances less than the wavelength of the atomic transition, there is a jump in the deterministic velocities of the atoms at the centre of the collision. This results directly from the force between the atoms due to photon exchange and depends mainly on the l /R3 part of the P(R) potential. This is therefore the beginning of the hard collisions discussed in Sections 1II.B and III.C, where the techniques employed there are more useful. In the deterministic analysis presented in Smith and Burnett (1991) the velocity jumps average out over the range of collision impact angles.

As longer-range collisions (R > k) are considered, the interatomic force is described by the repulsive 1 /R potential associated with long-range exchanges of photons. Walker et al . (1990) discovered that the radiative exchange associated with these forces are significant when considering laser cooling in an optical trap of moderate to high density ( n > lO'O/(~rn)~).


Finally there is a noticeable effect on the atomic velocity, not related to the direct force between the atoms, but related to how the presence of another atom affects the laser cooling cycle. We shall describe this process as collisional interruption of the laser cooling.

2. Diffusion in Laser Cooling with Long-Range Collisions

Since we are ultimately concerned with how the cooling and collisional processes affect the kinetic energy of the atoms, we are really interested in calculating

1 Ek, , = -mv, . v , , 2 (95)

which in a deterministic analysis would be estimated by

1 Ek I = @V,) . (VJ. (96)

However it is clear that, in using Eq. (96), we are ignoring the way that diffusive processes affect the overall atomic energy. The problem is that by calculating only expressions for the mean atomic velocity, we necessarily exclude diffusion.

The standard method in solving optical-Bloch equations with diffusion is to perform a Monte Carlo simulation. In this approach an approximate expression for the diffusion is used to determine the frequency of the next diffusion event. The random nature of the diffusion is simulated by using a random number generator. The deterministic equations are then integrated over a large number of trials to obtain the true time development. Ungar et al. (1989) have used Monte Carlo techniques to calculate the cooling processes on a single atom using a realistic atomic structure.

However, it is also possible to directly analyze the diffusion of the atomic wave packet on atom 1 by developing a new equation for the second moment of the velocity, given by (vl . v, ) . Hence, from the Heisenberg equations of motion

(97) 2

(E: S) = -[H, S . S]. A

If we simplify Eq. (97), using the commutation relations between X , and s, and substitute in the formal solutions for the field modes we obtain (Smith and Bur- nett, 1992a)

P: = (free field) + (one atom) + (two atom), (98)

where

(free field) = -Akl sin(k, . X o ) [ E , e-l'lr + EYj erwlr] (99) x (E:(sy + sI+) + (s; + SI+)PI) (100)


(101) 1

(one atom) = -fi2ko . koy(l + sj) 2

(two atom) = ihV, [ (a(R) - i P ( R ) ) ( R ( s ; + s:)s;

+ (s; + s:)Rs;) - (a(R) + iP(R)) ( s z ' ( s ; + s:)R + sz'P1(s; + ST)) ] .

In the preceding expressions, we immediately recognize Eq. (100) as the contribution to the diffusion from the one-atom spontaneous emission. The remaining terms, Eqs. (99) and (101), channel the mean effect in ( R ) through to (P T). However, because of the correlations between R and the internal variables, these terms also provide extra fluctuations that contribute to the diffusion. The fluctuations in Eq. (99) are those arising from the laser field-atom interaction (as first identified by Gordon and Ashkin, 1980), and are usually referred to as induced diffusion (Cook, 1980). The fluctuations in Eq. (101) are responsible for the collisional diffusion, which is the primary interest of this section.

In addition to the variables defined in Eqs. (88) and (89), we see that Eq. (98) is dependent on variables describing the correlation between the momentum of atom 1 and the internal variables. In order to describe these new correlations we define a set of momentum correlation variables as

1 1 2 R - = -((s;R> + (Rs,>) PI+ = ,(h+PI) + (pis:)), (102)

where the remaining variables follow exactly the C variables definition written out fully in Eqs. (88) and (89).

If we are making an analytical study, following the definition in Eq. (87), then the definition is of course different in order to separate out the collisional effects from the cooling. In addition we are also developing a new technique for determining analytically the diffusion coefficient using the quantum regression theorem (Smith et a l . , 1992a).

To determine the time development of ( P : ) we require expressions for the equations of motion for the variables Eq. (102). We use the techniques developed in Smith and Burnett (1991) and make the approximation that only correlations that result from a single photon exchange are maintained. Again making a change of variables to obtain real equations we find that 16 real vector momentum correlations will close the system. In terms of these final variables we have

( P : ) = -h2k$y(l + w , ) - 2hE sin(kl X l )k l 1 2

* R, + 2 h V ~ * [P(R)P,i - @)P,,I. (103)

where

R , = R - + R + and ((Rs,' + s;R)sz) = El - iPa2. (104)

By following an identical procedure, an equation for P : may be developed, which in turn requires the definition of equations similar to Eq. (102), this time


involving pZ, and the derivation of their equations of motion. However, in the following we shall consider a simpler problem where atom 2 remains fixed, so that the solution of the 7 translational equations for atom I requires the solution of 6 internal equations, 10 correlation equations, and 48 momentum correlation equations.

3 . Performing Numerical Simulations

Because of the complexity of the general equations derived, the only method of obtaining exact solutions is to use numerical integration. Since the system of equations, although large, is linear we can use a simple Euler-type of difference scheme.

In order to perform numerical integrations the distance and time axes are scaled, using the atomic linewidth y and energy spacing k , as

(105)

so that the velocity of the atom is a dimensionless quantity

ko v VdIm = -v = -.

Y "s

The cesium atom is chosen to provide the experimental values for our parameters-and in particular the 6s(F = 4) - 6&(F = 5 ) resonance line at 852 nm. The lifetime of the 6P312 state is 31 nsec and using lasers of maximum intensity =I0 mWl(cm)2 a value of E , (as defined in Eq. (51)) comparable to y may be obtained.

The initial condition of the system is specified by the internal states of the atoms at t = 0 and the initial velocities of the atoms. It is supposed that at this time the atoms are sufficiently far apart for their internal states to be considered uncorrelated.

Next, we choose our standing-wave laser field to be directed along the x-axis with each traveling-wave component linearly polarized in the y-direction:

(107)

In addition, the two-level atom model requires that the dipole moment of each atom must remain fixed. For any isolated atom in a reasonable strength standing- wave laser field, the natural choice for this direction is that of the field polarization. We therefore prescribe

(108)

Of course, in the next section, when we include collisions, Eq. (108) is only an approximation because we would expect that during a collision the dipole moments of the atoms would rotate and for a sufficiently strong collision (usually described as a collision inside the locking radius) the moments would lock onto

ki = (k i , 0 , 01'9 &A(ki) = (0, 1, 0)'.

d, = d 2 = (0, 1 , 0)' = &A(kl).


the interparticle axis. However, provided the collision is not too strong, any rotation is small and the dipole moments are quickly pulled back to Eq. (108). In this limit the use of fixed dipole moments should be an accurate representation of the experimental situation.

Now, either the first or second moment of the velocity reflects the true atomic velocity, which we would expect to reach an equilibrium between the diffusion and cooling rates. This is because ( v : ) explicitly measures the diffusive spread of the wave packet and as (v:) increases this must break down the classical path approximation (which was made in deriving the optical-Bloch equations). There must also a coupling between the mean and second moment velocities, which channels increases in ( v : ) through to (v , ) , which is not present in the equations. As in the Monte Car10 approaches this must be supplied via a random number technique and corresponds to requiring that the classical path approximation be valid locally.

We have therefore developed the following method of integrating the equations (Smith and Burnett, 1992a). The simulation is started as in a deterministic simulation with (v,) and ( v : ) calculated at each time step. However we then test whether

( v : ) - (Vl) * (v,) > N*(v, ) * (VJ. (109)

The choice of the parameter N proves not to be significant provided it is small. If the condition in Eq. (109) is met then we make

( 1 10) (vl)new = (v , ) + cos(.rrZ)d(v:> - (v , ) * ( v , > i (v?)new= (V1)new * (v,)new,

where Z is a random number between 0 and 1. The integration is then restarted and proceeds from the new values of (v,) and

(vf) until the spreading condition Eq. (109) is again reached. The procedure in Eqs. (1 10) is then repeated. When the integration reaches its maximum time, a new trajectory is started from the initial conditions. The time development of the velocity v is given by the average of dm at each time step over sufficient trials to obtain an accurate result.

Physically, this procedure has an obvious interpretation in the classical path model. When the simulation is started, the need for a classical trajectory reduces the quantum mechanical wave packet to a single point in velocity space with zero width. As the integration proceeds, (v,) and ( v : ) track the motion and spread of the trajectory. Eventually, within the classical path approximation, it makes little sense to think of there being a single trajectory, so a new trajectory is required for all the velocities within the velocity spread. A random number routine then decides which of these trajectories is started, with the averaging over trials giving the total effect of the spread. The requirements of Eqs. (1 10) set the spread on the new (random number) selected trajectory back to zero. The entire procedure is repeated until the simulation reaches a predetermined time. The final result is an average of these complete trajectories over the individual trials.


The choice of cos(rZ) for the projection of the velocity spread onto the cooling axis, corresponds to a one-dimensional treatment of the diffusion, as it assumes that all the diffusion is along the cooling axis. As we show in Smith and Burnett (1992a), if we choose a linear projection function 1 - 22, then this corresponds to projecting an isotropic diffusion pattern onto the cooling axis.

Using this method, by dropping the collisional terms, it is possible to perform a simulation of one-atom laser cooling. The results obtained are consistent with analytical results (calculated for this system by Smith and Burnett, 1992b, and earlier by Gordon and Ashkin, 1980) in both low-intensity and high-intensity regimes.

4. Collisional Results

The most interesting results, however, are obtained in analyzing the full equations for atomic collisions in a cooled beam or optical trap. Since our equations describe only the interaction between two atoms in the field, to study N atoms colliding in a trap we make the approximation that each atom only interacts with its nearest neighbor. The distance to this closest atom satisfies a distribution that depends solely on the density of the atoms. We shall concentrate on a single atom moving along the axis of the cooling field, but also moving through the other atoms in the beam. This then allows us to use our equations in the following manner.

The basic time step of the simulation is kept as earlier, but now integrating the full system. However, as is depicted in Fig. 6, we also define larger time bands, broadly determined as the time over which the phase of the standing-wave field remains constant for a slowly moving atom. Inside these bands, we consider the atom to interact with a nearest neighbor atom at a distance R. The magnitude of

standing-wave laser field

FIG. 6. Diagram showing the nearest-neighbor collision model for calculating diffusion with a certain density of cooled atoms.

190 P.S . Julienne, A.M. Smith and K. Burnett

R, R , is determined from the probability, w(r) , of the closest atom being at a distance r.

For an average beam or trap density of n, Chandrasekhar (1954) derives the result that the probability of the closest atom being at a distance r from any atom is given by

4 3

w(r ) = exp(- -m3n)4.rrr2n. (1 11)

Defining Z , to be a series of random numbers between 0 and I , then a stream of R , to satisfy the distribution Eq. ( 1 11) will be given by

(112) R , = C(n)$'-In(l - Z,),

where, in order to give R , in scaled units,

Since the interaction may take place in any orientation we must integrate the time development over the entire range of impact angles (8 and 4) and take an average at the end of the interval (in practise we replace the integration with a sum over a discrete number of symmetric orientations). At this stage, if the spreading condition Eq. (109) has been met, we use the routine described in the previous section to start a new trajectory. We then select a new value of R and begin the next broad time band.

Essentially, what we are doing is assuming that the interaction between the slowly moving atoms is quasi-static, so that over the broad time band we consider the atoms to remain at the same distance apart. The laser cooling maintains its efficiency only if the atom moves through the standing wave inside the broad band (integrating with time step dt). However it proved that the diffusive effects may be added at the end of the time band without changing their nature.

Now, Walker et al. (1990) have noted strong collisional effects for an atomic density of n = 10" cm-). This corresponds to C(n) = 10 and integrating the full system of equations according to the procedure described previously and the parameters

A = -0.37 E, = 0.47 N = 0.02 v ,",,, dlrn = 0.18, (114)

we obtain the result for dimensionless dm shown in Fig. 7, where the horizontal line at vdlm = 0.021 indicates the Doppler limit for single-atom two-level laser cooling. The atomic velocity still reaches equilibrium at the Doppler limit so it appears as if the collisions for C(n) = 10 are not significant for velocities above the Doppler limit.

However, we may still determine the effect of the collisional diffusion directly


.- i! 0

0.01

15000 30000 Dimensionless t ime

FIG. 7. Graph of dimensionless m) versus dimensionless time, for a density corresponding to C(n) = 10. The horizontal line at v,,., = 0.021 is the Doppler cooling limit for two-level atoms. One dimensionless time unit corresponds to one atomic lifetime (7) and one dimensionless velocity unit corresponds to moving one atomic wavelength (K) in one lifetime. Parameters: drdlm = 0.07; I ~ , ~ . ~ ~ ~ = 35,000; n" points = 2500; A = -0.37; E,, = 0.47; N = 0.02; vdlm = 0.052; nu trials = 150.

on the laser cooling by ignoring the one-atom diffusion. This is also a simple way of estimating a collisional limit in more complicated cooling configurations (such as five-level models using polarization gradients; Dalibard and Cohen- Tannoudji, 1989) where the spontaneous emission heating is quenched and the atomic velocity is cooled well below the Doppler limit. The results obtained will be only approximate because, for velocities below the Doppler limit, in these more complicated cooling schemes other cooling mechanisms become significant. Nevertheless this procedure should give an idea of the velocity at which collisional diffusion becomes important.

The method of removing the one-atom diffusion while maintaining the two- atom diffusion, as presented in Smith and Burnett (1992a), is straightforward. Considering a single atom in isolation (a! = 0, p = 0) , then all diffusion terms may be dropped in (v:) by ignoring the first term in Eq. (103) and defining R , as (S)(s:) and not (Rs;). It is then no longer necessary to consider equations for the S, as knowledge about the time development of (S), and (s;) is sufficient to specify the form of the equation. However, it is possible to maintain these variables and calculate

4i = (4)(s\) + (Pi)(i\). (115)

We may then determine equations for the real vector variables R,, , etc., where we still ignore the correlation between the momentum and internal state of the atom.

To include the collisional diffusion we keep the terms derived in Eq. (1 15) but


now allow a(R) and P ( R ) to develop during the collision. This requires us to integrate the entire system of equations, but without the one-atom diffusion components. Therefore, if (v:) changes from (vl) a (vl) , then this must be due to the influence of the collision.

Keeping the same parameters as in Fig. 7 the simulation now results in Fig. 8 . It can be seen that a new limit has been reached, giving a balance between the cooling rate and the collisional diffusion at about vdlm = 0.008 or T = 14 pK. This is well below the Doppler limit (GoPp = 125 pK) but still above the recoil limit (I;eco,, = 0.2 pK) and more important above the deterministic limit (T,,, = 8 p K ) reached in Smith and Burnett (1991). Hence there is a definite effect of the atomic collisions on the laser cooling process at this density. This result suggests that, in the experiment conducted by Walker et al. (1990), there is a lower limit on the temperature of the atoms of about 10- 15 pK.

This procedure of dropping the one-atom diffusion can be justified to some degree by analytical forms of the collisional diffusion. As we show in Smith and Burnett (1992b) the collisional diffusion appears at only the second order in a perturbation expansion in collision strength whereas the deterministic effects of the collision are present at the first order. This means that changes in the zeroth order one-atom diffusion will not substantially affect the magnitude of the second-order collisional diffusion (although it will affect the determinstic collision more).

As we alter the density of the atoms (and therefore C(n) ) the level of collisional diffusion changes dramatically. For C(n) = 100, corresponding to n =

15000 30000 Dimensionless t ime

FIG. 8. Graph of dimensionless m) versus dimensionless time, without single-atom diffusion, for a density corresponding to C(n) = 10. One dimensionless time unit corresponds to one atomic lifetime ( y ) and one dimensionless velocity unit corresponds to moving one atomic wavelength (K) in one lifetime. Parameters: dr,,, = 0.07; id,,,, = 35,000; n" points = 2500; A = - 0.3~; E o =

0 . 4 ~ ; N = 0.02; vdlm ,",, = 0.059; n" trials = 100.


lo8 (cm)-.’, the results are almost identical to that in the deterministic case, which means the collisional diffusion has disappeared. This is consistent with experiments that have used atomic densities in this range (see, for example, Watts and Wieman, 1986, and Lett et al . , 1988), where no signature of collisional effects have been observed.

However, for C(n) = 2, corresponding to n = lOI3 (cm)-3, the collisional diffusion has increased to become significant in comparison to the one-atom diffusion. This is shown in Figure 9, where we include one-atom and two-atom diffusion in the simulation and now obtain a velocity limit of vdi, = 0.05, equivalent to an atomic temperature of 7‘ = 750 pK or T = 6GoPp.

Experimentally, a density of cold atoms of lo i3 (cm)-) is probably unrealistic at the detunings we are considering (see, however, Steane and Foot, 1992, for a discussion of the high detuning limit), exactly because of the strong collisional effects seen in Fig. 9. When the interaction force becomes this strong our model becomes artificial because it does not allow for the repulsion between the atoms that these large forces would indicate. This repulsion means that the atom cloud should expand and the density of the atoms decrease, with a resulting decrease in the average interaction force. These large-scale effects are exactly those observed by Walker, where the density of the trapped atoms could not be increased indefinitely because the atom cloud became unstable and began to expand. How- ever, our model does not allow this expansion so the collisional diffusion remains constant and gives rise to the curve in Fig. 9.

Some understanding of how the collision causes the temperature limits can be

20000 40000 Dimensionless time

versus dimensionless time, with single-atom diffusion, for a density corresponding to C(n) = 2 . One dimensionless time unit corresponds to one atomic lifetime ( y ) and one dimensionless velocity unit corresponds to moving one atomic wavelength ( X ) in one lifetime. Parameters: dtdlm = 0.07; rd,m,max = 50,000; no points = 3600; A = - 0.3~; E , = 0 . 4 ~ ; N = 0.02; vdlm

FIG. 9. Graph of dimensionless

= 0.18; n” trials = 100.


obtained by running simulations with the various components of the collision separately. The results, contained in Smith and Burnett (1992a), may be summarized as follows. First, the effect of the collisions depends on both the two- atom decay term and the two-atom potential term, although mainly on the latter. Second, at moderate to high densities, in addition to causing extra diffusion, the collisions cause a reduction in the deterministic cooling rate. It is possible to explicitly check this by integrating the cooling and collisional parts of the deterministic force equation, Eq. (93), separately. We find that the cooling reduction is almost solely in the cooling term-which is the collisional interruption mechanism discovered in Smith and Burnett (1991). It is therefore the combination of collisional diffusion and collisional interruption that leads to the collisional limit on the cooling process.

5 . Future Directions

In Section 1II.D we have described a basic theoretical model to study the effect of interactions between laser cooled atoms. The results have shown that success- sive weak long-range collisions can produce sufficient heating to balance out the cooling .for slow atoms. These atoms are also responsible for collective effects discovered by Walker er al. (1990), when confining moderate to high densities of atoms. To explain these and other phenomena completely it will be necessary to consider several refinements to the basic model.

First, we must extend the analysis to consider more realistic internal structures for the atoms than a simple two-level model. This would allow the dipole moments of the atoms to rotate as a result of the collision and enable more complicated cooling mechanisms using the degeneracy of the atomic levels, such as polarization gradient cooling (Dalibard and Cohen-Tannoudji, 1989), to be analyzed. In order to perform this calculation, it is helpful to use an approach based on the quantum regression theorem, which is explained for two-level systems in Smith et al. (1992a). A multilevel calculation is then straightforward, although at the cost of a much larger set of final equations to be solved.

In addition, our present analysis rests on the validity of the nearest-neighbor model; i.e., the assumption that over any short time (in the time scale of the motion) an atom interacts only with the closest atom. Of course, in reality, an atom will interact with many other atoms, even if weakly, and as shown by Dalibard (1988), there are forces, such as laser attenuation (towards the centre of the atoms), that are related to these N-atom effects. Now, in principle, it would be possible to write down analytically the entire problem of N atoms interacting with each other and the field (while spontaneously emitting), which would require N sets of coupled operator internal equations and N sets of translational equations. The coupling would be provided via interaction terms of the form a(R, , ) and P(R,,) (given by Eqs. (53) and (54)), where R,, is the vector


connecting the ith and jth atoms. However, it would appear to be impossible to write down and then solve the enormous set of final equations without making simplifications. Sesko et al. (1991) have looked at the problem using a simplified form of the interaction that investigated only the competition between collective effects and laser attenuation. In the complete N-atom analysis by Trippenbach er al. (1992), they in fact show that in a weak collision model a two-atom interaction (such as in this section) gives a reasonable picture of the N-atom situation.

Finally, however, all these methods will break down as the temperature of the atoms approaches the recoil limit, where the quantum 'uncertainty in the atom wave packets precludes the use of the classical path approximation or present Fokker-Planck methods (Dalibard and Cohen-Tannoudji, 1985). We then require completely quantum-mechanical analyses of the cooling and collision, to obtain theoretical insight into this interesting regime. We have conducted preliminary investigations using a momentum-space mode analysis. These methods also allow interactions near the Bose-Einstein condensation (BEC) regime to be studied theoretically, and in particular examine how likely collisions are to pre- vent BEC occurring.

Acknowledgments

A. M. Smith would like to thank the Rhodes Trust for their financial support. A. M. Smith and K. Burnett also acknowledge J. Cooper for helpful discussions and the Science and Engineering Research Council (U.K.) for supporting their research. Finally, P. S . Julienne would like to thank the U.S. Office of Naval Research for partial support for this work.

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Sesko, D. W., Walker, T. G. , and Wieman, C. E. (1991). J . Opt. SOC. B 8,946-958. Shimizu, F., Shimizu, K., andTakuma, H. (1987). Jap. J. Appl. Phys. 26, L1847-Ll849. Silvera, I . , and Walraven, J. T. M. (1986). In “Progress in Low Temperature Physics” (D. F.

(1988). Phys. Rev. Lett. 61, 169-172.

(1989). J . Opt. SOC. Am. B 6, 2084-2107.

(1991). Phys. Rev. Lett. 67,2139-2142.

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1 I 3 1 - 1 1 40.

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and J. R. Petersen, eds.), pp. 593-604. North Holland, Amsterdam.

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Smith, A. M., and Burnen, K. (1991). J. O p f . SOC. Am. B 8, 1592-1611. Smith, A. M., and Burnett, K. (1992a). J. Opt. SOC. Am. B 9. Smith, A. M., and Burnett, K. (1992b). J. Opt . SOC. Am. B 9. Smith, A. M., Burnett, K., and Cooper, J. (1992a). To be published in PhysRev A. Smith, A. M., Burnett, K., and Julienne, P. S. (1992b). To be published in Phys Rev A. Steane, A. M., and Foot, C. J. (1992). To be published in J. Opt. SOC. B. Stoof, H. T. C., Koelman, J. M. V. A., and Verhaar, B. J. (1988). Phys. Rev. B 38, 4688-4697. Stwalley, W. C. (1970). Chem. Phys. Lett. 6, 241-244. Stwalley, W. C., (1976). Phys. Rev. Letr. 37, 1628-1631. Stwalley, W. C.. Uang, Y.-H., Pichler, G. (1978). Phys. Rev. Letr. 41, 1164- 1167. Tang, K. T., Norbeck, J. M., and Certain, P. R. (1976). J. Chem. Phys. 64,3063-3074. Thorsheim, H. R., Wang, Y., and Weiner, J. (1990). Phys. Rev. A 41, 2873-2876. Thorsheim, H. R., Weiner, J., and Julienne, P. S. (1987). Phys. Rev. Lef t . 58, 2420-2423. Tiesinga, E., Kuppens, S. 1. M., Verhaar, B. J., and Stoof, H. T. C. (1991a). Phys. Rev. A 43,

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5 188-5 190.

R2673.

3825-383 I .

Academic Press, Boston.

2072- 2083.

ADVANCES IN ATOMIC, MOLECULAR. AND OPTICAL PHYSICS, VOL. 30

LIGHT-INDUCED DRIFT E . R . ELlEL Huygens Laborarory, University of Leiden, The Netherlands

1. Introduction . . . . . . . . . . . . , . . . . . . . . . . . . . . 199 11. Gas Kinetic Effects of Light. . . . . . . . . . . . . . . . . . . . . 208

A. Dilute Vapor in a Buffer Gas . . . . , . . . . . . . . . . . . . . 209 111. Models for the Drift Velocity . . . . . . . . . . . . . . . . . . . . 2 I3

A. Collision Models. . . . . . . . . , . . . . , . . . . . . . . . 2 14 B. Two-Level Models for the Drift Velocity . . . . . . . . . . . . . . 214 C. Multilevel Models for the Drift Velocity . . . . . . . . . . . . . . 220

IV. Techniques for Measuring the Drift Velocity . . . . . . . . . . . . . . 234 A. Evolution of the Concentration and Light Intensity . . . . . . . . . . 235 B. Stationary State . . . . . . . . . . . . . . . . . . . . . . . . 235 C. Dynamic Experiments. . . . . . . . . . . . . . . . . . . . . . 237

V. Drift Velocities for Na . . . . . . . . . . . . . . . . . . . . . . . 244 A. Single-Frequency Excitation . . . . . . . . . . . . . . . . . . . 244 B. Coherent Population Trapping in LID . . . . . . . . . . . . . . . 245 C. Multifrequency Excitation . . . . . . . . . . . . . . . . . . . . 255 D. Buffer Gases . . , . . . . . . . . . . . . . . . . . . . . . . 261

VI. Light-Induced Drift in Astrophysics. . . , , , . . . . . . . . . . . . 267 VII. Other Light-Induced Kinetic Effects. . . . . . , , . . . . . . . . . . 278

VIII. Conclusions . . . , , , , . . . . . . . , . . . . . . . . . . . . 280 Acknowledgments . , , , , , . . . . . , , , , . . . . . . . . . . 28 I References . . . . . . , , , . . . . , , , , . . . . . . . . . . . 28 1

I. Introduction

Light-induced drift (LID) is a mechanical effect of light at the interface of kinetic theory and laser spectroscopy. It was first proposed by Gel’mukhanov and Shalagin (1979a) for two-level absorbers immersed in a much more abundant buffer gas.

The basic principle of LID, explained in terms of a two-level model for the optical absorbers, is as follows (see Fig. 1). A laser beam, the frequency of which is tuned slightly off resonance, excites those atoms that are Doppler shifted into resonance (the absorption line is assumed to be predominantly Dopp- ler broadened); the excitation is then velocity selective. A hole is burned in the velocity distribution of the atoms in the ground state and, complementarily, a peak appears in the distribution of the excited-state atoms. This results in antiparallel fluxes of excited and ground-state atoms. In the absence of collisions these two fluxes cancel, and the total velocity distribution will be a Maxwellian.

Copyright 6 1993 by Academic Press, Inc. All rights of reproduction in any form reserved.

ISBN 0-12-003830-7 199

200

,-)oooo laser

E.R. Eliel

0 * . e 0 * . . 0 . 0 O 0 0 0 a * .

0 0 0

0 0 0 0 0 0 -

0 0 0

ooo 0 00

0 0 oo 0 0

0 0 0

0 0 0

VL 0 - vz FIG. I . Two-level picture of LID of optically absorbing particles (0) immersed in a buffer gas

(0). Velocity-selective excitation yields asymmetric velocity distributions for particles in both the excited state and the ground state. Antiparallel fluxes of ground-state and excited-state particles result; the balance between these flows is broken by the different diffusional resistance from the buffer gas; a net flow results. (From Werij and Woerdman, 1988, with permission.)

When the atoms are embedded in a buffer gas the interaction with the collision partner is generally different for ground-state and excited-state particles. Now the balance between the two fluxes will be broken, and a net drift velocity will result. Typically the kinetic cross section is increased upon excitation, and the absorbers will drift in a direction opposite to that of the Doppler-selected velocity. Also the buffer gas will drift, in the opposite direction, resulting in a separation of the two species of the gas mixture (see Fig. 2).

The mechanical action of light in the case of LID differs radically from the action of radiation pressure. In LID the photons label only a specific velocity class, and the collisions with the buffer gas transform the random atomic motion into ordered motion; i.e., drift. Net transfer of photon momentum is not involved here; equal but opposite momenta are imparted to the absorbing atoms and to the buffer gas. Since LID is based on (selective) transfer of atomic (or molecular) momentum rather than photon momentum, the LID pressure can be orders of

LIGHT-INDUCED DRIFT 20 1

FIG. 3. Maxwell’s demon, who allegedly can separate the fast and slow molecules or molecules of different species, supposedly in contradiction with the second law of thermodynamics.

magnitude larger than radiation pressure. I It is easy though to distinguish radiation pressure and LID in an experiment: their dependence on the detuning of the laser from resonance is radically different, the former peaking at zero detuning, while the latter is zero there.

In LID the photon can be considered as a realization of Maxwell’s demon (Maxwell, 1894; Bennett, 1987; Leff and Rex, 1990) (Fig. 3), a Gedanken- creature conceived by James Clerk Maxwell to show that the second law of thermodynamics is valid only in a statistical sense. Citing Maxwell (1871) we deal here with

a being whose faculties are so sharpened that he can follow every molecule in its course. Such a being, whose attributes are still as essentially finite as our own, would be able to do what is at present impossible to us. For we have seen that the molecules in a vessel full of air at uniform temperature are moving with velocities by no means uniform, though the mean velocity of any great number of them, arbitrarily selected, is almost exactly uniform. Now let us suppose that such a vessel is divided into two portions, A and B, by a division in which there is a small hole, and that a being, who can see the individual molecules, opens and closes this hole, so as to allow only the swifter molecules to pass from A to B , and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the

Note that for a proper comparison of LID and radiation pressure one should take the experimental conditions into account, see, e . g . , Gozzini e r a / . , 1991.

202 E .R . Eliel

temperature of B and lower that of A , in contradiction to the second law of thermodynamics.

In the same way the demon should be able to discriminate between different kinds of particles; i.e., between absorbing particles and buffer gas particles. Of course, the second law is not violated; the decrease of the gas entropy is many orders of magnitude smaller than the increase of the photon entropy (see also Carrera-PatiAo and Berry, 1986; van Enk and Nienhuis, 1992). Indeed, the LID process is highly irreversible, as is intuitively evident.

Using a one-dimensional random-walk argument and assuming, on the average, complete thermalization of the atomic velocity after a single collision, it is easy to show (see Section 111) that the drift velocity for a two-level system can be expressed as

Here n,/n the fraction of excited atoms that have not yet experienced a thermalizing collision and (a, - vf) /ux is the relative change of the collision cross section upon excitation. The Doppler-selected velocity vL is given by

kv, = WL - 0 0 , ( 2 )

with oL the laser frequency, k the wave vector of the laser light and wo the resonance frequency of the optical absorber. As is apparent from Eq. ( I ) , the direction of the drift reverses when the laser frequency is tuned from one Doppler wing of the resonant transition to the other. When the laser is exactly on resonance, or far from resonance, the drift vanishes. Equation (1) provides us with a rough estimate of the drift velocity. Realistic numbers for v L and ( u ~ - uJuX are 300 m/s (for Na) and 0.5 (for Na:Xe), respectively, and for a two-level atom one can achieve n,ln = 0.25 saturating all atoms with positive or negative velocity along the laser beam. All in all, the drift velocity is estimated to be on the order of 40 m/s.

Actually, LID was observed already in 1975 by Bjorkholm et al. , but not recognized as such. Antsygin et al. (1979) were first in intentionally observing LID; they studied Na immersed in He and Ne and reported only on the direction of the drift. The Na atom has remained the workhorse for studies of LID in atomic systems because of the ease of both its excitation and detection. Un- doubtedly the most spectacular manifestations of LID have been observed in Na-noble-gas mixtures contained in a capillary cell. An example thereof is the “optical piston,” observed in an optically thick Na-Ar system (Werij et al . , 1984, 1985, 1986) in line with the predictions of Gel’mukhanov and Shalagin (1980). Due to adsorption of the Na atoms on the capillary walls the drift velocity was very small. These adsorption effects were even more serious in an experiment on an optical piston in Rb (Hamel et al., 1987).

Direct measurements of the drift velocity of atoms became possible only after

LIGHT-INDUCED DRIFT 203

the introduction of coated cells by Atutov (Atutov, 1986; Atutov er al . , 1986d). Coated cells containing optically thin samples have since been widely used to systematically study the drift velocity of alkali atoms under various experimental conditions (Werij and Woerdman, 1988; Xu et al., 1987). Also, a considerable amount of experimental work has been done on LID of molecular gases using vibrational excitation by a COJaser. The initial impetus to this work originated in the potential of LID for molecular isotope separation (Chapovsky et al., 1981). More recently, molecular systems have shown themselves to be particularly fertile and new light-induced kinetic effects have been observed (for a review see Hoogeveen, 1990; Hermans, 1992; see also Section VII).

Electrons in a semiconductor present a totally different system where light- induced drift was predicted to occur (Skok and Shalagin, 1980; Dykhne et al . , 1980; Luryi, 1987; Woerdman, 1987; Stockman et al . , 1990; Grinberg and Luryi, 1991). Here electron drift is expected when the electrons in ground and excited states obey identical dispersion laws; i.e., when the electronic energy bands are parallel. Situations where this occurs are Landau levels in a magnetic field or subbands of a two-dimensional electron gas. If the momentum relaxation is subband dependent we have a near-perfect analogy to LID in a binary gas mixture. So far only a single observation of this effect has been reported, in a sample of nondegenerate InSb (Kravchenko et al., 1983). Recently a controversy seems to have arisen whether one can really distinguish light- induced drift of electrons and the photon-drag effect (Grinberg and Luryi, 1991).

A sizable body of theoretical work followed the first description of LID of Gel’mukhanov and Shalagin (1979a). Almost all this work (Dubetskii, 1985; Gel’mukhanov and Shalagin, 1980; Zielinska, 1985; Nienhuis, 1986; Gel’mukhanov er al., 1986a, Gel’mukhanov 1986b, 1987; Nienhuis and Kryszewski, 1987; Kryszewski and Nienhuis, 1987b) was analytic and employed a two-level description of the absorbers and/or a rather simplified collisional model (i.e., either weak or strong collisions). Though yielding important qualitative understanding of the observed phenomena, such as the dynamics of the optical piston (Nienhuis, 1985, 1986) this work failed to give quantitative predictions for LID in alkali-noble-gas mixtures. For a quantitative calculation of the drift velocity of alkali atoms it was important to incorporate the details of the level structure of the alkali atom, the most important aspects being the hyperfine structure of the atomic ground state and the fine structure in the p-excited stated. In particular the ground-state hyperfine structure is a complicating factor, due to the associated pumping mechanism, in both the experiment and the theoretical description. Haverkort et al. (1988, 1990) incorporated the multilevel aspects in the theoretical description, using a four-level rate-equation model. The collisions are described by a suitable chosen kernel allowing for collisional interactions of any strength. This model, having no adjustable parameters, was highly successful in predicting drift velocities (Werij and Woerdman, 1988).

The extensive work on LID gave rise to the prediction and observation of

204 E.R. Eliel

many related phenomena, of which a few are: light-induced current (Atutov et al., 1984, 1986b), light-induced diffusive pulling (Atutov et al., 1986a, 1986~; Wittgrefe et al., 1991), light-induced viscous flow (Hoogeveen et al., 1989) and surface-light-induced drift (Ghiner et al., 1983; Hoogeveen et al., 1987; see also Section VII). Many of these ideas were first formulated and worked out in a rather ad hoc manner. Recently they were put into a consistent theoretical framework (Nienhuis, 1989; van Enk and Nienhuis, 1990a, 1990b; Nienhuis and van Enk, 1991). In this work light is considered as a thermodynamic force, affecting the velocity distribution of the gas (mixture) in a similar way as do gradients of temperature and pressure. We will discuss some of these ideas in Section 11.

The buffer gas plays a very important role in LID, transforming the label put on the absorber by the exciting light into the drift motion. The noble gases, having a closed-shell electronic structure, were considered the obvious collision partners in LID of alkali atoms. Na:Xe used to be the favorite combination of experimentalists until it was shown that simple molecular buffer gases gave rise to even higher drift velocities under comparable experimental conditions (de Lignie and Woerdman, 1990). That work was inspired by the idea that the molecular collision partner could quench the atomic excitation, transforming (part of) the excitation energy in translational energy of the collision partners. If the post-collision velocity has forward or backward preference, quenching could potentially lead to large LID effects. The experimental results showed that in all cases studied quenching collisions had a negative effect on the drift velocity (de Lignie and Woerdman, 1990) (see Section V). Inelastic collisions have also been shown to be important in light-induced kinetic effects in molecular systems, giving rise to an effect that has been called velocity-selective heating. This “heating” creates an anisotropy in the transport coefficients and therefore new transport phenomena (Hoogeveen and Hermans, 1990, 1991; Van Enk and Nien- huis, 1991).

A field where LID has shown promise in terms of applicability is isotope separation. Clearly, this transport phenomenon separates the active absorbers from the buffer gas: under the influence of the light one component drifts in one direction and, as a result of momentum conservation, the other component of the binary gas mixture drifts in the opposite direction. Soon after the transport phenomenon was first suggested it was realized that the two components of the binary mixture could also be two isotopes of the same species. In that case LID would separate the two isotopes. This was confirmed in an experiment on an isotopic mixture of CH3F, using a C0,-laser for (rotational-vibrational) excitation (Chapovsky et a/., 1981; Folin and Chapovsky, 1983; Chapovsky and Shalagin, 1987). Isotope separation of atoms has also been proven viable, the species being here rubidium immersed in a buffer gas (Streater et al., 1987, 1988). The experiment is shown in Fig. 4. When the optical shutter diode laser

LIGHT-INDUCED DRIFT

"Rb(F= 2 ) =Rb( F= 3) =Rb(F=2) "Rb(F= 1) .

I . . . , . . . , . . . I . . * .

205

temperature control

probe probe ,n , ,

diode laser 4

B (a) I computer I

loo I 80

P

8 60

e, Po

2 40 m

e,

L

a 20

FIG. 4. (a) Experimental arrangement for observing isotope separation in Rb. (b) Percentage of 87Rb observed in the probe capillary as a function of the frequency of the central mode of the spectrum of the optical shutter diode laser (power output 3.3 mW). The horizontal line indicates normal abundance. (From Streater et al . . 1988, with permission.)

206 E.R. Eliel

is appropriately tuned the Rb atoms are imprisoned (by LID) in the optical shutter capillary. At a different tuning of the shutter laser there is no imprisonment by LID and, depending on the tuning, either one or both isotopic species of Rb can appear in the probe capillary. Figure 4 also shows the isotopic composition of the Rb vapor in the probe capillary when the central frequency of the optical- shutter diode laser (3 mW optical power) is varied.

LID can also be applied to separate the ortho and para nuclear-spin modifications of a molecular gas (Krasnoperov et al . , 1984; Chapovsky et al., 1985; Bakarev and Chapovsky, 1986). Here we deal with one molecular species, one isotope, but still a binary mixture because of the presence of two nuclear-spin modifications. The ortho-para label is directly connected to the set of rotational quantum numbers of the molecule, and by selecting a specific rotational- vibrational transition, one excites either ortho or para molecules. LID can then be used to separate the two species, enabling the study of ortho-para conversion and thus contributing to the understanding of intramolecular relaxation processes (Chapovsky, 1990, 1991).

In recent years some astrophysical implications of LID have been suggested, in particular in connection with “chemically peculiar stars” (magnetic Ap and Bp stars) and the anomalous distribution of the deuterium/hydrogen ratio across the planets of our solar system (Atutov and Shalagin, 1987, 1988; Atutov 1988). For the chemically peculiar stars it was suggested that the thermal emissionfrom the core of the star could cause LID in the stellar atmosphere. A prerequisite is that Fraunhofer absorption occurs in the stellar atmosphere, modifying the local optical spectrum of the light. If the local spectrum displays asymmetry across an absorption line of some species that is present in the stellar atmosphere, LID could occur, causing a drift of that species. This type of LID was called white- light-induced drijii and discussed in terms of two-level absorbers (Popov et d . , 1989). To explain the anomalous distribution of the deuterium/hydrogen ratio in our solar system (see Fig. 5 ) in terms of LID, one can use a simple mechanism based on line emission and absorption. Here the process would have taken place in the early developmental stage of the solar system while the protosun was heating up as a result of gravitational contraction. Molecular species in the protosun would then emit in the infrared, the emission being absorbed by molecules in the protoplanetary cloud. However the emission and absorption lines were shifted with respect to one another as a result of the gravitational redshift of the emission line by the massive sun, providing the required velocity selectivity for LID. Assuming that the protosolar system was initially homogeneous in terms of molecular concentrations, it is clear that the line emission by molecules (e.g.. water molecules) containing a deuterium atom (HDO) was many orders of magnitude weaker than the emission of equivalent molecules containing only H atoms (H,O). Assuming, in addition, that the kinetic cross section increases upon excitation, LID then provides a mechanism to drive the all H-species out to the perimeter of the solar system enriching the D content near the center.


T h

L 4

10" 10l2

R (m) FIG. 5 . Deuterium/hydrogen ratio across the planets of our solar system. Data for Venus (a) are

from (Donahue et al., 1982), for Mars from (*) (Pinto et al . , 1986) and (A) (Owen et al., 1988) for Jupiter, Saturn and Uranus from (0) (Geiss and Reeves, 1981) and (0) (Owen et al., 1986). The cosmic background value approximately equals [D]/[H] = 2 X

The shape of the excitation spectrum has recently drawn interest (Popov et af., 1981; Werij et al., 1987; Gabbanini et af., 1988; Gozzini et al., 1989; de Lignie and Eliel, 1989; de Lignie et af., 1990). Partially this was related to the specifics of the canonical optical absorber Na; i.e., to alleviate the consequences of optical hyperfine pumping when a Na vapor is illuminated by light from a single-mode laser (Werij et af., 1987; de Lignie and Eliel, 1989). From the point of view of LID a more interesting perspective was the use of spectral shaping to increase the excited-state population without losing velocity selectivity, by spec- trally covering half the Doppler-broadened absorption profile (Popov ef af., 1981). We will return to this point in Section V.

In Section I1 we will shortly review gas kinetics in a light field for the simple case of a two-level absorber immersed in a buffer gas. Here one has a new thermodynamic force, i.e., the light, in addition to the more common thermodynamic forces like pressure and temperature gradients. In Section 111 we describe theoretical models for LID ranging from simple two-level models to more realistic multilevel models for the atomic dynamics. Techniques for measuring the drift velocity are discussed in Section IV, and experimental results are discussed in Section V. The body of the work discussed in that section pertains to LID of Na where the Na atom experiences a multifrequency laser field. We discuss coherent population trapping in LID, the application of broadband excitation and the use of molecular buffer gases. Section VI contains a discussion of some varieties of LID that may have astrophysical implications and a discussion of first experimental efforts to test these ideas in the laboratory. In Section VII LID is placed in the wider framework of light-induced kinetic effects, and Section VIII contains our conclusions.

208 E.R. Eliel

II. Gas Kinetic Effects of Light

The prediction of light-induced drift by Gel’mukhanov and Shalagin (1979a) prompted a flurry of suggestions for new light-induced kinetic effects. The element common to these suggestions was that there was a new tool to directly modify the velocity distribution of the gas and thus create nonequilibrium conditions (for the translational degrees of freedom) in a gas without the use of gradients of temperature, concentration or pressure. The factors essential to almost all light-induced kinetic effects are velocity-selective excitation and state- dependent relaxation characteristics.

Velocity-selective excitation is achieved using the Doppler effect and near- resonant excitation by light with beam-like properties; i.e., having a well-defined directionality. In most discussions of light-induced kinetic effects the light is characterized by a well-defined k-vector. This choice makes the problem essentially one-dimensional, and one needs only to consider a single Cartesian velocity component v, . Velocity-selective excitation by itself does not alter the total velocity distribution f( V J ; it is still an equilibrium distribution. The velocity distributions fe( v,) and f,( v,) of the excited and ground-state particles are obviously highly nonequilibrium. The state-dependent relaxation characteristics (collision rates or kernels, or accommodation coefficients) then break the symmetry between the direction parallel to k and antiparallel to k and the total velocity distribution f( v z ) is no longer a Maxwellian.

When the absorbers are embedded in a buffer gas of much higher density, the description of the kinetic effect, i.e., light-induced drift, is relatively simple (Gel’mukhanov, 1985; Gel’mukhanov et al . , 1986b; Nienhuis, 1985; Nienhuis and Kryszewski, 1987; Kryszewski and Nienhuis, 1987b; Van Enk and Nien- huis, 1990b). ‘The buffer gas acts as a heat bath remaining in thermal equilibrium. The velocity distribution of the active particles evolves rapidly under the influence of the radiation and collisions; this evolution is local. The density of these particles is the only conserved quantity.

The radiation field does not only change the steady-state velocity distribution but it also affects the transport properties like the heat conductivity or the viscosity. For one, the directionality of the light breaks the usual spherical symmetry of the microscopic evolution into one of cylindrical symmetry. Hence, we expect a difference between longitudinal (11 k) and transverse (I k) components of the transport coefficients (Nienhuis, 1989). Another consequence of the reduction of the symmetry of the system is a large increase in the number of transport coefficients.

Nienhuis and van Enk (Nienhuis, 1989; Van Enk and Nienhuis, 1990a, 1990b, 1991) set up a general theoretical framework to describe gas kinetics in

* Note that gas-kinetic transport coefficients are already modified when the excitation is broad band. The effect of light-induced diffusive pulling provides an example of a light-induced kinetic effect that can occur under broadband irradiation.


a light field. Many of the effects they discuss had been treated before, in particular in the Soviet literature (Folin et al., 1981; Ghiner, 1982; Ghiner et al., 1982; Gel’mukhanov, 1985; Gel’mukhanov et al., 1986b; Atutov et al . , 1986a). The work of Nienhuis and van Enk (as almost all theoretical work) is based on a two- level description of the active particle. The evolution of the velocity-dependent density matrix for the internal state of the atom is then described by the Bloch-Boltzmann equations, from which the rapid evolution is eliminated.

A. DILUTE VAPOR IN A BUFFER GAS

As the focus is on light-induced drift we will limit the discussion to the kinetics of a dilute vapor in the presence of a much more abundant buffer gas.

I , Microscopic Evolution

For a dilute vapor in a buffer gas, the system in which light-induced drift occurs, the microscopic state of the system can be described by the distribution functions fp(v, r, t) for the excited-state particles and f,(v, r, t) for the ground-state particles.

Under conditions where the bandwidth of the radiation field is larger than the homogeneous width of the optical transition the coherence between the two atomic levels can be neglected, and one arrives at rate equations for the velocity distributions

with A the spontaneous decay rate and h(v, r) the velocity-dependent stimulated transition rate, proportional to the local intensity f(r). The collision terms J , , J , are commonly expressed as

J,v,) = - r , ( w x V ) + I d V r ~ , ( v ’ + v)f,(v’).

I dvK,(v’ .-+ v) = r,(v’) .

(4)

Here i = e, g and r, is the (velocity-dependent) rate for velocity-changing collisions. K,(v’ + v) is the associated collision kernel, and because of particle- number conservation, one has

( 5 )

Instead of the rate-equation approach, as adopted here, the full Bloch equations for the density matrix for the internal state can be used (Nienhuis, 1985; Nien- huis and Kryszewski, 1987).

210 E.R. Eliel

In order to obtain transport equations for the active atoms one now introduces a separation of time scales. The radiative (A, h) and collisional ( J ) terms in Eqs. (3) describe the rapid evolution of the system, whereas the macroscopic gradients determine the slow evolution. These equations can be written formally as

(: + v * V)f = & - I ( % + 3)

where f is a vector containing the distribution functions

f = Cfp f A (7)

and % and 3 stand for the collisions and radiative transitions, respectively. The distribution functions are expanded according to

(8) f = f, + &fl + . . .

2. Macroscopic Quantities

We are not really interested in the velocity distributions fe and f, as we are studying a macroscopic phenomenon to be described by equations for macroscopic quantities. Obviously the latter quantities are linked to the velocity distributions. A macroscopic quantity is a quantity that is conserved when the slow terms are left out of the transport equations; i.e., it does not change under the action of radiative transitions or collisions with the buffer gas. As the active particles can freely exchange energy and momentum with the buffer gas, there is but one such quantity; viz., the active particle density n(r , t ) .

Integrating the sum of the two equations (3) over v one obtains

with

n(r, t ) = J dvf(v, r , t ) (10a)

(lob)

the particle density and flux, respectively. Equation (9) is nothing but the continuity equation for the particle density.

j(r, t ) = J dv v j ~ v , r, t ) ,

3 . Drqt Velocity

The zeroth order distributions fco and fpo are determined by the E - I term in Eq. (6)


%, + 9, = 0. (11)

The collisions and the interaction with the light field determine, for a given local particle density n(r, r ) , the zeroth-order steady-state solutions for the velocity distributions fe0 and f n o . These velocity distributions in turn determine the zeroth-order flux

One introduces the drift velocity Vdr by

.io(r) = n(r)vdr(r). (13)

We see that the drift velocity appears as a result of the modification of the zero- order solution.

4. Dirusion Coeficient

To order E O Eq. (6) becomes

($ + v * V)f0 = (el + 3,. Here, fo depends on r , and on r through the density distribution and the local intensity I @ ) . Writing

af, af, an at an at ’ - - _ _ -

and

Of, = Vn- af0 + VI-- , afo

an ar we obtain for the left-hand side of Eq. (14)

where df,/an = f,/n. The first-order velocity distributions f e , and fgl are determined by the gradients of the light intensity and the particle density. The first- order flux is given by an expression in the form

j , = - D . V n + n C * V I (1 8a)

= - D - V n + n u l . (18b)

Here D and C are cylindrically symmetric tensors, each with a longitudinal and a transverse component. The first-order flux is driven by the gradients of the

212 E.R. Eliel

particle density and of the light intensity. We see that the second term in the first- order flux has the appearance of a correction to the drift velocity (Nienhuis and Kryszewski, 1987). Contrary to the drift velocity proper, which is always parallel to the wave vector k, the gradient velocity u, can have components parallel and perpendicular to k as it is driven by the gradient of the light intensity.

A simple interpretation of the gradient velocity is given by Nienhuis and Kry- szewski (1987) assuming that the diffusion tensor is isotropic but state dependent. The diffusion tensor and gradient velocity then take the form

with p i the probability to find the active particle in state i :

and

P e + Pn = 1.

For the gradient velocity we then can write

We see that the gradient velocity actually gives rise to a diffusivelike flow. If D , > D , the gradient velocity is parallel to the gradient in p e , causing particles to flow to regions of high intensity. In steady state this will give rise to an increased particle density in these high-intensity regions. This is the effect of light- induced diffusive pulling (Gel’mukhanov and Shalagin, 1979b; Atutov ef al., 1986a, 1986c; Wittgrefe et af., 1991a). Atoms in the excited state have a larger kinetic cross section and consequently a smaller diffusion coefficient than atoms in the ground state. When the particle density is initially uniform, the diffusional flow out of a high-intensity region is smaller than the diffusional flow into it, because the inward flow consists of ground-state atoms only and the outward flow is a mixture, with, on average, a smaller diffusion coefficient. This unbal- ance of diffusional fluxes drives a density gradient that compensates for the gradient in diffusion coefficient. We see that the gradient velocity, which appears in the particle flux in first order, is essential in describing this effect.

Explicit results for the diffusion coefficient have been obtained for some simple model systems (Kryszewski and Nienhuis, 1987a; Nienhuis and van Enk, 199 1). Assuming weak velocity-selective excitation, the stimulated transition rate can be written as

(23) h(v, r) = h0W + &hl(v, r),

and in the hard-collision approximation the expression for the diffusion coefficient becomes


1

0.9

b

0.8

0 1 2 3 4 5

B J / A FIG. 6. Reduction factor u for the diffusion coefficient as a function of the stimulated transition

rate &f/A = ho/A for atoms immersed in a buffer gas. The excitation is broad band and T,/T, = 3/2, r, = A. Here r, represent the rates for velocity-changing collisions in state i and A the Ein- stein A-coefficient. The high-intensity limit of the reduction factor u is given by 2r , / (A + re).

kT D = -U

mr,

where u is a factor parametrizing the reduction of the diffusion coefficient. The dependence of u on the light intensity, parametrized by BI,IA = ho/A (see Eq. (23)) is shown in Fig. 6 . Note that in this model of weak velocity selectivity the diffusion coefficient is a C-number.

Also the results of Kryszewski and Nienhuis (1987a) on the diffusion tensor were calculated in the hard-collision approximation, using a step function for the spectral distribution. Explicit expressions for the parallel and transverse (to the k vector of the incident light) diffusion coefficients were given as a function of laser intensity and of the position of the step in the excitation spectrum.

III. Models for the Drift Velocity

There are two dominant currents in the theoretical description of light-induced drift, those in which a two-level description of the optical absorber is taken and those where the multilevel properties of the optical absorbers are considered essential. A two-level description is well suited for a qualitative description of the vast majority of the observed phenomena. For a quantitative comparison of experimental data with models for light-induced drift a multilevel description of the optical absorber has been found to be essential for most atomic systems studied.

All models use a one-dimensional description of LID. Therefore there is velocity selectivity for only one component of the velocity, which is chosen to be the z-component. This approach is also adopted in the present work.

214 E.R. Eliel

A. COLLISION MODELS

Two collision models have found widespread application in the description of light-induced drift; i.e., the strong-collision model and, to a lesser extent, a model using Keilson-Storer kernels. In the strong-collision model the collision term for atoms in state i obeys the relation

Z , f I ( V J = T,[n,W(v,) - f l ( V J 1 9 (25)

where the population n, of level i is given by (see Eq. (10))

n, = 1 dVzf,(V;), (26)

and r, is the rate of strong collisions, assumed to be velocity independent. The term W( v,) is the one-dimensional Maxwell distribution

W ( V ~ ) = ( v o f i ) - l exp[-(vi/vi)], (27)

with vo = V‘m, the most probable velocity.

equivalent of Eq. (4): For the Keilson-Storer collision model we have to use the one-dimensional

The- Keilson-Storer kernel is of the form (Keilson and Storer, 1952)

with

fff = ( I - (a,KS)2)v6

r, = ry. and

The parameter aKS, with 0 S aKS < 1, measures the strength of the collision; i.e., the average change of the velocity in a collision: a = 1 corresponds to soft collisions where the velocity does not change appreciably in a single collision and a = 0 corresponds to the strong-collision limit; i.e., the limit where the postcollision distribution is a Maxwellian.

B. TWO-LEVEL MODELS FOR THE DRIFT VELOCITY

An intuitive picture of light-induced drift, with monochromatic excitation, is provided by a one-dimensional random-walk model (see Fig. 7). Due to the


VL vz -VL 0

U L

FIG. 7. One-dimensional Maxwell distribution. Particles in the velocity class around vL (resonant with the exciting laser) and - vL (nonresonant) have mean-free paths equal to e , and 4,. respectively.

excitation the symmetry of the velocity distribution is broken for the pair of velocity classes around v = IvLI. At the resonant velocity v = v L there are n, particles in the excited state; their step length equals the mean-free path in the excited state C,. The same number of particles at the velocity v = - v L have a step length equal to the mean-free path in the ground state C,. For each particle the number of steps per second equals IvLllC with C = 1/2(C, + CK). The net distance covered in a step (on average) equals 1/2(C, - C,) and the number of participating particles equals 2n,. The flux is then given by

and the drift velocity by

with n the total number of active atoms. From Eq. (33) we see the essential properties of the drift velocity. The quantitative value is determined by the fraction n,ln of particles that are in the excited state (and have not undergone a velocity-changing collision), by the tuning of the laser (through vL) and the change in kinetic properties of the absorbing particle upon excitation (through the factor (C, - C J C ) . As in most cases C, > C, the drift velocity is antiparallel to the selected velocity. Lawandy (1986) has given a more elaborate treatment of the use of a one-dimensional random-walk model for light- induced drift.

216 E.R. Eliel

1. Analytic Models f o r the Drift Velocity

An analytic expression for the drift velocity of two-level atoms, based on the Bloch-Blotzmann equations, has been found for both a strong-collision model (Nienhuis, 1985, 1986; Gel'mukhanov et a l . , 1987) and a model using Keil- son-Storer kernels to describe the velocity-changing collisions (Kryszewski and Nienhuis, 1987b).

Atomic Evolution. The generalized Bloch equations, describing the evolution of the velocity-dependent matrix can be cast in tensorial form (Werij and Woerd- man, 1988; Haverkort and Woerdman, 1990; Werij et a l . , 1987)

1 ( i t - + v,- :J p = --w, ti p1 - d p + 2 p , (34)

where d and 2 are tetradic operators describing the spontaneous emission and the collisions, respectively. The Hamiltonian X has matrix elements

(35)

where Aw, is the energy of level i and d,J is the transition dipole moment. E,, = [El is the amplitude of the oscillating electric field

(36)

at frequency wL = w,. The atomic transition frequency is given by w , = ( E , - E,)/A in terms of the energy difference between levels i and j . For a two- level atom, Eq. (34) translates, in the rotating-wave approximation, into

X,, = A o J , ~ , , - d , J * E ,

[E(z, t)l = E , C O S ( ~ Z - w L t ) ,

with

Here I' = I ' P h + 1/2A is the collision-broadened homogeneous width with Ph the rate of phase-intemption collisions. The terms 2 , , p , , ( v L ) represent the velocity-changing collisions in level i and R is the Rabi frequency


d, * E a = - h ' (39)

In steady state one can eliminate the coherences prg(vr ) and p,,(v,) from Eqs. (37) and obtain a set of rate equations for the populations in the ground and excited states:

- APee(Vz) + ~ e e c ~ e X v z ) ~

which is of the same form as Eq. (3) when we identify the diagonal elements of the density matrix p, , (v , ) with the velocity distributions f i ( v , ) . A is the spontaneous emission rate and the excitation function h( v,) is given by

with A = wL - w,, the detuning of the laser. The steady-stare solution of the rate equations yields the velocity distributions; this solution has been given in the strong-collision approximation (Eq. (25)) for both arbitrary values of the rates for velocity-changing collisions r, and r, (Gel'mukhanov et af., 1986b, 1987) and in the limit (r, - T,) << r, (Nienhuis, 1986). In the strong-collision approximation the drift velocity, in zero order in the sense of Section 11, is given by (Gel'mukhanov et af., 1987)

Note that the LID function "(A) differs from the detuning function +(A) used by many authors (Mironenko and Shalagin, 1981; Gel'mukhanov, 1985, Van der Meer et al., 1989). The latter is given by the ratio of the LID function "(A) and the Voigt absorption profile V(A).

218 E.R. Eliel

0.4

0.2

0 0 1 2 3

U k V O FIG. 8. Scaled LID function E(A) = (k/fir)'P(A) as a function of the normalized detuning

A/(kvo) for various values of the normalized homogeneous linewidth (Tlkvo). These curves display the expected frequency dependence (at constant optical power) of the drift velocity for a two-level atom in the low-intensity limit.

a Voigt line profile. Here

is a saturation-broadened Lorentzian. K = I / I , is the inhomogeneous saturation parameter and I, is the saturation intensity. The latter takes the form (Gel'mukhanov er al. , 1987)

with

A f g g = - 27r (47)

the homogeneous absorption cross section for zero pressure and detuning. The factors 2r lA and 2/Ar, represent the modification of the saturation intensity as a consequence of collision broadening and velocity-changing collisions in both ground and excited states. The time constants T, and T* appearing in Eqs. (42) and (46) are given by


2 7 2 = - - 71. A

In the limit re = I?, these times are simply twice the residence time in the resonant velocity class in the excited state (T~) and twice the time the particle hangs on in the excited state after a velocity-changing collision (T~). In the case that is relevant for LID, re + r, there is no such simple interpretation of these times (Gel'mukhanov et al . , 1987).

Equation (42) fully describes the dependence of the drift velocity on the detuning, the radiation intensity I and the buffer-gas pressure p through the factors "(A), K and Y(A) . In the low-intensity limit (large K - I ) Eq. (42) can be simplified to

showing that the drift velocity is proportional to the light intensity.

weak-field absorption spectrum: It can be shown that the LID-function * ( A ) is simply the derivative of the

i.e., in the low-intensity limit the dependence of the drift velocity on the detuning is dispersivelike.

In the Doppler limit ((r2 + A2) << kv,) Eq. (42) reduces to (Gel'mukhanov et af., 1987)

where the function Z(K, A) describes the dependence of the drift velocity on detuning and intensity

There are two saturation parameters in the Doppler limit (Nienhuis, 1986). In addition to the inhomogeneous saturation parameter, K , for the resonant velocity class there is a second saturation parameter, [ K , describing the saturation with

This relationship is sometimes referred to as the Bakarev-Folin theorem (1987)

220 E.R. Eliel

respect to the total velocity distribution. The latter is a direct consequence of the velocity-changing collisions.

For a description of light-induced drift in an optically thick system one also has to take along an equation describing the evolution of the local intensity; i.e., Beer’s absorption law. The absorption law has the general form (Gel’mukhanov et al . , 1987)

_ - aK A Y ( 4 - -nu:- az 2r 1 + K - ’ + ~(L\)T~/TI’

( n is the particle density) and in the Doppler limit it reduces to

aK A 71 _ - - -nu1 - -Z(K, A). az 21- 72

(53)

(54)

Combining Eqs. (42) and (53) yields an expression for the drift velocity in terms of the decrease of the radiation intensity

re - r, 1 1 *(A) a1

A + r, nhwo Y ( A ) az Vdr = r,

where n,/n is the fraction of particles in the excited state that have not yet col- lided. In the Doppler limit Eq. (55) becomes

r, - r, 1 1 ar A + r, nAoo az

V,Jr = -vL - - - r,

Although the latter equations are appealing in their simplicity, no insight is obtained on the variation of the drift velocity with experimental parameters.

C. MULTILEVEL MODELS FOR THE DRIFT VELOCITY

A two-level model, as presented in the previous section, provides fruitful insight in the variation of the drift velocity when various experimental parameters, such as buffer gas pressure, detuning and the intensity or the spectral distribution of the radiation field are varied. Yet nature does not provide atoms or molecules that, in the presence of collisions, can adequately be described as two-level systems. In LID of molecules the excitation is rotational-vibrational, and inelastic (rotational and, to a lesser extent, vibrational) collisions vastly enlarge the number of levels involved in the LID process. LID in atoms has so far been restricted


to the study of alkali atoms, i.e., Na and Rb; in these systems the fine, hyperfine and Zeeman substructure of the levels that are coupled by the radiation field destroys the two-level simplicity.

1 . Analytic Models

The importance of the multilevel aspects of the active particle in LID was realized almost from the outset, and the first theoretical study of LID in multilevel systems was presented in a paper by Mironenko and Shalagin (1981). Using a strong-collision model for the velocity-changing collisions they obtained an expression for the drift velocity in a molecular system having the same generic form as Eq. (42). Compared to the result of Eq. (42) the drift velocity is reduced by a factor representing the thermal population of the rotational-vibrational level from which the excitation starts. This latter factor appears quite naturally: the large number of rotational levels in the vibrational ground state dilutes the number of particles that can participate in LID. The ratio (see Eq. (42)) can be very large as T~ depends on the vibrational relaxation time, which, in many cases, is two or three orders of magnitude larger than the rotational or kinetic relaxation time. This also reduces the drift velocity; again the fraction of molecules that can participate in the LID process is reduced. Note that the residence times T~ and T~ differ from the explicit form of Eqs. (48).

For atoms Mironenko and Shalagin (1981) considered a three-level system (see Fig. 9), where two of the levels are connected by the radiation field and the third level is coupled to either the upper or the lower level by (inelastic) collisions. For the “V-type’’ three-level system an expression for the drift velocity is found that is very similar to that of Eq. (42). Here it is assumed that the inelastic collision fully thermalizes the velocity distribution. For the ‘‘/\-type” system an extra term appears in the expression for the drift velocity, due to the fact that the velocity distribution in level 3 is not a thermal distribution but bears the imprint of the velocity distribution in the excited state. The drift velocity is obtained by the following substitution:

where A3+, represents the spontaneous emission rate for the 3 + g transition. Note that LID can appear even when rR - r, = 0, as implied by the second term in Eq. (57).

The effect of level degeneracy and the polarization of the exciting light on the drift velocity in molecular systems was investigated by (Gel’mukhanov et al., 1986a) using a semiclassical description of the rotational motion. The polarization and the degeneracy of the magnetic sublevels affect the drift velocity via the LID function *(A) (see Eq. (42)). Within the strong-collision approximation the effects are relatively small (10-20%).

E.R. Eliel 222

3-

(a) (b) FIG. 9. (a) “V-type” and (b) “A-type” three-level systems, excited by a single-frequency laser.

The first study of the effect of an atomic hyperfine splitting in the ground state on LID was contained in the analytic work of Par’khomenko and Shalagin (1986) employing a rate-equation model with strong collisions to describe the evolution of the system. In the presence of a resonant monochromatic radiation field, velocity-selective optical hyperfine pumping will occur, transferring, without a velocity change, atoms from the hyperfine level, where they resonantly interact with the laser field, to the level where they interact nonresonantly. If the atoms experience a collision when they are in the excited state, the optical pumping is non-velocity selective. If the ground-state hyperfine splitting hahfs is on the order of the Doppler width (as is the case for Na) the laser can be resonant with both hyperfine transitions, creating a Bennett peak and dip in the velocity distributions of the atoms in both lower levels (see Fig. 10). It is clear that this three-level model is appreciably more complex than the two-level model discussed before. In order to be able to use a rate-equation model it was assumed that the Bennett peaks and dips do not overlap; this uncoupling condition corresponds to the condition ( 1/2fi)2 << (rAuhfs), with Auh,s, the ground-state hyperfine splitting, fi, the Rabi frequency (Eq. (39)) and I‘, the collision-broadened homogeneous linewidth (Werij and Woerdman, 1988; Haverkort et al., 1988; Haverkort and Woerdman, 1990). An additional approximation was made; namely, the assumption of fast collisional relaxation between the two lower levels. However, for alkali-noble-gas systems this assumption is generally invalid since the cross section for relaxation between the ground-state hyperfine levels ranges between

cm2 (Happer, 1972). The last assumption seriously reduced the impact of the work of Par’khomenko and Shalagin on the quantitative treatment of LID in alkali atoms.

2 . Numerical Models

A fundamentally different approach was taken by the Leiden group renouncing the analytical approach. Rather, a numerical approach was adopted (Haverkort


2

-1000 0 1000 (b) vz (m/s)

FIG. 10. (a) Level scheme and (b) velocity distributions of the particles in the various levels.

et al . , 1988; Haverkort and Woerdman, 1990), enabling a quantitative treatment of LID for alkalis and a comparison with experimental results on the drift velocity in Na-noble-gas mixtures, results that were obtained concurrent with the development of the model. Their approach is based on the work of Berman (Berman, 1972, 1978; Berman et al . , 1979, 1982), incorporating the effect of collisions in the optical Bloch equations, with a collision kernel to describe the velocity-changing and state-changing collisions. This model allows for arbitrary intensities (Bloch equations) and contains a collision model, the Keilson-Storer

224 E.R. Eliel

4 -

hl,

1 -

k

a

L

f

-3

- 2

FIG. 1 1 . Simplified energy-level diagram of Na. Levels 1 and 2 are the F = 1 and F = 2 hyperfine levels of the 3s 2S,,2 ground state and levels 3 and 4 the resonant and nonresonant fine- structure levels of the 3p *P state. The terms h , are the velocity-selective excitation rates, A,, the spontaneous-emission rates and 7, represent the velocity-changing collisions. Fine-structure mixing is indicated by FSC.

model (Keilson and Storer, 1952) (see Eq. (29)), that had proven its value in the analysis of optical double-resonance experiments. A four-level description of the Na atom was chosen, two hyperfine levels in the ground state and two fine- structure levels in the excited state. This description is both sufficiently transparent and sufficiently complete to describe the essentials.

The simplified energy level diagram of Na with the radiative and collisional couplings is shown in Fig. 11; levels 1 and 2 represent the ground state 3s 2S,,2 (F = 1) and 3s *S,,,(F = 2) hyperfine levels, respectively. Level 3 is the fine- structure level with direct access by the laser (mostly 3p 25,2), and level 4 is the nonresonant fine-structure level. The hyperfine and Zeeman structures in the excited state are neglected. This is well justified5 in experiments using monochromatic excitation at sufficiently high intensities and buffer-gas pressures p > 1 Tom, as the rates of collisional F and m, mixing are rather high then (Papp and Franz, 1972; Gay and Schneider, 1976).

Equation (34) again provides a full description of the evolution of the system but the explicit form is much more involved than given in Eqs. (37). Apart from velocity-changing collisions in each of the levels 1 through 4, there are collisions

Note that for polychromatic excitation this is not always true, as discussed in Section V.


that induce a transition between the two excited states (fine-structure changing collisions). The effect of collisions on the coherences pjk with i = 1 , 2 and k =

3, 4 is to destroy them; the decay of these coherences is described by the rate of phase-perturbing collisions r$. For Na colliding with a noble-gas atom, the ground-state coherence plz is not destroyed in the collision because the collisional interaction is identical for both hyperfine levels. The Bloch equations then yield a set of 10 coupled differential equations for the time derivatives of the populations pII through p4, and for the coherences p13, pZ3 and pI2 (and their complex conjugates). The Leiden group made an ad hoc substitution for the free- flow term

to account for equilibrium atoms diffusing into the illuminated volume. The coefficient Tr was called the transit relaxation rate. In most cases the free-flow term can be neglected altogether, an approximation that is adopted also here.

Rate Equations. An appreciable simplification is achieved when the evolution of the system is described by rate equations rather than by the Bloch equations. Compared to the 10 coupled equations of the Bloch model, there are only four equations for the velocity-dependent populations p, , (v I ) , i = 1-4 in a rate-equation model. This step was made (Haverkort et al . , 1988; Haverkort and Woerd- man, 1990; Werij and Woerdman, 1988) by neglecting the ground-state coherence p 1 2 and eliminating the remaining coherences from the Bloch equations. One can justify neglecting the ground-state coherence when a single-frequency laser field is used at power levels such that R2 << L\w,r, with R the Rabi flop- ping frequency (Eq. (39)), Aq , s the ground-state hyperfine splitting and r the collision-broadened homogeneous linewidth. In all experiments on LID of Na using a single-frequency laser field this condition is fulfilled as it corresponds to a laser intensity I << 30 W/cm2. In essence the preceding condition implies that the laser should interact with nonoverlapping velocity classes in levels 1 and 2. Then only two levels are coupled by the laser field for each of these velocity classes, allowing one to eliminate the interlevel coherences plk , i = 1 , 2, k = 3, 4. Note that these arguments can be invalid when a two-frequency field is applied, as will be discussed in Section V in connection with coherent population trapping in LID.

The resulting rate equations for the velocity-dependent diagonal elements6 of the density matrix p, (vz ) = p, , (v , ) are

6 Note that the elements p , ( v . ) are nothing but the velocity distributions f,(b':) of the two-level model.

226 E.R. Eliel

- A3P3 + 3 3 3 P 3 - 3 3 4 P 3 + 3 4 3 P 4

-A4p4 + 3 4 4 ~ 4 - 3 4 3 ~ 4 + 3 3 4 ~ 3 7 (594

where g, is the degeneracy of level i and Ak = A l l + A,,, k = 3 , 4 are the spontaneous emission rates. The fine-structure changing collisions are represented by the terms proportional to 334 and 343 in the last two equations of Eq. (59). The velocity-selective excitation rates are given by (see Eq. (41))

_ - aP4 - at

(60) IB r cT (wL -

h,,(v,) = - F N , ~ + r2’

with I the intensity of the laser field, B,, the Einstein B-coefficient, defined as in (Loudon, 1983), r = Ph + 1/2A the homogeneous linewidth, wL the laser frequency and hw,, the energy splitting between levels i and j. As we are interested in steady-state phenomena the time derivatives in Eq. (59) are set to zero. Collisions. The terms representing velocity-changing collisions are expressed in terms of a collision kernel (see Eqs. (4) and (28)) for which the Keilson- Storer kernel (Keilson and Storer, 1952) was chosen (Haverkort et al., 1988; Haverkort and Woerdman, 1990) (see Eqs. (29) and (30)); this choice has been extensively motivated (Haverkort and Woerdman, 1990).

In order to properly describe the velocity distributions-which were measured in a set of laser-spectroscopic experiments (Haverkort et al., 1987)-a composite collision kernel was chosen to encompass both small-angle (SAS) and large- angle (LAS) scattering:

K,,(v: + v,) = K f S ( v : + v,) + K Y S ( v : + v J , (61)

each of these kernels being a Keilson-Storer kernel. In the two lower states only velocity-changing collisions come into play. Of

the four parameters describing the effective kernel, two can be fixed; i.e., aLAS by the hard-sphere value (Borenstein and Lamb, 1972) and rLAs by the diffusion coefficient D , which can be expressed in terms of the Keilson-Storer parameters

(62)

with mA the mass of the Na atom and k, Boltzmann’s constant. Values for the Na-noble-gas diffusion coefficient with the Na atom in the ground state are

[(I - a L A S ) r L A S + (1 - a s ~ ~ ) r s ~ ~ ~ , - - kBT - r d i W = m*D


available in the literature (Hamel et a l . , 1986). The contribution of the weakly velocity-changing collisions a:As = 1 were taken into account using a single parameter riyS;Sf = ( 1 - aSAS)rSAS to fit the velocity distributions.

In the excited state, fine-structure mixing collisions occur in addition to the velocity-changing collisions; actually these collisions should be treated hand in hand, a result of the nonadiabaticity of the collisions. The sudden approximation was chosen to describe the fine-structure mixing, yielding the following relations between the transfer rates:

For Na the exponential factor exp( - (E, - E 4 ) / k B T ) = 1. All details of the excited-state velocity distributions can be described in terms of a composite Keil- son-Storer kernel as in Eq. (61) (see Haverkort et al . , 1987). Again we have four parameters aLAS, rLAS, aSAS and PAS to be determined. For aLAS the hard- sphere value, which depends only on the mass ratio of the active atom and its collision partner, was chosen (Borenstein and Lamb, 1972); the sum rpLAS + rSAS = r? is fixed by the total transfer rate rp, for which experimental data are available (Gay and Schneider, 1976). An additional constraint comes again from the diffusion coefficient (see Eq. (62)). Values for the Na-noble-gas diffusion coefficients, in the sudden limit, with the Na atom in the excited state, are given by Hamel et al. (1986). Finally aSAS was determined from a fit to the excited-state velocity distributions.

The measured parameters for the Keilson-Storer kernels can be found in Table I (for more details, see Table 1 of Haverkort et al. (1987) and Table l c of Haverkort and Woerdman, (1990). The strength of the method chosen by the Leiden group is that for Na-noble-gas systems all parameters in the model for LID are known, and consequently it has quantitative predictive power (see later).

The preceding discussion might leave the impression that a large set of kernel parameters needs to be known in order to predict the drift velocity for a specific alkali-noble-gas combination. Fortunately, numerical studies showed that the drift velocity is determined only by large-angle scattering and, in addition, that the drift velocity does not depend on rkAS and akAS separately, but only on the product (1 - a,4As)r,4As; i.e., on the diffusion coefficient for state i (see Eq. (62)) (Haverkort et a l . , 1988; Haverkort and Woerdman, 1990), in accord with results from two-level theory (Kryszewski and Nienhuis, 1987b).

Drijt Velocities, Numerical Results. Figure 12 shows the dependence of the drift velocity of Na on the tuning of the single-frequency laser for Xe pressures of 1, 3 and 10 Torr, as calculated with the rate-equation model. Note that the absolute values of the drift velocity are on the order of a few m/s, appreciably smaller than the rough estimate (for a two-level atom) of Section I. The culprit here is hyperfine pumping, an effect that becomes less important for increasing

TABLE I

Na-Ar 0.2 0.975 6.84 20.9 8.61 26.3 34.9 61.5 50.3 0.10 Na-Xe 0.0 0.975 8.58 8.11 12.8 12.1 25.0 58.8 51.8 0.088

Parameters specifying the collisional interactions for Na-Ar and Na-Xe at a gas pressure of I Torr and a temperature of 400 K. The rates r[FS and T,u" for the ground state are for truly velocity-changing collisions (large-angle and small-angle scattering, respectively). The excited-state rates describe both changes in velocity and in fine-structure state. Note that rp = r!? + r,?'. The rates for phase- perturbing collisions are indicated by V, and T r represents the rate for transit relaxation, two orders of magnitude smaller than the other rates. All rates are in units lo0 S K I .


5

1

G f 9-10

-2 0 2 4 Detuning ( G H z )

(b) Detuning (GHz) FIG. 12. (a) Calculated drift velocity of Na in Xe as a function of the detuning of the single-

frequency laser for the *S, ,*(F = 2) + 28n transition. The laser intensity equals I = 10 W k m 2 and the Xe pressure equals 1 , 3 and 10 Torr. Heavy bars indicate the centers of the F = 1 and F = 2 Doppler profiles. (b) Solid curve as in (a) for 10 Torr Xe; other curves show the contributions to the drift velocity from atoms in the four atomic levels. Note that the various contributions to the drift velocity are large and cancel each other almost completely. (From Werij and Woerdman, 1988, with permission.)

230 E.R. Eliel

buffer gas pressure, as is evident in Fig. 12. The shape of the curve is not perfectly antisymmetric as for a two-level atom; it bears the imprint of the hyperfine structure of the 2S,,, state.

The importance of the four-level description for LID of Na becomes particularly clear when we look at Fig. 12(b). Here are shown the individual contributions of the four levels to the drift velocity

together with vdr itself (10 Torr Xe). The contributions of the individual levels are large, but cancel each other almost completely. Full cancellation, and therefore zero drift velocity, occurs ~ 0 . 5 7 GHz above the transition starting from the F = 2 level. Note that the total drift velocity always has a sign opposite to that of the atoms in the excited states; i.e., the atomic flux is dominated by the ground-state flux, a result that is not at all surprising as the atoms in the ground state have a larger mean-free path. The variation of the drift velocity, for single- frequency excitation, as a function of a large set of experimental parameters are to be found in Werij and Woerdman (1988) and Haverkort et al. (1988).

Strong-Collision Model. The velocity distributions pertaining to each individual atomic level are clearly sensitive to the detailed aspects of the collisional interaction. The numerical work of Haverkort et al. (1988) showed that the dr$t velocity is highly insensitive to these details. Largely, this reflects the insensitivity of the drifty velocity to small-angle scattering, in obvious contrast to the velocity distributions of atoms in a specific level. So the drift velocity is essentially determined by the large-angle scattering collisions, more precisely by products of the type r( 1 - a) , quantities determined by the diffusion coefficient for that atomic state (Haverkort et al . , 1988; Haverkort and Woerdman, 1990; Kryszewski and Nienhuis, 1987a, 1987b, 1989). This observation was the starting point for the work of Streater and Woerdman (1989) on a strong- collision model for LID, an approach that requires much less numerical effort than the model with Keilson-Storer kernels. The disadvantage of the strong- collision model is that the predicted velocity distributions are of little value.

The strong-collision model of Streater and Woerdman is a straightforward extension of the model described previously. In the strong-collision limit aLAS + 0 and aSAS + 1; i.e., small-angle scattering (SAS) does not lead to any change of velocity of the Na atom in a collision. For the ground states the effective strong-collision rate is then directly deduced from the diffusion coefficient (Eq. 62). In the excited state the SAS term in the rate equations does not go to zero but represents fine-structure mixing collisions that do not alter the velocity of the Na atom. The LAS term represents both fine-structure mixing and velocity-changing collisions; the respective strong-collision rates are simply related. In the sudden approximation we have, for the excited-state LAS rates,


Y43 - Y33 - g3 Y34 744 g4'

where we have employed the fact that (E4 - E 3 ) = 17 cm-I << k,T for Na at room temperature. To distinguish the rates in the strong-collision model from the rates of the Keilson-Storer model the rates of the strong-collision model are denoted by y for LAS and x for SAS. The SAS and LAS parameters are related by

y43 x 4 3

Y34 x 3 4

-

rp = + X U . (66b)

The absolute value of the various strong-collision rates for the excited state are then determined by the excited-state diffusion coefficient, in combination with Eqs. (65) and (66).

In steady state the rate equations now read

+ r4(n4W(v,) - P4) - (Y43 + X431P4 + y43n3W(vz) + x 3 4 p 3 r

where we have also allowed for excitation to level 4. Parameters for Na-Xe are given in Table 11.

Recognizing that Eqs. (67a-67d) can be written in matrix form

0 = Ap + BnW(v,) (68)

232 E . R . Eliel

TABLE 11

A , , = A,, = 23.4 YI = y? = 8.78 A,? = A,? = 39.1 y , = y,, = 4.37 r, = 90.1 y, = yu = 8.74 r, = 83.1 xu = 7.93

x,, = 3.97

Na-Xe parameters ( in units IOh s - I ) used in the hard- collision version of the rate-equation model at a Xe pressure of I Torr and a temperature of 400 K. The labels I through 4 refer to the ?SII(F = I ) , 'SI,(F = 2), ?PI? and 'PlZ states. A refers to the spontaneous emission rate. rl , to the homogeneous linewidth on the D,- and D?-line, respectively, y represent rates for velocity and state-changing collision, whereas x represent the rates for state-changing collisions without velocity change.

a simple formal solution of the rate-equation model can be obtained (Streater and Woerdman, 1989). As shown in Fig. 13, the results of the strong-collision model for the drift velocity are in very good agreement with the results of the full Keilson-Storer model. Figure 14 shows a result of the strong-collision rate- equation approach. Here the dependence of the drift velocity of rubidium atoms in argon on the tuning of a single-frequency laser is shown. A three-level approximation has been made (y34 = x3., = 0), well justified for Rb, as the splitting between the 2p1,2 and 2&2 levels is large compared to thermal energies. Figure 14 shows that, for a certain range of laser frequencies, the two natural isotopes 85Rb and *'Rb (abundance 72% and 28%, respectively) will drift in opposite directions; i.e., they can be separated by LID, as discussed in Section I .

Extensions to the Strong-Collision Model. Up to this point we have discussed only single-frequency excitation notwithstanding the fact that, due to the severe effects of optical hyperfine pumping, multifrequency excitation is an attractive approach. For most cases multifrequency excitation can easily be implemented by modifying the expression for the excitation rate (Eq. (60)) into

with I(w)dw the intensity within a frequency band d o (Streater and Woerd- man, 1989). In one application of a multifrequency optical field to the study of LID a frequency-modulated laser has been used (de Lignie et al . , 1990) (see Sec- tion V). Here the magnitude of the electric field is written as

E ( t ) = Re E O n e ( ( W n r - k n z ) . , (70) n


200

100 h .rz E 0 Y

-200

-2 0 2

Frequency (CHz) FIG. 13. Results for the frequency dependence of the drift velocity for the strong-collision model

(full curve) and the full Keilson-Storer model (squares). The triangles indicate the results for the Keilson-Storer model in the limit for strong collisions. Single-frequency excitation of Na in 1 Torr argon ( T = 400 K ) at I = 10 Wicm?. (From Streater and Woerdman, 1989, with permission.)

50

-50

t 0 5

Frequency (CHz)

FIG. 14. Calculated drift velocity as a function of tuning of a single-frequency laser (I = 10 W/ cm*) for Rb atoms in 10 Torr argon ( T = 363 K , DL-line). The tick marks along the horizontal axis indicate the centers of the various hyperfine transitions X7Rb (F = 2), RsRb ( F = 3). "Rb ( F = 2) . *'Rb ( F = I ) . (From Streater and Woerdman, 1989, with permission.)

234 E.R. Eliel

where we include the possibility of excitation by an additional laser with its k- vector antiparallel to the k-vector of the light of the frequency-modulated laser. The excitation rate can then be written as (including effects of excited-state hyperfine structure (de Lignie et al . , 1990))

with W ; the transition frequency from level i to the hyperfine sublevel F of level j and ff the normalized line strength of the transition at frequency W ; ( Z F f; = 1). The hyperfine quantum number F has, for Na, the values F = 1 and F = 2 for excitation to the ?e,? level and runs from 0 to 3 for excitation to the 2&,2 level (de Lignie et al . , 1990).

LID using two-frequency excitation can usually be described by an excitation rate of the form of Eq. (69) with an appropriate form for I(w). When the “uncoupling” condition does not hold, i.e., the two optical fields interact with the same velocity group in the excited state, the ground-state coherence p , 2 can no longer be neglected (de Lignie and Eliel, 1989). Then, in the strong-collision approximation, rate equations can still be used (Haverkort and Woerdman, 1990). In that case there are six equations, four for the velocity distributions p,(vr) and one each for the real and imaginary part of the coherence P , ~ ( v , ) (de Lignie, 1991).

IV. Techniques for Measuring the Drift Velocity

Two routes have been followed to measure light-induced drift: experiments where a stationary state is reached and dynamic experiments. In the first approach a closed capillary tube contains the gas mixture that is illuminated by the near-resonant radiation field. The active component of the gas mixture will drift to one end of the capillary, creating a concentration gradient that in turn creates a diffusive counterflow. If there is no loss of particles, e.g., by chemical reaction with impurities in the buffer gas, the net flow equals zero in steady state but a concentration difference across the cell has arisen (see Fig. 3). This concentration gradient can be measured and provides a measure for the drift velocity. The steady-state approach has been the preferred mode of operation in experiments where LID is a very small effect; i.e., in molecular systems and in experiments using incoherent light sources (see Section VI). In the other approach, applicable when the drift velocity is high (as in atomic systems interacting with resonant laser light) the evolution of a spatial variation in the concentration is measured in real time. The local fluorescent intensity provides here an almost ideal probe of the local concentration of active particles.


A. EVOLUTION OF THE CONCENTRATION AND LIGHT INTENSITY

The evolution of the local density n(z , t ) of active particles is governed by the diffusion equation (in one dimension)

an a2n a - = 0,- - -(nv,,) - y n , at az2 az

where the three terms represent the diffusive flux, the drift flux and a loss term. The factor D, is the diffusion coefficient for particles in the ground state and n is the density of active particles. In writing Eq. (72) we have taken a zero-order approach in the sense of Section 11; i.e., the diffusion coefficient is not affected by the light field and a zero-order drift velocity is introduced. The drift velocity vdr depends on the radiation intensity I that, in an optically thick system, itself depends on the position

(73) ar _ - - -n(z , t )ma( f ) l ( z , t ) . az

Here u,(I) is the absorption cross section, which is intensity dependent in strong fields.

B. STATIONARY STATE

In steady state the situation is rather simple. The light entering the cell will generate a drift of the active particles towards either the one or the other end of the cell, depending on the sign of the drift velocity. The resulting concentration gradient will induce a (locally) balancing diffusive flux. If the vapor is optically thin ( Jdz n(z)u,(I) << l ) , the drift velocity is position independent, and the steady-state distribution of active particles n(z ) is the sum of two exponential distributions with characteristic lengths e , and Cz:

In the limit that v:,/4D, >> y (low-loss limit) we have

V d r - Y

236 E.R. Eliel

In that limit the “chemical length” e2 is much larger than the “drift length” el, and in essence, the density distribution is exponential with a characteristic length equal to the “drift length,” C , . In molecular LID the drift velocities are small (typically vdr = 0.5 cm/s) and the chemical loss can be neglected. For a typical value of the diffusion coefficient (D, = 150 cm2/s) the LID length equals e , = 300 cm. The experimental cell has to be quite long in order to measure an LID effect. For atomic LID, using laser excitation, typical values for the various parameters are Vdr = 10 m/s, D, = 100 cm2/s, y = 50 s-I, yielding C , = 0.1 cm and e2 = 20 cm. Here the density distribution is very sharply peaked (Atutov, 1991; Atutov et al., 1991b).

The exact particle distribution depends on the boundary conditions. In loss- free experiments (y = 0), the cell contains a fixed number of particles and the flow is zero everywhere (van der Meer et al., 1989). In other experiments, an infinite reservoir of particles is connected to one end of the experimental cell, fixing the particle density at that end. At the same time the flux is required to vanish at the other end of the cell (Chapovsky et al. , 1985; Chapovsky and Shalagin, 1987; Atutov et al., 1991b).

Actually, in steady state neither the density distribution nor the drift velocity is sought. Rather, it is the integrated effect of light-induced drift, i.e., the concentration difference over the length of the tube, that is important, e.g., to determine the relative change in collision cross section upon excitation (van der Meer et al., 1989). In these cases the induced change in density can be directly related to the change in the radiation intensity, a relationship that continues to hold when the vapor is no longer optically thin. For a loss-free stationary situation we can equate the drift and diffusive flux

which, in integrated form, reads

When the concentration variations are small we can put n(z) = n in Eq. (77) and, using Eq. (56a), we arrive at a straightforward expression (in the Doppler limit) for the density difference across the cell in terms of the absorbed laser power per unit area:

r, - r, vL 1 1 An = -M. r, D , A + r,tiw,


C. DYNAMIC EXPERIMENTS

When the drift velocity is sufficiently high-as in atomic systems-and when the transport effect is visible with the naked eye (as in Na), LID can give rise to dramatic dynamic effects.

I . Optically Thick Regime

The most spectacular demonstration of LID is, without doubt, the optical piston in Na (Werij et a l . , 1984, 1985, 1986; Werij and Woerdman, 1988; Nien- huis, 1985) shown in Fig. 15, as predicted by Gel’mukhanov and Shalagin (1980). This phenomenon, where the light shovels the atoms together and sweeps this collection of atoms through the buffer gas, can occur only when the vapor is optically thick. In that case the light is absorbed in a small region of space, and the drift velocity is highly z-dependent. In Fig. 15 one distinguishes three zones: (i) the left side of the capillary that has (almost) been swept clean of Na atoms and consequently has a low fluorescent yield, (ii) the bright spot where the Na density is strongly peaked, and (iii) the right side of the capillary that has not yet been touched by the optical piston; no photons reach this side of the capillary and consequently it is dark. Under typical experimental conditions (initial Na density = 10l2 ~ m - ~ , Ar density = 3 x lo1’ ~ m - ~ , I = 3 W/cm2, the laser tuned 1 GHz below the *S, , , (F = 2 ) + 2P,,2 transition) the peak density in the piston is increased by a factor 500 and the density in the entrance reservoir reduced by a factor 30.

The velocity v , with which the piston propagates is very low (= 1 mm/s) as is evident from Fig. 15, much lower than expected.’ This has been attributed to surface effects: a large fraction of the Na atoms is adsorbed on the wall of the cell and is not in the vapor. When the atoms in the vapor are pushed or pulled by LID, the volume density is replenished by atoms desorbing from the wall. This continues until the wall density achieves equilibrium with the new bulk density. Clearly, this slows down the dynamics of the optical piston; more specifically the drift and piston velocity as well as the diffusion coefficient are reduced by the factor (1 + K ) . K is related to the residence time at the wall T , ~ ,

the radius of the cell R and the most probable thermal speed vo (Werij and Woerdman, 1988)

’ Note that this piston velocity v,, is, even under ideal experimental conditions, appreciably lower than the drift velocity (Werij et nl.. 1984, 1986; Werij and Woerdman, 1988).

238 E.R. Eliel

0

20

O ' O E

0'9 E 40

100

120

140

3.6

4.5

160 (b) L 1

0 z (cm) 15 0 z (cm) 15

Laser beam Laser beam - ____,

FIG. 15. (a) Example of an optical piston. At r = 0 a laser beam ( I = 3 Wlcm2) is admitted to the cell, which is initially uniformly filled with Na (density 10l2 cm-') and Ar (density 3.2 X

101' cm-I) and slowly sweeps the Na atoms through the capillary, away from the entrance reservoir. The laser frequency is tuned ==I GHz into the red wing of the Na 2Sl,2(F = 2) + transition. (b) The corresponding evolution of the density profile. (From Werij and Woerdman, 1988, with permission. )


Actually the factor I / ( 1 + K) = 0.01 represents the fraction of atoms residing in the gas phase. A detailed analysis of surface effects in LID is presented in Nienhuis (1987).

A semi-direct measure of the drift velocity is obtained from the density distribution. The characteristic lengths at the light L , and dark L2 sides of the piston profile are inversely proportional to Vdr and vp , respectively, according to

D L , = 2 Vdr.0

where v ~ ~ . ~ is the drift velocity at the entrance of the cell. After correction for surface effects drift velocities on the order of 1-2 m/s were obtained under typical experimental conditions.

Piston action has also been observed in rubidium (Hamel et a l . , 1987). Here the effects were much less spectacular than in Na, wall adsorption having an even more pernicious effect. Even under ideal circumstances the piston velocity in Rb would be smaller than in Na as the drift velocity in Rb is smaller than in Na under comparable experimental conditions. In Na the optical piston has also been studied in a setup where different boundary conditions prevailed than in the experiment shown in Fig. 15. Basically the optical piston was created in a sap- phire capillary connecting two infinitely large reservoirs of Na (Werij et al., 1988). The evolution of the Na density profile in the capillary was remarkably sensitive to the Na density no in the reservoir where the light entered, heaping up at either the one or the other end of the capillary. A sharply defined density no separated the regimes where, in steady state, the density peaked at the entrance or at the far end of the capillary.

2 . Optically Thin Regime

A breakthrough in atomic LID was brought about by the introduction of coated cells (Atutov, 1986; Atutov et a l . , 1986d), eliminating the surface effects that were discussed earlier. In the first experiment diffusion-pump oil was used to coat the cell walls (Atutov, 1986), replaced by paraffin in later work (Atutov et al., 1986d). The settling time, i.e., the time required for the system to come to a steady state, was claimed to be reduced by a factor los over experiments using uncoated cells. Also an elegant technique was introduced to measure the drift velocity, i.e., measurement of the time of flight of optically thin clouds of sodium moving through a capillary cell (Atutov et u l . , 1986d). Werij et ul. (1987, 1988) improved on the experimental setup and performed a systematic experimental investigation of the drift velocity of Na in various noble gases. The si-

240 E.R. Eliel

multaneous development of the multilevel rate-equation model of Section I11 enabled a critical test of that model, showing it to be highly successful. This work has been further extended by de Lignie et al. (de Lignie and Eliel, 1989; de Lignie and Woerdman, 1990; de Lignie et al . , 1990; de Lignie, 1991) to molecular buffer gases, broadband excitation and a study of coherent population trapping in LID. At the heart of the experimental setup is a cross-shaped capillary cell made of Pyrex glass (see Fig. 16). The main (long) capillary (40 cm long, 1.5 mm diameter) is used for the actual measurement. Two multipurpose side arms are connected to it. One of the side arms has an appendix, partially filled with metallic sodium. This same side arm also connects to the pumping system and is used to admit buffer gas to the cell. The other side arm has an appendix filled with paraffin. Optically flat BK7 windows, at a slight angle with respect to the capillaries, close off the various ports of the cell. The paraffin is used only to initially coat the cell walls.

Figure 17 shows the experimental setup of the Leiden group (de Lignie, 1991). In the majority of their most recent experiments two dye lasers are used in conjunction and in most experiments the two beams counterpropagate through the cell and are orthogonally polarized. The two laser beams are led through both the main capillary and the side arms.

The following procedure is used to measure the drift velocity. Laser A is tuned to push and laser B to pull the Na atoms; the Na atoms cannot diffuse from the Na reservoir into the main capillary; the laser beams act as an optical shutter for the Na atoms. Both laser beams are intercepted repeatedly, for a few milli-

FIG. 16. Sketch of the glass cell in which light-induced drift velocities in Na are measured. Details are explained in the text.

r" laser

LIGHT-INDUCED DRIFT I: laser

24 1

FIG. 17. Experimental setup for measuring drift velocities with two lasers. Horizontal and ver- tical polarization of the laser beams are indicated with arrows ( $ $ $ ) and dots (ow), respectively. PBS stands for polarizing beam-splitter cube, and PD for photodiode. The glass cell that is at the heart of the setup is shown in Fig. 16 in detail.

seconds, by the blade of a chopper. During this time the Na atoms are no longer imprisoned in the side arm and can freely diffuse into the main capillary. A small cloud of Na vapor is created at the intersection of the capillaries. When the laser beams are readmitted to the cell, this cloud shoots through the main capillary as a result of LID. A photodiode located somewhere along the main capillary moni- tors the fluorescent light of the cloud when it passes by. As the system is optically thin this signal directly yields information on the evolution of the Na density at the position of the photodetector. Gathering this information at various positions along the capillary then yields a complete picture of the evolution of the Na density, as shown in Fig. 18.

Three aspects can be seen in Fig. 18. (i) The mountain ridge has a specific orientation relative to the space and time axes, yielding the drift velocity. (ii) As a function of time particles are lost; i.e., J dz n(z , t ) decreases (chemical loss) and (iii) the cloud broadens as a function of time due to diffusion. These aspects are all included in the evolution equation of the density, Eq. (72), which has a solution of the form

242 E.R. Eliel

A Na density

FIG. 18. Evolution of the Na cloud as it drifts (vdr = 17 m/s) through the capillary. Buffer gas: C2Ha at 4.9 Tom. Note the diffusive spreading of the cloud and the decrease of the total number of Na atoms due to chemical losses during the travel of the cloud.

where it is assumed that all atoms are bunched together at a single position at time t = - to (Werij and Woerdman, 1988). This solution can be used to fit the experimental density profiles n(z) at fixed times to obtain the drift velocity Vdr,

the diffusion coefficient D, and the chemical loss rate y. The fits are excellent (Werij and Woerdman, 1988). Once D, and y are known for a specific setup a simplified approach can be used, where only a single detector position is used, yielding a time-of-flight value of the drift velocity. A simple correction procedure, important only for relatively small drift velocities (vdr < 5 m/s), to extract the real drift velocity is described by Werij and Woerdman (1988).

Coated cells for LID were independently developed by the Pisa group (Xu et al., 1987; Mariotti et al., 1988; Gabbanini et al . , 1988; Gozzini et a l . , 1989). Instead of a paraffin coating, a silane coating, obtained from an ether solution of dimethylpolysiloxane (Mariotti et al . , 1988), was used. This type of coating, identical or closely related to the coatings based on the polymerization of di- chlorodirnethylsilane on a glass surface (Camparo, 1987), has the advantage that it can be used at higher temperatures, allowing an investigation of LID up to intermediate optical densities.

In the experimental setup of the Pisa group a laser beam longitudinally tra- verses the coated cell. An appendix containing liquid Na is attached to one end of the cell, which is initially homogeneously filled with a Na-buffer-gas mix-


ture. Taking the evolution of the fluorescence as a probe for the evolution of the density, the drift velocity is determined in a time-of-flight type manner. The solution of the evolution equation of the density (Eq. (72)) for small optical thickness and negligible chemical loss is approximately given by

with

and initial conditions

n(z , 0) = no (84a)

the latter implying a vanishing flux at the cell entrance. When the laser is switched on the Na is swept through the capillary, the density profile exhibiting a steep front separating the swept and unswept parts of the capillary. The slope of the front decreases as a result of diffusion. Yet, when an observation point at position z* along the capillary is chosen the drift velocity is simply given by the time-of-flight relationship Vdr = z* / t* , where t* is the time at which the fluorescence at point z* is halfway between the initial and final value (Gozzini et al., 1989).

Although the drift velocity is now a well-measurable quantity, serious discrepancies exist between the values for the Na drift velocity reported by the various groups. The drift velocities obtained by the Pisa group are sizably larger than the results obtained by the Leiden group under comparable experimental conditions. Also the first measurement of the Na drift velocity (in Novosibirsk) (Atutov et af., 1986d) yielded a high value. The results of the Leiden group are uniformly consistent with the rate-equation model of Section 111 and are therefore internally consistent. The mutual inconsistency of the various experiments has not been explained so far. Of course, the quality of the coating of the cell is a crucial factor in all experiments, yet there is no indication that in any of these experiments adsorption of Na on the wall plays a role. The high values for the drift velocity obtained in Pisa would seem to indicate that a silane coating is “better” than a paraffin coating; this suggestion is untenable (Camparo, 1987; Frueholz and Camparo, 1987).

244 E.R. Eliel

V. Drift Velocities for Na

Almost all the quantitative results on light-induced drift in atomic systems per- tain to Na immersed in a variety of buffer gases. Obviously this directly relates to the convenience of exciting Na atoms with a tunable dye laser. For this same reason a large body of data on the collisional interaction between Na and noble- gas atoms exist, extremely useful for the quantitative comparison between experiment and model description. The other alkali atom that has been used to study LID is rubidium; here only one publication has appeared reporting on the drift velocity of Rb in argon (Wittgrefe et al . , 1989).

Much of the recent work on LID of Na has been motivated by, on the one hand, the wish to investigate whether dramatic increases in the drift velocity could be achieved by specifically designed experiments and, on the other hand, by the question whether LID is a phenomenon that possibly plays or played a role outside the laboratory; i.e., in an astrophysical setting. The main route followed to examine the first issue is to tailor the spectral distribution of the exciting laser to optimize the fraction of Na atoms in the excited stated (see Eq. (1)). Alternatively it has been tried to find a buffer gas, for which the relative change in kinetic cross section (cr, - crg)/crx is very large. These efforts are the subject of this section. The investigation of modes of LID that could play a role outside the laboratory are discussed in Section VI.

A. SINGLE-FREQUENCY EXCITATION

The work on LID of Na immersed in noble gases using a single-frequency laser has been reviewed by Werij and Woerdman (1988). A typical result of such a measurement is shown in Fig. 19. Apart from the excellent agreement between the experiment and the model description (solid line) one notes that the drift velocities are on the order of 5 m/s, much less than the rough estimate (40 m/s) of Section I. As mentioned in Section 111 the relatively small values of the drift velocity reflect the deleterious effect of optical hyperfine pumping in these experiments.

An obvious remedy lies in reducing or eliminating optical hyperfine pumping, e.g., by introducing a second laser field to pump the atoms back to the depleted ground-state level (see, e.g., Strohmeier, 1990). Indeed this is an efficient way to enhance the drift velocity (Werij et af., 1987; Werij and Woerdman, 1988). Alternatively, an radio-frequency (RF) field oscillating at the ground-state hyperfine frequency can be used to induce transitions between the F = 1 and F = 2 lower levels. The latter approach has not been pursued; it requires an RF

LIGHT-INDUCED DRIFT

5 1 " ' . 1 ' ' ' ' l ' . ' ' l . " ' l . . ~ .

245

- 1 0 1 2 3 Detuning (GHz)

FIG. 19. Drift velocity for Na in 2 Torr Xe for single-frequency excitation on the D,-line. I =

12 Wkm2 and the detuning is shown relative to the 2S,12(F = 2) -+ 2f$2 transition. The solid curve shows the results of the rate-equation model of Section 111. (From Werij and Woerdman, 1988, with permission.)

setup where the magnetic and electric fields should be well separated spatially to avoid igniting a discharge in the atomic vapor.

B. COHERENT POPULATION TRAPPING IN LID

With the Na atom being excited by two coherent laser fields with frequencies differing by an amount approximately equal to the Na ground-state hyperfine splitting, new effects can arise that go beyond the elimination of hyperfine pumping. These effects, arising from a nonzero steady-state coherence p12 between the levels 1 and 2, can actually cause the excited-state population and thus the drift velocity to vanish. This applies for a A-type three-level system, resonantly excited by two laser fields (see Fig. 20). Such a three-level system has been the subject of extensive laser-spectroscopic studies (Alzetta et al., 1976; Gray et al., 1978; Feld et a f . , 1980; Thomas et al., 1982; Arimondo and Orriols, 1976; Dalton and Knight, 1982) and has recently attracted renewed attention in connection with laser cooling (Aspect et a f . , 1988, 1989) and lasing without inver- sion (Scully et a f . , 1989). Coherence effects have been shown to occur when the frequency difference (oA - 08) between lasers A and B equals the splitting AohJy between the two lower levels; i.e., when a Raman resonance is excited. Then a

246 E.R. Eliel

FIG. 20. A-type three-level system near-resonantly excited by two optical fields A and B with frequencies w, and w B . The fields have detunings A, = w, - (w3 - w l ) and A B = wB - ( w , - wl) , respectively. The various levels have energies hw,; R, and R, represent the Rabi frequencies associated with the laser fields at frequencies w A and w B , respectively.

narrow' dip appears in the absorption spectrum, known as a black resonance (Alzetta et al . , 1976). At this stimulated Raman resonance the atoms are pumped into a coherent superposition state that is immune to excitation by the combined laser fields (Gray et al . , 1978). This effect is usually referred to as coherent population trapping (CPT), and the superposition state of the hyperfine states is called the nonabsorbing state. The CPT resonance is inherently narrow; it has a spectral width limited only by the decay of the ground-state levels and the applied pump intensities.

The origin of the CPT resonance can easily be understood by considering the interaction between the three-level atom and two optical fields A and B:

with d the atomic electric-dipole operator, E A and E, the electric-field amplitudes, w A and w, the frequencies of the optical fields and h.c. the Hermitian- conjugate term. Consider now the superposition state


with R, = d,, * E,/A and R, = d 13 * E,/h the Rabi frequencies on the transitions 2-3 and 1-3, respectively; 11) and 12) are the two ground-state levels with energies Awl and Aw, (see Fig. 20). One can check now that ( J I N A ) is a stationary nonabsorbing state if o, + O, = 0, + w,; that is,

= 0, (87b)

where we have neglected the counterrotating terms. Spontaneous emission provides a channel to populate this nonabsorbing state in which the atoms remain trapped. Thus, at the Raman resonance, in steady state, all atoms reside in the nonabsorbing state and no atom can reach the excited state: the atoms are coher- ently trapped.

When no atoms can reach the excited state the drift velocity must be equal to zero, a direct consequence of the fact that the drift velocity is proportional to the fraction of Na atoms in the excited state (Eq. ( 1 ) ) . Obviously this simple rea- soning is valid only when collisions do not destroy the coherence in the system; i.e., when a collision does not perform a measurement in the sense that it per- forms state reduction (Berman et al., 1982). This condition is fulfilled for the ground-state coherence pI2 of Na when the collision partner is a noble-gas atom. Therefore, CPT in Na can also occur under experimental conditions typical for LID (de Lignie and Eliel, 1989; Eliel and de Lignie, 1989a, 1989b). Obviously Na is not a three-level system, and it comes as no surprise that the results are not as simple as mentioned here. Actually the situation is somewhat involved as will be discussed later.

The numerical model of Section 111 can still be applied to the description of LID of Na in the CPT limit provided that the ground-state coherence p , , is not neglected. The essence of the method (Haverkort and Woerdman, 1990) has been worked out recently (de Lignie, 1991).

In the experiment (de Lignie and Eliel, 1989) two separate free-running lasers were used, each having a linewidth of =1 MHz. In contrast with most other experiments copropagating laser beams were used in the experimental setup. In this way the Raman condition (0, - W , - (k, - k,) - v = AW,,~) is fulfilled for all atomic velocities (Ik, - k,( = 0). For counterpropagating laser beams this is not the case, and a velocity-changing collision will eject an atom out of the nonabsorbing state. The results of the experiment with the lasers tuned to either the D,-line or D,-line, are shown in Fig. 21. Note that the effect of coherent population trapping on the drift velocity is clearly observable on the D ,-line and only marginally so on the D,-line. Excited-state hyperfine structure plays a fundamental role here.

248 E.R. Eliel

n

-0: E

W

h c, .d

0 0 3

Q, 3

k n

-1 0 1 2 3

Detuning laser B (GHz)

n rA \ E'

W

h -4 .r(

0 0 Q, 3

3 3

k 2 n

-1 0 1 2 3

(b) Detuning laser B (GHz) FIG. 21. The drift velocity of Na in 1.5 Tom Xe as a function of the detuning of laser B: (a) for

the D,-line and (b) for the &-line. Laser intensities are (a) I, = 3.9 W/cmz, Is = 2.8 W/cmz and (b) I, = 3.2 W/cmz, Is = 2.6 W/cmz. The F = 1 and F = 2 resonance frequencies are indicated by bars. The solid line in (a) represents the result of a model calculation. In both cases laser A is tuned 650 MHz in the red Doppler wing of the LS,, ,(F = 2) + ' P transition. (From de Lignie and Eliel, 1989, with permission.)


1. Excited-State Hyperfine Structure

The notable difference between Figs. 21(a) and 21(b), cannot be understood in terms of the simple three-level picture that we have used so far. For the D,-line, where the dip in the drift velocity at the Raman resonance is quite pronounced, the excited state has two hyperfine sublevels that both are connected to the two ground-state levels through allowed transitions (see Fig. 22). Thus this four-level system can be viewed as a superposition of two separate A-type three-level systems (see Fig. 23(a)). Note that here there are two nonabsorbing states, which can be written as

- - where fi A and

The fact that we deal with an inhomogeneously broadened system now turns out to be an advantage. When the (power-broadened) homogeneous linewidth is smaller than the excited-state hyperfine splitting (this condition is fulfilled in the

8 are effective Rabi frequencies.

[ F'=3 t 60 MHz - 35 MHz - 16 MHz

-

. . - - DI 02

(5896 i) (5890 i) FIG. 22. Schematic energy-level diagram for the D,- and DJines of Na. Relative values of the

line strengths are indicated as well as the level splittings in the ground and excited state.

250 E.R. Eliel

F’=2 F=3 I F’=2

F=2

(a) D,-line F= 1 (b) D2-line F= 1 FIG. 23. Schematic hyperfine energy diagram of the (a) z S , / ~ + zS,2 and (b) 2Sl,2 + z&2 transition

in Na. Relative line strengths are indicated.

experiment) there is no overlap between the velocity groups that have been pumped to l$LF;=l)) and IJIL%=*)), respectively. Then the two three-level systems are distinguishable and the naive three-level picture holds: coherent population trapping remains effective.

Hyperfine structure affects CPT on the D,-line much more drastically. Here the excited state is split in four levels, two of which (F’ = 0 and F’ = 3) are connected with a single lower level only (selection rules, see Fig. 23(b)). The other excited-state sublevels (F’ = 1 and F’ = 2) are again involved in separate A-type three-level systems. In this case however, the excited-state hyperfine splittings are on the order of the homogeneous linewidth and the four excited- state levels all connect with the same group of velocities. So there is no stationary nonabsorbing state, and it is always possible to excite a Na atom to the *Pj i2 fine-structure level. In short, the excited-state hyperfine structure inhibits CPT when the lasers are tuned to the D,-line. A small effect of the Raman resonance on the excitation probability of the atom may remain as may be present in Fig. 21(b).

We will now discuss coherent population trapping in LID on the Na D,-line in some more detail; in particular we will discuss the effect of velocity-changing collisions and of fluctuation in the frequencies of lasers A and B.

2. Velocity-Changing Collisions

Velocity-changing collisions affect CPT in Na on two different levels. There is already an effect of velocity-changing collisions on CPT when the Na atom is described as a three-level atom, and in addition, there is an effect when the


excited-state hyperfine structure is taken into account. We will assume these effects to be independent.

Velocity-Changing Collisions in a “Three-Level Nu Atom. ” The rate-equation model is a helpful tool to gain insight in the effect of velocity-changing collisions on CPT in Na when the excited-state hyperfine structure is neglected. The rate- equation model makes use of the hard-collision approximation to describe the velocity-changing collisions, an approximation that is reflected in the resulting velocity distributions.

Within these approximations CPT in Na is affected by velocity-changing collisions as a result of the overlap of the 2Sl,z(F = 2) + zSi2 and 2Sl,z(F = 1) +

2fi,2 transitions; i.e., of the fact that the hyperfine splitting Aohfs is on the same order as the Doppler width AoDoppler. Laser B, which is tuned in between the two transitions, excites atoms with both positive and negative velocities, while laser A excites only atoms with negative velocities. Coherent population trapping can occur only for the atoms with negative velocities as only they are resonant with both lasers. Figure 24 shows the velocity distributions p I( v , ) through p4( v , ) for the four levels of Na and the real and imaginary part of the ground-state coherence p l z ( v I ) = p l z ( v , ) + i q l l ( v r ) , using parameters corresponding to the experimental situation of Fig. 21. In Fig. 24(a) the lasers are tuned close to the Raman resonance whereas Fig. 24(b) displays the situation at the Raman resonance. In Fig. 24(a), where wB - wA # Aohfs, the excited-state population pg(v,) is large for a negative velocity v - = - 3.7 x lo4 cm/s. Hyperfine pumping is a marginal effect as the atoms with velocity v , = v - are resonant with both lasers. In addition there is a much smaller peak in p3(v , ) near the positive velocity v + = 7 x lo4 cm/s. This resonance is due to excitation of atoms by laser B in the blue Doppler wing of the 2Sl12(F = 2) + 2S,2 transition; here there is hyperfine pumping indicated by the dip in p 2 ( v , ) and the peak in pl (v , ) . The ground-state coherence pI2 (v ) is very small for all velocities and has little influence accordingly. The fact that it is included in the model description has little impact here.

The situation is dramatically different in Fig. 24(b), exactly on the Raman resonance, although the tuning of laser B has been changed by only 100 MHz. The total excited-state population is much smaller than in the upper figure because a significant fraction of the atoms has been trapped in the nonabsorbing state. This is reflected in the large negative value of Jdv ,p12(vZ) . Atoms can get trapped if they have a velocity near v - where they are resonant with both lasers. However if they have a velocity near v + they are resonant with laser B only and they can be excited, to be optically pumped to the F = 1 level where they are no longer resonant with any laser.

As a result of velocity-changing collisions the atoms travel back and forth between negative velocity space, where they suffer coherent populating trapping, and positive velocity space, where they suffer optical hyperfine pumping. The

252 E.R. Eliel

E 0 4 \ rn

W

rn g 2

a -2.

.3

4 7

2 Ll 0 3 rn .- n -2.

0 1 o5 5 -1. 10

(b) Velocity (cm/s) FIG. 24. Velocity distributions for the ground-state levels 1 ( F = 1) and 2 ( F = 2). the excited-

state levels 3 (2&) and 4 ( 2 f i , 2 ) and the real and imaginary part of the ground-state coherence p12(v2) = pn(v,) + iq12(v , ) of Na for two different situations: (a) o8 - oA = how> + 2n X

100 MHz, and (b) w g - oA = Ao,. All other parameters are the same as in Fig. 21(a).


effectiveness of both processes is thus reduced. The resulting excited-state population is small, but the vapor does not become completely transparent nor does the fluorescence vanish.

The solid line in Fig. 21(a) represents the result of the rate-equation model of Section 111, with the ground-state coherence included. The agreement with the experimental data on the D,-line is very good despite the fact that the excited- state hyperfine structure was neglected in the model.

Velocity-Changing Collisions and Excited-State Hyperfine Structure. As discussed before, the presence of two hyperfine levels in the excited state implies that there are two velocity classes v ( F = ' ) and v(!'=*) (in negative velocity space) for which CPT occurs. Velocity-changing collisions cause a transfer of atoms between these velocity groups, and as the corresponding nonabsorbing states are different (see Eq. (87)), this mechanism provides an escape out of both nonabsorbing states. This effect of velocity-changing collision is not included in the model description used to generate the solid line in Fig. 21(a).

3. Laser Frequency Fluctuations

It is well known that fluctuations in the frequencies of lasers A and B influence the effectiveness of CPT (Dalton and Knight, 1982). This is immediately clear with the help of Eq. (87): an inadvertent change of either phase factor ( w 2 - w A ) t or (w, - wB)t destroys the destructive interference between the two contributions to the matrix element (3 /V( t ) /$NA) . A rough estimate of the fluctuations in (wA - w B ) comes from the linewidth of each laser individually: roughly 1 MHz. Using a heterodyne technique the fluctuations in w A - wB were shown to fall in a band of approximately 3 MHz width (FWHM). It can be shown that this fluctuation bandwidth directly determines the decay rate of the ground-state coherence pI2(v , ) (de Lignie, 1991). The rate of decay r,* of the ground-state coherence due to frequency fluctuations is given by (Dalton and Knight, 1982)

r 1 2 = A, + Ass - 2AM, (89)

where A, and Ass are the HWHM bandwidths of lasers A and B and A A B is the cross-correlated bandwidth of the two lasers. Here it is assumed that the frequency fluctuations of the laser fields are delta correlated in time; that is,

(Aw,(t)A~,(t')) = A,,&[ - t'),

with i , j E ( A , B ) . The frequency spectrum of the fluctuations of each laser separately is certainly not flat; i.e., the fluctuations are not delta correlated. The correlation time of the fluctuations of a dye laser is on the order of microseconds (Salomon et a l . , 1988), i.e., on the order of the Na(3p) spontaneous lifetime, supposedly short enough to consider the spectrum of fluctuations to be flat. With two independent lasers A A B = 0 and the decay rate for the ground-state coher-

254 E.R. Eliel

ence is given by r,* = A, + A,, = 1.5 MHz. This value for the decay rate of p l Z has been used in the rate-equation calculation that results in the solid line in Fig. 21(a).

It is possible to avoid fluctuations in ( w A - o8) altogether. Then one should use a single laser and acousto-optic or electro-optic modulation techniques to generate a sideband on the optical carrier, displaced by AvhfS = 1.77 GHz. The fluctuations in (oA - w,) are then determined by the phase noise of the radiofrequency generator that drives the modulator; this noise can be arbitrarily small. The results of the rate-equation model for negligible loss of coherence are shown in Fig. 25, showing the Na drift velocity as a function of the tuning of laser B (frequency of laser A fixed). The striking feature of this prediction is that the drift velocity is predicted to change sign at the Raman resonance, a feature not observed in the experiment. At the Raman resonance coherent population trapping is effective for the velocity class around v - (see Fig. 24) but not for the velocity class around v + ; that is, resonant with just laser B . Now that the coherence decay rate is negligible, there is only excited-state population at velocities around v, , giving rise to a negative drift velocity. As soon as the Raman condition is no longer fulfilled, the velocity class around v - will contribute to the drift, dominating over the contribution of the atoms at velocity v + . An experi-

n 4 < E

- 2

-1 0 1 2

Detuning laser B (GHz) FIG. 25. Calculated drift velocity of Na as a function of the detuning of laser B on the D,-line

with perfectly correlated laser frequencies. Intensities are I , = 2.0 W/cmz and fB = 6.0 Wlcmz. The tuning of laser A relative to the F = I and F = 2 spectral components (black bars) is indicated. (From de Lignie and Eliel, 1989, with permission.)


ment along these lines has shown that the prediction of the rate-equation model is incorrect. Notwithstanding the used of phase-locked modes, no sign reversal in the drift velocity around the Raman resonance was observed (de Lignie et al., unpublished). This is an indication that the effects of excited-state hyperfine structure pose a real limitation to CPT in Na under conditions of LID.

C. MULTIFREQUENCY EXCITATION

For two-level atoms the road to high drift velocities is quite clear: maximize the set of resonant velocity classes and maintain full velocity selectivity. In other words, the light source should resonantly excite all atoms in exactly half of velocity space and drive those atoms into saturation. It is not possible to achieve this with monochromatic light as one then relies on power broadening or pressure broadening to obtain a large velocity coverage. This unavoidably imposes Lor- entzian tails onto the Bennett features in the velocity distributions, tails that extend into the “wrong” half of velocity space and reduce the velocity selectivity of the excitation.

A broadband laser with a suitably tailored spectrum does allow one to combine large velocity coverage with velocity selectivity. The available light intensity is then spread out over half of the Doppler profile and no broadening mechanism needs to be invoked. The excited-state velocity distribution then closely resem- bles the product of the excitation spectrum and a Maxwell distribution. For a sharp-edged broadband spectrum few particles are then excited in the wrong Doppler wing, and a large velocity asymmetry and consequently a large drift velocity can be achieved (Popov et al., 1981).

Additionally, broadband lasers are attractive for LID of real atoms because they offer the opportunity to eliminate optical hyperfine pumping, so deleterious when a single-frequency laser is used. When the width of the laser spectrum is on the order of the ground-state hyperfine splitting, particles from both hyperfine levels can be excited. For Na, where the ground-state hyperfine splitting and the Doppler width are almost equal, this can be achieved without losing the velocity selectivity required for large LID effects. More attractive is a setup with two broadband lasers (counterpropagating beams), each covering an outer wing of the Na weak-field absorption spectrum, as shown in Fig. 26; this approach should result in the highest possible drift velocities (Werij et al., 1987; Werij and Woerdman , 1988).

These ideas were picked up by groups in both Pisa and Leiden resulting in a set of partially overlapping experiments with remarkably different results. The experimental tool of the Pisa group was a long-cavity multimode laser (“lamp laser”) to optimize the coupling between the laser and the atomic vapor (Moi, 1984; Liang et al., 1984; Liang and Fabre, 1986; Weissmann et al., 1984; Xu

256 E.R. Eliel

F=2

L a s e r 2 Laser 1 - -

2 p1/2 n L a s e r 1

L a s e r 2

- F=2

FIG. 26. Scheme for obtaining the ultimate limit to the drift velocity for Na. Laser 1 resonantly excites all atoms with positive velocity on the 'S , , , (F = I ) + ?S,, transition, with laser 2 exciting all atoms with positive velocity on the ZS, , , (F = 2) + transition. The laser beams are counterpropagating and are therefore tuned to the outer wings of the weak-field absorption spectrum. The lower half of the figure shows the velocity distributions in the various levels for Na in 1 Torr Xe gas at I , = 1, = 5 Wlcm,. (From Werij and Woerdman, 1988, with permission.)


and Moi, 1988; Eliel, 1988). The improved coupling between the atoms and the laser radiation results from the fact that many closely spaced modes (axial mode spacing 10- 100 MHz) of the laser oscillate, exciting atoms in many velocity classes. Implicit here is the assumption that all cavity modes oscillate simultaneously.

The route followed by the Leiden group (de Lignie et al . , 1990) was to employ a frequency-modulated (FM) dye laser (Kane et al. , 1986a, 1986b; Harris and McDuff, 1965; Siegman, 1986; Bramwell et al., 1987; Ferguson, 1987). Here a single-frequency laser is converted into a multimode laser by introducing a phase modulator inside the laser cavity and driving the phase modulator at a frequency v, = v,,, where the latter represents the axial mode spacing of the laser. The output field of the FM laser can be described by a superposition of plane waves:

where E, is the amplitude of the electric field, w,/27r is the laser frequency, om = 2m,, M is the effective modulation index and J , ( M ) is the nth order Bessel function. In short, the frequency spectrum of the output consists of a series of equidistant spikes (separated by v,) with well-defined amplitude. Herein lies the fundamental difference between an FM laser and a free-running multimode laser. For the former detailed information regarding the spectral components (and their correlation) is available, which is not the case for the free- running laser.

In an FM laser the modulation index M is a sensitive function of the modulator detuning

where 6 is the single-pass phase retardation, directly connected to the properties of the modulator and the strength of the RF field that drives the modulator. Equation (92) would seem to indicate that arbitrarily large values for the effective modulation index M can be achieved when v, + vax. However below a certain value of Iv, - vaa/ the FM operation becomes “unquenched” or the laser starts to operate in a pulsed mode (Harris and McDuff, 1965). Within the FM regime of operation the amplification factor vaa/lvm - va,I can reach values of ~ 3 0 0 (de Lignie et al . , 1990).

Also for the case of an FM laser an improved atom-field coupling arises only

* This not a priori ensured in view of mode-competition effects that arise in the homogenoysly broadened gain medium of a dye laser. Experiments on deceleration of atomic beams suggest however, that on a sufficiently long time scale all cavity modes will partake in the lasing process (Liang and Fabre, 1986).

25 8 E.R. Eliel

when the spacing between the oscillating modes of the laser is sufficiently small; i.e., on the order of the (pressure-broadened) homogeneous linewidth. This au- tomatically implies that the laser cavity is rather long ( ~ 3 - 5 m) and the modulation frequency conveniently low (60- 100 MHz). The attractive points of the FM laser approach are that (i) the laser spectrum can simply be varied from being single-mode to quite broadband (==lo GHz) and (ii) that its spectrum is well defined in every aspect. However an FM laser is a more complex piece of equip- ment than a free-running multimode dye laser.

For a 8- 10 m long broadband dye laser (spectral width =2 GHz) drift velocities up to 30 m/s have been reported by the Pisa group (Gozzini et a l . , 1989), approximately twice as large as the drift velocity obtained with a single-mode dye laser having comparable intensity (5 W/cm*). Both these values are in con- flict with the predictions of the four-level rate-equation model, an issue that has not yet been resolved. A 10 m long cavity was found to yield the best results although the drift velocity did not critically depend on the length L of the laser cavity for L > 5 m (Gozzini et a l . , 1989). This result was ascribed to a peculiar property of this type of laser; i.e., that for given output power the number of oscillating modes remains approximately constant. This implies that the spectral output of the laser narrows when L is increased (Gabbanini et a l . , 1988).

Systematic measurements of the drift velocity as a function of the bandwidth of an FM laser were obtained by the Leiden group (de Lignie et a/ . , 1990). Figure 27(a) shows the results for Na in Xe (2 Torr), with Fig. 27(b) showing a 2.5 GHz wide optical spectrum of the FM laser relative to the Na weak-field absorption spectrum. For this bandwidth the drift velocity reaches a maximum (vdr = 12 m/s). Passing from single-mode operation of the laser over to FM operation the drift velocity is seen to increase by a factor of four. This increase mainly reflects the 'reduction of hyperfine pumping when the width of the laser spectrum is increased: the laser becomes resonant with atoms in both ground- state hyperfine levels. The supposed effect of improved velocity coverage on the drift velocity cannot be distinguished. For bandwidths larger than 2.5 GHz the drift velocity is seen to decrease again. Figure 27(b) contains the key to the understanding of this last point. Here we see the laser spectrum relative to the Na weak-field absorption spectrum for the optimal bandwidth and detuning. One sees that the FM-spectrum is symmetrically positioned with respect to the 2S, ,2(F = 2) + 25/2 transition (no contribution to LID) and highly asymmet- rically relative to the 'S, , , (F = 1) + 25/2 transition, covering only the low- frequency Doppler wing. To maintain velocity selectivity the FM spectrum should not penetrate into the high-frequency wing of the latter transition, thereby pushing the low-frequency part of the spectral intensity outside the absorption spectrum and reducing the effective atom-field coupling.

In this experiment two factors that determine the dependence of the drift velocity on the bandwidth of the laser radiation are entangled. On the one hand, there is the effect that one wants to study; i.e., the increased velocity coverage.


-2 0 2 4 (b) Detuning (GHz)

FIG. 27. (a) Experimental results for the drift velocity as a function of the bandwidth of the exciting FM laser ( I = 3.2 W/cmz, mode spacing v,, = 81.5 MHz) for Na in 2 TOIT Xe; three different values of the modulation frequency v, were used. (b) 2.5 GHz wide optical spectrum of the FM laser relative to the Na weak-field absorption spectrum. (From de Lignie er al . . 1990, with permission. )

On the other hand, the effectiveness of optical hyperfine pumping is influenced by the spectral coverage of the laser. A more clearcut experiment, in that sense, is an experiment using two lasers, an approach that a priori leads to a severe reduction of optical hyperfine pumping. Obviously the optimum approach is that of Fig. 26, using two broadband or FM lasers. Figure 28 shows the results for

260 E.R. Eliel

I " " " ' ' ' ' ' ' ' ' ' ' ' ' ' ' I

5 1 .,l : v,= v,,

k * ,2 : vm=2v,, n * ,3 : v,=3va,

(a) Bandwidth ( G H z )

I

-2 0 2 (b) Detuning (GHz)

laser J 4 FIG. 28. (a) Experimental results for the drift velocity as a function of the bandwidth of the

exciting FM laser (I = 2.8 W/cm2, mode spacing wax = 81.5 MHz) combined with a single-mode laser (I = 2.0 W/cm2), for Na in 2 Torr Xe for excitation on either the D,- or &-line; three different values of the modulation frequency w, were used. (b) 1 GHz wide optical spectrum of the FM laser relative to the Na weak-field absorption spectrum. (From de Lignie ef al. , 1990, with permission.)

the Na drift velocity in a somewhat simpler approach, using an FM laser and a single-mode laser in conjunction (de Lignie et al . , 1990). Compared to Fig. 27 we see a much smaller effect of the laser bandwidth on the drift velocity but this effect now represents the relevant effect of the laser bandwidth in almost pure form. As there is very little hyperfine pumping, the spectrum of the FM laser


does not need to cover more than the low-frequency wing of the 2S , ,2 (F = 2) +

2 p j , 2 transition. For larger widths the drift velocity is reduced again, part of the laser spectrum extending beyond the Na absorption spectrum resulting in a decreased Doppler-averaged atom-field coupling.

Figures 27 and 28 show results for three different modulation frequencies; i.e., v, = u,, (squares), u, = 2u,, (triangles) and u, = 3u,, (asterisk) with v,, =

81 MHz. No significant difference is observed indicating that the spacing between the oscillating modes can be appreciably larger than the homogeneous linewidth without affecting the drift velocity. The excited-state hyperfine structure is an important ingredient in the explanation of this behavior (de Lignie er a/ . , 1990).

Figures 27 and 28 also show the results of the rate-equation model of Section 111, and we see that the agreement between the experiment and the model is excellent. For this study the four-level model was slightly modified to effectively account for the hyperfine structure in the excited state by modifying the excitation functions (see Eq. (71)) (de Lignie er al. , 1990).

In the experiments discussed here the output of the laser itself is multimode. Alternatively one can use a single-mode laser in conjunction with a device that generates a comb of sidebands on the carrier. The simplest but hardly interesting example is a single-mode laser in combination with an extra-cavity phase modulator. A combination of two acousto-optic frequency shifters in a ring configuration behind a single-mode laser has been shown to yield a broad spectral intensity distribution with sharp cutoff on either the low- or high-frequency side of the spectrum (Kristensen, 1991). Similar results can be obtained with a dye laser possessing frequency-shifted feedback (Littler and Bergmann, 199 1). These latter approaches have the advantage that one can use commercial lasers rather than modify an existing device. In addition the sharp cutoff in the spectrum is a property that would be of great use in the context of multifrequency LID.

D. BUFFER GASES

As mentioned in the introduction, the buffer gas plays a vital role in LID by introducing a differential resistance to the opposing fluxes of ground and excited state atoms. In the actual expressions for the drift velocity the buffer gas enters through the relative difference in kinetic cross section (uc - uR)/uR or, equivalently, the relative difference in collision rate (r, - T,)/T,. For convenience sake we write

262 E.R. Eliel

1 . Atomic Buffer Gases

Traditionally the noble gases have been the preferred collision partner for Na in the study of light-induced drift. Na and noble-gas atoms form an excellent combination due to the chemical inactivity of the noble-gas atoms, directly related to their electronic shells being completely filled. The noble-gas atoms also do not carry intrinsic angular momentum (except nuclear spin for some isotopic species) and thus the number of interatomic potentials relevant for a description of the dynamics of the Na-noble-gas collision is limited. For Na in the 3s ground state the collision dynamics are dictated by the X2CIl2 interatomic potential; for Na in the excited state three potentials come into play: A211,,, connecting to the atomic 2P;,2-state, and A2II,,, and B 2 C I , , connecting to the atomic 2fii2- state. Aulu then differs for excitation on the D,- and D2-lines (Hamel et al . . 1986):

1 -[~(A’lI3,2) + u ( B ~ C , , ~ ) ] - U ( X ’ C , , ~ 2

d X 2 C 1/21 (94b)

If collisional fine-structure mixing is important a statistically averaged value for Aulu has to be used

The relevant parameter determining the importance of collisional fine-structure mixing is the Massey parameter Ao/,T,, with Awfs the fine-structure splitting and 7,- the collision time. When Aw,,T,. is small compared to one (the “sudden limit”), the fine-structure levels are efficiently mixed. For Na-noble-gas systems the sudden limit is considered appropriate, and the value of Au/u according to Eq. (95) applies. In contrast, the Rb fine-structure levels 5p2P;,, and 5 ~ ~ 5 , ~ lie sufficiently far apart that fine-structure changing collisions can be neglected (Aw/,T,. >> 1, adiabatic limit). Thus for Rb one has to distinguish between D,- and D2-line excitation. Values for A u l u for Na-noble-gas combinations are shown in Table 111.

2 . Molecular Buffer Gases

For Na-noble-gas systems the collisions are elastic (except for the state-mixing collisions Na(3p 2P;12 S 3p 2fi,2)). In collisions between alkali atoms and poly-

TABLE 111

Na Rb

AuIu (Aul)Di (AU/)Dz ( A d ) a,r buffer

gas Expt. Calc. Expt . Calc. Expt . Calc. Expt . Calc.

He 0.12 2 0.02* 0. I4 -0.17 2 0.02 -0.11 0.00 2 0.06 0.10 -0.06 2 0.05 0.03

Ne 0.03 ? 0.005* 0.04 -0.18 ? 0.01 - 0.40 0.17 ? 0.01 0.27 0.05 2 0.01 0.05 Ar 0.29 ? 0.02* 0.26 0.19 ? 0.04 -0.12 0.29 ? 0.02 0.41 0.26 t 0.03 0.23

Kr 0.37 2 0.060 0.42 0.49 2 0.05 0.09 0.28 2 0.05 0.46 0.35 2 0.05 0.34 Xe 0.37 2 0.060 0.49 0.35 ? 0.20 0. I8 0.19 t 0.03 0.44 0.24 ? 0.09 0.35

-0.33t 0.38t 0.14t

0.13 ? 0.02$

0.49"

Measured and calculated values of A u l u for Na and Rb immersed in noble gases. For Na only the statistically averaged value (see Eq. (95)) is given; the calculated values for Na are from Werij and Woerdman (1988); Hamel et a/. (1986). For Rb the collisional interaction distinguishes between the D,- and &-lines. Both the experimental and calculated values for Rb are from Wittgrefe er al. (1991a).

*From Atutov era/ . , 1986b. @From Atutov er a/ . , 1987. "From Werij and Woerdman, 1988. ?From Parkhomenko, 1988. $From Wittgrefe er al . , 1989.

s X

264 E.R. Eliel

atomic molecules inelastic processes play an important part as the molecules can efficiently quench the electronic excitation of Na. In such a quenching collision the electronic energy (2.1 eV for Na(3p)) is converted into both translational energy of the collision partners and internal energy of the molecule. For some collision partners a large fraction of the Na(3p) electronic energy E,, reappears as translational energy E,,,,,, (superelastic collisions) (Hertel, 198 1). The accompanying change in velocity is interesting from the point of view of LID (de Lignie and Woerdman, 1990). Obviously not all collisions are inelastic; the quenching cross sections are typically on the order of the kinetic cross section and thus elastic and inelastic processes can both contribute to LID.

An interesting aspect of quenching collisions is that there may be an anisotropy of the E,, + E,,,,,, conversion process. To illustrate this point we consider two-level atoms immersed in a heavy molecular buffer gas where inelastic (superelastic) collision dominate; i.e., the postcollision velocity v 2 of the atom is much larger than the laser-selected velocity vL (see Fig. 29). As the molecule is assumed to be much heavier than the two-level atom the center-of-mass system is effectively fixed to the heavy molecule.

Isotropic Scattering. If the scattering is isotropic (in the center of mass system) the average postcollision velocity of the two-level atom will be zero in that system and therefore also in the laboratory frame. We thus have a thermalization process that results in zero average velocity.

Forward Scattering. The case of forward scattering is shown in Fig. 29(a), assuming that the postcollision velocity v 2 is much larger than the laser-selected velocity v L and that the forward preference is complete; i.e., the collision kernel is a 6 function of the scattering angle. The wavy arrow indicates the absorption and stimulated-emission processes, the solid arrow, the quenching collisions (cross section CT,,), and the dashed arrow, the thermalizing collisions of the atoms in the ground state (cross section CT,). We also assume that the quenching is effective so that the excited-state population can be neglected. The light-induced drift effect will now be determined by the velocity dependence of C T ~ . For a l / r4 potential (“Maxwell molecules”) u, v - I so that the rate of thermalizing collisions, r,, is independent of v. In this case the area under the superelastic peak in the velocity distribution is equal to the area in the Bennett hole; the vapor will drift as v 2 >> vL and does so in a direction opposite to the direction for elastic LID. For a hard-sphere potential u, is independent of the velocity and r, is proportional to v. In this case the ratio of the area of the superelastic peak to the area of the hole in the ground-state distribution equals v l h 2 and the “reversed” LID effect vanishes. Obviously for real atoms the situation is more complex: the scattering anisotropy is far from perfect and the excited-state population is not negligible nor is the interaction potential that simple. Elastic LID will occur as well and the drift velocity will have a contribution from both elastic and inelastic collisions.


V2 - VL

t fe

FIG. 29. Ground-state and excited-state velocity distributions for two-level atoms undergoing velocity-selective excitation (at velocity v,) and quenching collisions (heavy collision partner). The scattering has forward preference in (a), backward preference in (b), and is isotropic in (c). After the quenching collision the atoms have a velocity - v 2 along the axis of the laser beam [case (a)], a velocity + v 2 along the axis of the laser beam [case (b)], or have their velocities symmetrically distributed around zero velocity [case (c)]. (Adapted from de Lignie and Woerdman, 1990, with permission.)

E.R. Eliel

Backward Preference. The velocity distributions for this case are shown in Fig. 29(b). Quenching now always would result in LID parallel to the conventional LID effect. Backward preference would thus be a process that could yield very large drift velocities. However, anisotropic scattering with backward preference does not occur in quenching collisions of Na(3p). Note that an early paper on the effect of quenching collisions on LID is in fact restricted to this unrealistic case (Kalyazin and Sazonov, 1979).

3. Experimental Results

The results of a series of experiments on the Na drift velocity are compiled in Table IV. These results are quite interesting: molecules that are known to provide a large quenching cross section yield low values of the drift velocity. In particular for C2H, and C6H6, having the largest quenching cross sections of the molecules listed in the table, one obtains drift velocities that are much smaller

TABLE IV

Buffer Gas u,,( cm') I J , , ~ ( C ~ / S )

< lo-* I6* 39: 930

0.35$ 920

12. I 10.3 8.7

16.0 3.4

17.7 14.0 14.2 3.8

12.9 12.4

-

Quenching cross sections u,, and drift velocity of Na in a variety of buffer gases at a pressure of 2 Torr using two single-mode lasers tuned away from the Raman resonance. Laser A ( I = 4.6W/cm2) is tuned 650 MHz in the blue wing of the 'SI ' ( F = 2) + ?PI/? transition and laser B ( I = 3.1W/cm2) roughly 550 MHz in the blue wing of the 'SI ? ( F = I ) + ?PI,? transition. The relative error in the drift velocities is 2 3%.

* From Kibble et a/ . , 1967. @From Earl er a/ . , 1972. f From Tanarro e t a / . . 1982. $ From Norrish and Smith, 1941. The values for the drift velocity are from an

experiment by de Lignie and Woerdman (1990).


TABLE V

Rb (D,-line) Rb (DL-line) Na

0.49 rf: 0.05 0.35 2 0.20 0.45 2 0.10 0.53 2 0.08

0.28 zk 0.05 0.19 2 0.03 0.46 2 0.05 0.62 t 0.09

0.42 2 0.05 0.49 ? 0.05 0.64 4 0.10 0.68 f 0.10 0.56 ? 0.10 0.56 2 0.10 0.52 ? 0.10 0.50 2 0.10

Values of A o l o for Rb and Na in various buffer gases. The values for Rb are from Wittgrefe et al. (1991a) and the values for Na:Kr and Na:Xe are from Werij and Woerdman (1988). The remaining values for Na are based on the ratio of, on the one hand, the experimental value of the Na drift velocity in a molecular buffer gas and, on the other hand, the drift velocity of Na in Xe. The Rb values are valid for a somewhat lower temperature (300 K) than the Na values (390 K).

than those measured in kinetically similar saturated hydrocarbons CH4, C,H,, C,H 14 (n-hexane) and C,H (cyclo-hexane). Anisotropic scattering with forward preference is the most probable explanation for this inverse correlation between the drift velocity and the quenching cross section (de Lignie and Woerd- man, 1990).

It came as somewhat of a surprise that the drift velocity of Na in saturated hydrocarbons is on the same order or larger than that in Xe, measured under the same experimental conditions (see Table IV). As there is no excited-state quenching with methane and ethane (probably also for the other saturated hydrocarbons) the large drift velocities must be due to a very efficient elastic process; i.e., a large value of AuIu. Similar results, but now in an experiment on light- induced diffusive pulling (measuring Au/u directly), were obtained for Rb (Wittgrefe et al., 1991a). Using Xe as a reference one can translate the drift velocities of Table IV into values of Au/u. A compendium of the results is shown in Table V.

VI. Light-Induced Drift in Astrophysics

It is well known that radiation pressure plays an important role in astrophysics. Radiation pressure generates a flow of particles away from the light source by imparting, on the average, the photon momentum to the gaseous absorbers. Atu- tov and Shalagin (1988) argued that light-induced drift could also be an impor-

268 E.R. Eliel

tant driving force in stellar atmospheres competing with the radiation-pressure force or providing an alternative mechanism where radiation pressure could not provide an explanation for the observations. These thoughts were developed in particular in connection with abundance anomalies in “chemically peculiar” (magnetic Ap and Bp) stars (Cowley et a l . , 1986) and the anomalous distribution of the ratio [D]/[H] of the hydrogen isotopes across our solar system (see Fig. 5). Babcock’s star and the star 53 Camelopardalis are good examples of chemically peculiar stars (Landstreet, 1988; Landstreet et a l . , 1989). The former has an asymmetric dipole magnetic field with a strength up to 1 Tesla at the surface. In its atmosphere the global Fe abundance is =lo times larger than solar. Observations of the spectrum during a full revolution of the star (rotation period 8.03 days) revealed that the Ti abundance at the magnetic south pole is = 10 times larger than the solar abundance and = 10 times smaller at the magnetic north pole; for Ca the situation is precisely reversed. It is widely believed that these abundance anomalies result from a competition between gravitation and radiation pressure, modified by the magnetic field (Michaud et al . , 1981). As argued by Atutov and Shalagin (1988), light-induced drift could provide an alternative explanation for these concentration variations, even though the idea is surprising in many ways. For one, what mechanism makes the excitation velocity selective?

Assuming the spectral distribution of the light emitted by the core of the star to be white, Atutov and Shalagin argued that in an optically thick vapor where two species are embedded in a buffer gas, light-induced drift can occur when the two species have overlapping absorption lines.9 As the gas is optically thick the spectral distribution of the light depends on the distance from the source with ever deeper, overlapping Fraunhofer absorption lines (see Fig. 30). The fact that the Fraunhofer lines overlap is the root cause of the LID effect; it causes an asymmetry across the spectral lines of the two species. More exactly, there is a slight reduction of the excitation of species A in the high-frequency Doppler wing and similarly for species B in the low-frequency Doppler wing; here w A < wB with wA and wB the resonance frequencies of species A and B. If for both species the collision cross section increases upon excitation, the two species will drift in opposite directions; i.e., species A and B will be separated by the white light. For the case sketched in Fig. 30 species A will be pushed outward and species B pulled towards the stellar core. Species A and B may be different atoms or ions; they may also be different isotopes of the same species. Strong isotopic anomalies in stellar systems are well documented (Trimble, 1991). For instance, on the star x Lupi the elements Pt and Hg are, relative to the sun, a factor lo4 and lo5 overabundant respectively, the latter seemingly being 99% 204Hg (Fien- berg, 1991).

gas should be state dependent. Obviously the cross section for kinetic collisions between each of the two species and the buffer

LIGHT-INDUCED DRIFT

T

269

I I I I I I b

c T

A Frequency FIG. 30. Simple picture of light-induced drift in a stellar atmosphere. (a) White light emanating

from the core of a star has been partially absorbed by two species, A and B, which have overlapping absorption lines at frequencies v A and vB. (b) Both species are excited velocity selectively, since the spectral intensity is frequency dependent: species A is excited primarily in the low-frequency Dopp- ler wing and species B in the high-frequency Doppler wing. In the presence of a buffer gas (e.g., atomic hydrogen) both species will drift; separation will take place if for both species the kinetic collision cross section increases or decreases upon excitation.

A first quantitative treatment of this novel effect, called white-light-induced drift (WLID) appeared in 1989 for a one-dimensional system containing two isotopic two-level atoms in a buffer gas (Popov et al. , 1989). In that treatment white light enters a capillary cell at the point where a source of atoms imposes fixed boundary conditions on the particle densities N A ( z = 0) = N : , N8(z = 0) = N g . The optical field is assumed to be weak, i.e., no saturation, and steady-state solutions are sought; i.e., all particle flows are zero. The coupled equations governing the spatial variation of the atomic densities and light intensity are given by

270 E.R. Eliel

where D is the diffusion coefficient for species A and B in the buffer gas, cr, is the absorption cross section of species i, 4 i ( w ) is a normalized Voigt lineshape centered around o, and r is the homogeneous linewidth. The parameters A, B , r, and rx represent the Einstein A and B coefficients and the rates for velocity- changing collisions in excited and ground states, respectively. These equations can be rewritten in dimensionless units

where we have introduced a dimensionless length

5 = z m , ~ : : = z/e.bs, in terms of the absorption length tab, of species A. We also introduced densities n, , normalized in terms of the density of species A at z = 0:

n, = N , / N j , (99)

dimensionless drift velocities u , ( t ) in terms of the LID-length eLID and the diffusion coefficient D for species A and B in the buffer gas

( 100) 42 LID

u t ( 8 = V d r . r 7


and a dimensionless intensity in terms of the intensity I 0 at z = 0:

The LID length contains most of the physical parameters

~ + r , D D c,,, = ~ - - - - [re r ] [ BI, ] v,, - v H ’

where vH is the drift velocity for perfect velocity-selective excitation; i.e., when I(w) = IoO(o - oA) with 0 ( w ) the Heaviside function. Finally, the strength parameter a of the white-light-induced effect is defined as

( 103) abs a = - - - -

eLID ’

The remaining dimensionless variables are defined as follow

p ( x ) =

Y =

(x - X i ) =

a, =

y exp( -y2)aVl,rr3’* I dy (x - y)* + a: (104a)

( 104b)

( 104c)

( 104d)

a , is the Voigt parameter and vo = v‘m the most probable velocity. Note that the variable p(x) is directly related to the low-intensity limit of the LID function *(A) (Eq. 43) of Section 111:

(105) k

p(x) = - lim *(A). ,rrr I-o

For small optical depths, en, << 1, for instance close to the point where the light enters the vapor, the expression for u,(.f) can be linearized, yielding

uLS) = -5 1 h p ( ~ - xO[nX+(x - x A ) + n%#4x - xdl , (106)

where we have used the fact that the induced density changes can be neglected in zero order, The expression for u,(5) depends on a single integral

= -1 &+(x - xA)p(x - x B ) , (107b)

because +(x ) is an even function of x and p(x) an odd function of x. In the limit of small optical depth one can then write

uA(5) = -ngse6 (108a)

212 E.R. Eliel

( I08b)

( 108c)

1 2

AnB = -an4n$d52. (108d)

We see that AnB = -An,; i.e., the sum of the two concentrations is constant (for small optical depths). The result of a numerical solution of the coupled equations for a large range of optical depths is shown in Fig. 3 1 for a closed cell with fixed density at 8 = 0. Clearly a large change in concentration can be induced. In Fig. 31 we also see some of the characteristics of the small-optical- depth solution of Eqs. (108); i.e., a linear dependence of the drift velocities ui for small ,f, with the drift velocity for the less abundant species being largest.

A number of variations around this theme have been worked out. Arkhipkin et al. (1990) discussed white-light-induced drift (WLID) in spherically symmetric and cylindrically symmetric geometries, more appropriate geometries from the astrophysical point of view. This does not introduce new physics; the effects are just smaller. A new element was introduced by Streater (1990) realizing that WLID could also occur with just one species in a buffer gas; the atom should however be a A-type three level atom with different statistical weights for the lower levels, e.g. an atom with a hyperfine-split ground state. Here optical pumping and overlap of the spectral features are essential to the phenomenon.

To illustrate three-level WLID let us consider again a one-dimensional case where white light impinges on a capillary filled with three-level atoms immersed in a buffer gas. Again a Fraunhofer absorption doublet is burned into the white- light spectrum, the Fraunhofer lines getting more pronounced deeper into the gas. At the point where the light enters the vapor the spectrum is flat and no optical pumping occurs. Downstream, the spectral intensities at the two resonance frequencies w , and w 2 of the atom are unequal and optical pumping occurs. Thus at every point in the vapor the atomic populations are already redis- tributed due to optical pumping without the intervention of kinetic collisions. This is in stark contrast with the situation in two-atom-two-level WLID. What is similar to the case discussed earlier is that there are overlapping spectral profiles in the absorption spectrum and therefore in the local spectral intensity. Thus we expect velocity-selective excitation and two opposing drift fluxes, when, for convenience, one considers atoms in two hyperfine levels as different species. The question is whether there is a net atomic flux.

Most simply this can be answered under the same conditions where we worked before; i.e., small optical depths and small deviations from equilibrium. Now, however there is an a priori concentration profile, which, in this limit, can be

0.1 h

0 0 Q *

~

- g 0.05 L a a

s z a 0

-0.05

LIGHT-INDUCED DRIFT

0 5 10 15 20

Positiong

273

0 5 10 15 20

(b) Position 5 FIG. 31. Predicted variation of the densities (a) and the drift velocity (b) as a function of position

for two-level atoms A and B illuminated by white light. The reduced densities (Eq. (99)) and drift velocities (Eq. (100)) are shown as a function of the optical depth 4 (Eq. (98)). The prediction is given for a = e.bs/el.lD = 10, N i I N ; = 5 , w g - w1 = 2kvo. (Adapted from Popov ef a[ . , 1989, with permission.)

274 E . R . Eliel

taken to be linear: n , ( t ) = n? qt; nz(& = n4 - qt. We have taken a constant particle density in zero order. The following expressions for the drift velocities for atoms in levels 1 and 2 result:

( 109a)

(109b)

The total flux is now given by

j ( 0 = n ( O u ( t ) = nI ( tb l (4 ) + n2(tb42(5) (1 10a)

(1 lob)

with no = n? + n: = 1 the total (dimensionless) atomic density and 93 =

1 2

= -93noqt2,

s &+(x - X 2 ) P ( X - XI) .

The dimensionless drift velocity can then be approximated (n(6) = no) by

Comparing Eq. (1 11) for the drift velocity with the expression for the drift velocity for two-level atoms (Eq. (108)) we see that, for small optical depth, the drift velocity varies quadratically with t in the present case versus linearly in the two level case and that the drift velocity is not proportional to the level population but rather to the gradient of the level population. In this limit, one- species-three-level WLID is a much weaker effect than two-species-two-level WLID. At moderate optical depths the effects are comparable though. Detailed numerical results have been obtained (Arkhipkin et a l . , 1991; Atutov et al., 1991a). Recently a striking detail of three-level WLID was discussed (Atutov et al., 1991a): the drift velocity is not only position dependent but can also change sign along the spatial coordinate. We then encounter a situation where, near the entrance, the vapor is pulled towards the light source whereas far from the entrance, the vapor is pushed away. Optical pumping lies at the heart of this change of sign of the drift velocity.

A much simpler variety of light-induced drift that may have astrophysical implications is light-induced drift caused by line emission from the stellar core. In the simplest case one can think of atomic or ionic resonance light, emitted by the stellar core, being reabsorbed by identical atoms or ions in the cooler circum- stellar gas cloud. How about velocity-selective excitation, one may ask? It has been speculated that the gravitational red shift may be sufficient to ensure ve-


locity selectivity in the excitation. This variety of light-induced drift has been suggested in connection with the anomalous distribution of the hydrogen isotopes in our solar system (Atutov, 1988; Bloemink et al., 1992) (see Fig. 5).

There are as many varieties of light-induced drift as there are ways to create velocity-selective excitation, and all of them may or may not be relevant for astrophysics, even when on a laboratory scale the effects would be extremely small indeed. For instance, in a one-species-two-level system one can have velocity-selective excitation due to the collision dynamics itself or due to a temperature gradient; in both cases the zero-intensity velocity distribution is no longer a Maxwellian (Arkhipkin et al., 1992). On the experimental side of things there has been much less activity; this can be ascribed to the fact that the effects were expected to be very small, e.g., drift velocities on the order of 0.2 cm/s for spectral intensities of 1 mW/cm2/GHz. Popov et al. (1989) suggested that rubidium would be well suited for an experiment on two-atom-two-level WLID, having two isotopes with overlapping spectral features. The crucial point is to find a white-light source with sufficient spectral intensity in the appropriate wavelength interval. In a first experimental effort a very bright (luminance = 3500 cd/mm2) short-arc Xe discharge lamp, rated at 500 W power output, has been used to excite the two Rb isotopes 87Rb and S7Rb having normalized abundances of 0.2785 and 0.7215 respectively. The experimental setup is shown in Fig. 32. The atomic vapor is contained in the L-shaped cell; the light from the Xe lamp passes through one of its legs. A filter is included rejecting all wave- lengths A > 1 pm and A < 0.5 p m . The isotopic composition near the entrance and exit of the cell is measured with high accuracy using absorption spectroscopy

I

r 1 r

filter & A

PDY PDY, i I Ramp u Digital I Xe lamp I I generator I oscilloscope

FIG. 32. Experimental setup for measuring white-light-induced drift. I S 0 is an optical isolator, PD is a photodiode, VA is a variable attenuator and IF an interference filter.

E . R . Eliel

Laser current Fic. 33. Measured (data points) and calculated (solid line) transmission spectrum (20 GHz

width) on the Rb D,-line (lamp off) in an experiment on white-light-induced drift. The upper panel shows the scatter (around zero) of the difference between the calculated and experimental spectra, enlarged ten times.

with a tunable diode laser. The measured transmission spectrum is shown in Fig. 33, together with a spectrum based on known spectroscopic constants of Rb (hyperfine splittings and isotope shift). High accuracy in the determination of the isotopic composition could be achieved using a complex expression for the transmission- spectrum where, e.g., the nonlinearity of frequency sweep of the diode laser and mass-dependent Doppler widths were included. The isotopic composition could be determined with a statistical error of 0.0005; systematic errors were often appreciably larger. A source of systematic errors is the use of a capillary with a circular cross section: the transmission spectrum is an integral over absorption paths of different length; the exponential form of Beer’s law gives the short absorption paths an increased weight factor and leads to an underestimate of the optical depth. Of course this effect occurs for every absorption feature but affects the most abundant isotope most strongly. In short, the abundance of 8SRb tends to be underestimated. In the differential technique used in this experiment most of these systematic effects cancel. An analysis of these errors yielded a detection limit of 0.005 for compositional changes. A change in abundance at the exit window of the cell equal to 0.008 had been predicted by the two-level model (Popov et al., 1989) applied to the experimental conditions. In the experiment, however, no compositional changes were observed. New experiments


using more powerful white-light sources, such as mode-free lasers (Kowalski et al . , 1987; Littler et al., 1991) or amplified spontaneous emission devices (An- drews, 1986; Ewart, 1985), are necessary.

An entirely different experiment on light-induced drift with an astrophysical flavor is the experiment of Wittgrefe et al. (1992) using a Rb resonance lamp replacing the Xe lamp in a similar but less sensitive setup than the setup of Fig. 32. The idea in this experiment is that the emission spectrum can be shifted relative to the absorption spectrum using various combinations of buffer gases in the absorption cell and carrier gases in the lamp. In this way velocity-selective excitation can be ensured even though a pressure shift is unavoidably accompa- nied by pressure broadening. Here we do not expect a change in the isotopic composition; it is a pure two-level effect similar to LID induced by the gravitational red shift. The effect is quantified by the difference r] in the relative change of the Rb concentration across the cell:

where Anln = (noN - non)/noNr on and off indicating the burning of the discharge lamp. The results are shown in Fig. 34, where we have plotted the quantity r] versus the relative change in kinetic cross section A m l o (Wittgrefe et al . , 1991a). The correlation between the plotted quantities is quite high, indicating that this type of light-induced drift has been observed.

n

6\" W

F

0 0.5 A+

FIG. 34. The difference r) in the relative change of the Rb concentration across the cell in an experiment on light-induced drift of Rb with a Rb resonance lamp as a light source; r) is plotted as a function of the relative change in kinetic cross section A v / r for various buffer gases.

278 E . R . Eliel

VII. Other Light-Induced Kinetic Effects

Light-induced drift is just one of many phenomena that arise when light is introduced as a thermodynamic force, and many of these phenomena occur concur- rently. Some of the effects are more pronounced than others and well-designed experiments are required to bring forward the more subtle phenomena. For instance, as discussed in Section 11, light-induced drift is unavoidably accompa- nied by light-induced diffusive pulling. The latter effect, a result of the diffusion coefficient of resonantly excited atoms generally being smaller than the diffusion coefficient of atoms in the ground state, “traps” the atoms in the light beam. To unravel light-induced drift and light-induced diffusive pulling one can employ the fact that the former requires velocity-selective excitation in contrast to the latter (Gel’mukhanov and Shalagin, 1979b; Atutov et al . , 1986a, 1986c; Witt- grefe et al . , 1991a).

In conventional LID the collisions with the buffer gas break the symmetry between the two opposing flows of ground and excited-state particles. Ghiner et al. (1983) predicted that the role of the buffer gas could be taken over by the walls of the container. Then one relies on a difference in accommodation coefl- cient for tangential momentum between excited and unexcited particles. In a one- dimensional random-walk model (see Section 111) one can write for the drift velocity

in complete analogy with Eq. (1). Here n,ln is the fraction of particles in the excited state that have not experienced a thermalizing collision, a, and a, are the accommodation coefficients for tangential momentum for excited and ground-state particles, respectively, and vL is the laser selected velocity class. This phenomenon, dubbed suface-light-induced drqt (SLID), was extensively studied in molecular systems, in particular on the molecule CH,F undergoing rotational-vibrational excitation (Hoogeveen et al . , 1987, 1990a, 1990b). Sur- prisingly, the effect is sensitive only to the change in rotational state and not to the vibrational excitation. As a result of the insensitivity of SLID to the vibrational excitation, SLID provides a tool to study the role of the direction of the rotational angular momentum in the molecule-surface collision dynamics (“helicopter” vs. “cartwheeling” modes) (Broers et al . , 1991).

Light-induced drift is not expected to occur in a one-component gas at hydro- dynamic pressure, since this is excluded by momentum conservation. This is only valid when the system is truly one-dimensional. If, for instance, the illumination of the capillary tube that contains the vapor is nonuniform, new effects


may arise as first discussed by Ghiner et al. (Ghiner, 1982; Ghiner et al., 1982). In a typical experiment the light intensity peaks on the axis of the capillary, dropping off towards the capillary walls. Particles in the resonantly excited velocity class have a relatively large collision cross section, thus they transport their axial momentum less efficiently in the radial direction than their ground- state counterparts. Since the illumination of the tube is nonuniform, this will result in a net transport of the axial component of the momentum in the radial direction; i.e., a stress arises in the gas. This, in turn, will give rise to a nonuniform flow velocity in the tube. For realistic surfaces (nonzero momentum accommodation) a net flow of the gas results. For a stick-boundary condition (where the velocity near the wall is assumed to be zero) and under the assumption that the light intensity vanishes near the wall, one finds

7~ A u n,(r) V L , ” F ( T ) = -vc- -.

2 u n

Here AuIu represents, as usual, the relative change in kinetic cross section. Note that in the present case all particles are of a single species. This effect, called light-induced viscous flow (LIVF), has been well documented experimentally and theoretically (Hoogeveen et a f . , 1989).

In a simple picture such as used in Section I, light-induced kinetic effects originate in a modification, upon excitation, of some transport coefficient. In general transport coefficients such as the diffusion coefficient, the viscosity or thermal conductivity, can be written as some ratio of the mean thermal speed and an effective kinetic cross section. So far we have considered modification of the transport coefficients resulting from the change in the cross section. Recently a light-induced kinetic effect has been observed in a one-component gas that has been ascribed to a change in the thermal speed of the velocity-selected particles and not to a change in kinetic cross section (Hoogeveen and Hermans, 1990). As a result of an inelastic collision both collision partners get “hotter” or “colder,” resulting in an increase or decrease of their kinetic properties. This effect, called velocity-selective heating-cooling, has been observed in an experimental setup similar to the one used to observe light-induced viscous flow (Hoogeveen and Hermans, 1990, 1991; van Enk and Nienhuis, 1991). For nonuniform illumination, the expression for the drift velocity is identical to that of Eq. (1 14) with Aa/u replaced by the decrease in viscosity coefficient - A q / q .

All these effects derive from a modification of the first moment of the total velocity distribution. l o Light-induced kinetic effects (in a one-component gas) arising from higher moments of the velocity distribution should also exist (Folin

lo Note that light-induced diffusive pulling originates in a modification of the zeroth moment of the velocity distribution.

280 E.R. Eliel

er al . , 1981; Nienhuis, 1989). A nonequilibrium second moment gives rise to a light-induced anisotropy in the pressure tensor, whereas a nonzero third moment results in a light-induced heat flux. The latter is expected to be unobservably small (Hoogeveen, 1990), whereas the former should yield an effect of comparable magnitude as, e.g., light-induced viscous flow. Initial attempts to measure this effect have been unsuccessful so far but this effect has not been vigorously pursued (Hoogeveen et al., 1986). Note that all the light-induced kinetic effects discussed here have been observed only in molecular systems.

VIII. Conclusions

The field of light-induced drift has reached a certain level of maturity in the last five years. Many of the phenomena that were predicted in the early years of light- induced drift have been observed and studied in detail. On the theoretical side LID has become firmly embedded into gas kinetics and the differences and simi- larities between the various light-induced kinetic effects have been made clear. In atomic LID, which has been the focus of the present review, the detailed atomic level structure has turned out to be more important than initially appreciated. In particular the experiments on coherent population trapping in LID prove that LID is sensitive to details of the level structure that could be ignored in earlier treatments.

The rather new field of LID with classical light sources, inspired by specula- tions that LID could have important astrophysical implications, has resulted in a relative wealth of theoretical treatments, not quite balanced by experimental efforts. Although these experiments are difficult and rather unspectacular in their phenomenology, the underlying ideas are sufficiently surprising to warrant an increased experimental effort.

In view of the discussion of Section VII one is tempted to reverse the statement made in the introduction that atoms present the most attractive systems to study light-induced kinetic effects. Certainly the LID phenomenon has very spectacular manifestations in atomic systems. One can argue though that light-induced kinetics in molecular systems has a richness, far surpassing that of the atomic systems. This may be a direct consequence of the fact that the excitation is electronic for atoms and has, universally, been rotational-vibrational for molecules.

LID has been studied by only a limited number of groups, possibly as a result of its inherent quality of presenting a “marriage” between such disparate fields as laser spectroscopy and gas kinetics. Light-induced kinetic effects, applied to molecules, are slowly evolving into a tool to study other phenomena; e.g., small differences in collisional cross sections or momentum relaxation rates. It is a


useful tool to collect trace impurities to bring their local concentration up to a measurable value or separate isotopic components or separate the ortho and para varieties of a single molecular species. LID and light-induced kinetics in general have come of age.

Acknowledgments

The author gratefully acknowledges M. C. de Lignie, 3. P. Woerdman, L. J. F. Hermans, G. J. van der Meer and G. Nienhuis for their contributions and for critically reading the manuscript. The author also is grateful to D. A. van der Sijs for generating some of the figures in this paper. This work is part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) and was made possible by financial support from the Nederlandse Organ- isatie voor Wetenschappelijk Onderzoek (NWO).

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ADVANCES M ATOMIC, MOLECULAR, AND OPnCAL PHYSICS, VOL. 30

CONTINUUM DISTORTED WAVE METHODS IN ION-ATOM COLLISIONS D E R R I C K S . F. CROTHERS Department of Applied Mathematics and Theoretical Physics The Queen’s Universiry of Belfast Belfast, Northern Ireland

LOUIS J . DUBE Dipartement de Physique Universiti Lava1 Quibec, Canada

Not only time gives the brew its strength.

Johann Wolfgang von Goethe

1. Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . 287 11. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

A. Coordinates and Conjugate Momenta . . . . . . . . . . . . . . . . 290 B. Hamiltonians, Eigenfunctions and Energies . . . . . . . . . . . . . . 294 C. Coulomb Distorted Waves. . . . . . . . . . . . . . . . . . . . . 295

111. Time-Dependent Impact Parameter Formalism . . . . . . . . . . . . . . 296 A. General CDW Theory . . . . . . . . . . . . . . . . . . . . . . 296 B. Variational Multistate CDW Formulation . . . . . . . . . . . . . . . 300

IV. Time-Independent Wave Formalism . . . . . . . . . . . . . . . . . . 3 14 A. Full-House Wave Theory . . . . . . . . . . . . . . . . . . . . . 3 14 B. Half-Way House Wave Theory . . . . . . . . . . . . . . . . . . . 318

V. Conclusions and Future Perspectives . . . . . . . . . . . . . . . . . . 321 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 323 Appendix A: Recent Reviews of Ion-Atom Scattering . . . . . . . . . . . 323 Appendix B: Subject Oriented Index . . . . . . . . . . . . . . . . . . 324 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

I. Introduction and Overview

We haven’t seen everything yet but when we do it won’t be f o r thefirst time or the last, either. You know us.

J. Vinograd

Some thirteen years ago, Belkic, Gayet and Salin (1979, hereafter as BGS) reported a timely, penetrating and influential account of electron capture in high-

Copyright 0 1993 by Academic Press. Inc. All righls of reproduction in any rorm reserved.

ISBN 0-12-003830-7 287

288 Derrick S.F. Crothers and Louis J . Dub&

energy ion-atom collisions. in the context of some general and formal theory of rearrangment for heavy-particle collisions, they took upon themselves to isolate the necessary requirements that a consistent theory must fulfill. This led them to concentrate on distorted-wave perturbation expansions where proper account of the long-range nature of the Coulomb interactions was included explicitly from the outset. The formalism that emerged incorporated fully Cheshire’s (1964) ideas on Coulomb boundary conditions and provided a practical alternative to the Coulomb scattering operators and S-matrices of Dollard (1964; 1971).

More precisely, upon examining the undistorted first Born (B 1) and Oppen- heimer-Brinkman-Kramers (OBK 1) theories (that is, theories with and without internuclear potential, respectively), they showed why the latter two theories are never valid; namely, that the long-range Coulomb interactions may never be treated perrurbatively. A symptomatic manifestation in such undistorted approaches is the absence of Coulomb phases in the channel functions, resulting in incorrect boundary conditions, a direct consequence of which is the certain failure of any intended variational principle. Their remedy was simple yet sophisticated. They wrote down the first term of a boundary-corrected Born expansion. The BIB theory’ laid dormant for a number of years until given a new lease of life by Dewangan and Eichler (1985; 1986) and BelkiC, Saini and Taylor (1986a). Numerous applications and further theoretical developments (Bransden and Dewangan, 1988; Decker and Eichler, 1989c) rapidly followed suit, as indicated by Reference List 1.2 of Appendix B. (The lists of references from Ap- pendix B. will from here on be quoted as, e.g., B: 1.2.)

It is perhaps ironic that the first-order continuum distorted wave (CDW 1) theory (Cheshire, 1964; Gayet, 1972), a major topic of BGS’s review and to which they may be considered to have been leading disciples (see B: 1.5) was an existing theory that already satisfied these same correct boundary conditions. Nevertheless, other considerations had helped to undermine the CDW 1 theory. There were apparent CDW 1 infelicities regarding the asymptotic Thomas double- scattering total cross sections, and the CDWl differential cross sections were prone to unphysical dips at the critical Thomas angle. These difficulties have since been fully documented (asymptotic: BGS, Briggs, Greenland and Koc- bach, 1982a; Dub& 1984; Dewangan and Eichler, 1987b; Janev and Winter, 1985; McGuire et al . , 1984; differential cross section: McGuire et a l . , 1984; Rivarola and Miraglia, 1982; Rivarola and Salin, 1984). A CDW Born series merely lurked over the distant horizon, indeed its existence was generally doubted, not least because at a sufficiently low energy, well below asymptotic energies, its first term had the notorious tendency of producing unacceptably large cross sections.

It must not be too surprising, therefore, that in the interim decade, a rival

I Our favorite acronyms and competing synonyms are listed in Table B. I

CONTINUUM DISTORTED WAVE METHODS IN ION-ATOM COLLISIONS 289

usurper sprang forward to wrest the crown. An approximation, called the strong- potential Born (SPB) approximation, introduced by Macek and Taulbjerg (198 1) and further developed by Macek and Alston (1982) and Alston (1983), held sway until the rapier sword (Dewangan and Eichler, 1985) bared its noncompact kernel and exposed an elastic intermediate-state divergence, associated with the presence of disconnected diagrams in the theory. One might recall that, contrary to the SPB approach, the CDWl theory has a compact kernel since it may be derived (Gayet, 1972) as the first-order term of a multiple scattering theory (Dodd, 1970; Dodd and Greider, 1966; Greider and Dodd, 1966) specifically designed to avoid any disconnected diagrams. The basic idea of the SPB theory, which is very similar in spirit to the impulse (I) approximation (Briggs, 1977), was that, for asymmetric collisions with a target nuclear charge much larger than the projectile nuclear charge, the Born series and its convergence properties could be greatly improved by the use of intermediate Coulomb target states associated with the strong potential. The origin of its development may be found in an investigation by Shakeshaft (1980) on the divergence of the OBK-series as a function of increasing target charge. The conclusion of his study was to raise the point that the divergence was probably due to the use of the free Green’s function rather than the target Coulomb Green’s function. This was then first investigated by Macek and Shakeshaft (1980). In view of the formal difficulties of the SPB approach, it should be stressed that any general regard (Briggs, Macek and Taulbjerg, 1982b; Taulbjerg, 1983) “that several other models for electron capture, that have been developed over the past twenty years, can be derived from the SPB approximation by introducing further approximations of increasing degree of severity in it” (Dewangan and Eichler, 1985) ought to be treated scepti- cally. The basic CDW theory for one is not a subset of the SPB theory. Though it may be opportune to depose the usurper and return to BIB theory (see B: 1.2a) and even B2B theory (see B: 1.2b), the acclaim appears to us premature.

It is fair to say at this point that despite growing acceptance in the ion-atom community, the ideas connected with the importance or necessity of satisfying the correct boundary conditions have not reached a complete consensus. The veils have been lifted, and the opposing factions have recently stated and clarified their positions (Salin, 1991; Taulbjerg, 1990). On the one hand, the formal consequences of a strict recognition of the long-range nature of the potentials involved have been developed and are nicely reviewed in Bransden and Dewangan (1988). On the other hand, in response to the criticism, a distorted wave SPB (DSPB) formulation has appeared (Taulbjerg, Barrachina and Macek, 1990) to supersede the SPB approach (see B: 1.6 for coverage of the relevant extensions). The debate is in full swing and is bound to keep provoking inspired and creative impulses from both sides.

Against the background of these turbulent and exciting developments, this chapter is more modest in scope. Its purpose is to present the CDW theory as a

290 Derrick S . F. Crothers and Louis J . Dub6

versatile and principal contender for the crown, a multiple scattering theory for all seasons and occasions. In this light, the BnB theory will emerge as a re- spected courtier. In addition, we shall show that all of the preceding defects of the CDWl theory do not in fact occur in a more general setting and that the CDW Ansatz has a wide range of robust applications. Whereas the goal of BGS was to show that the CDW theory is the high-energy capture theory “which minimises the number of shortcomings,” our goal is to show that in principle there are no shortcomings, though in practice calculational time and effort are always limiting factors.

The content of this chapter is organised as follows. After introducing the necessary notation in Section 11, we present the time-dependent (Section 111) and the time-independent (Section IV) CDW formalism with special attention to a newly proposed variational multistate close-coupling formulation based on continuum distorted wave functions. Section V sums up our conclusions and indicates future possible avenues of research. In addition, and in recognition of the efforts of our colleagues in the field, we provide in Appendix A a list of recent reviews and progress reports in the field of ion-atom collisions, as well as an annotated index to the bibliography in Appendix B.

II. Notation

If you wish to converse with me, define your terms.

Voltaire

The notation presented in this section is appropriate to the treatment of charge exchange processes. The corresponding expressions for excitation or ionisation can easily be adapted from it (see, e.g., Dunseath, 1990). Consider the single electron capture to a final statefof a projectile P of charge Z, and mass M, from an initial state i of a target T of charge Z , and mass MT:

(1 ) Pzp+ + (Tzr* + e - ) , + (Pzp+ + e - ) , + Tzr+.

A. COORDINATES AND CONJUGATE MOMENTA

Let rr, r, and r be the position of the electron with respect to T, P and the centre of mass (CM) of ( P + T). Let R, R, and R, describe the position of P relative to T, the position of P relative to the CM of the initial subsystem ( T + e ) and the position of the CM of the final subsystem ( P + e ) relative to T. These Jacobi coordinates are displayed in Fig. 1 and compose the three sets of coordinates ( rr , Rr), (rp, R,) and (r, R), appropriate to the possible pairings of a general three-body problem. They are of course the three pairs of vectors that diagonalise the kinetic energies in the CM of all three particles. Their pairwise relationships are summarized in a compact form via the introduction of the column vectors

CONTINUUM DISTORTED WAVE METHODS IN ION-ATOM COLLISIONS 29 1

e a

FIG. 1. The three sets of Jacobi coordinates, ( r r , Itr) , ( r p , R,) and (r, R), appropriate possible pairings (two-particle centres of masses) of a general three-body problem.

FIG. possible

e a

1. The three sets of Jacobi coordinates, ( r r , Itr) , ( r p , R,) and (r, R), appropriate ! pairings (two-particle centres of masses) of a general three-body problem.

to the

and the matrix transformations

- Q T = MTPQP~ QP = M p T Q T = (ETP)-'QT (3)

- Q = ETQT, - Q = M P Q P . (4)

The transformation matrices are given explicitly by

1 - ab, a - M p = ( -bp 1)

in terms of the mass ratios* (along r T , rp and R, respectively)

(7) MT

a = MP u p = - MT a7 = ~

MT + m M P + m M T + M P

Although we are using atomic units throughout, we will, when judged necessary for the sake of clarity, keep noting by rn the mass of the electron.


with aT + bT = up + b, = a + b = 1. The reduced masses are then accordingly

(9) M P ( M T + m,

MT + M, + m PT = a T m Pi =

PP = aPm

Another “hybrid” transformation between coordinates is sometimes useful, namely,

(12)

(13)

In coordinate representation, these definitions allow us to write the kinetic energy operator, H,, in the total CM in three equivalent diagonal forms

r = arT + br,

rr = r + b R

R = r T - r,,

rp = r - aR.

The sets of conjugate momenta

transform among each other as

k T - r T + K i - R T = k p . r p + K , - R , = k - r + K - R

One notices also that Eqs. (14)-( 16) can be expressed succinctly by

(20)

as it should.


where Mi, M, and M are the diagonal matrices:

This formulation renders transformations between the representation of H o in different sets of coordinates (not necessarily the Jacobi pairs) a simple exercise in matrix multiplication. For example, it is at times of some use to introduce the pair Q' = ( r ' , R) where r' is the vector position of the electron relative to the middle of the internuclear distance. All that is then necessary is the matrix transformation between, say, Q' and QT; viz.,

in order to write

where explicitly

The kinetic energy operator in these coordinates is then simply

Ho(r', R) = -L(L + i ) V : , - -Vk 1

4 k T P 2 k T P

This procedure eases tremendously the stress involved in performing otherwise the appropriate partial differentiations.

In the CDW formulation, the physical content of the approximation (Sections 111 and IV) becomes more transparent when the vectors ( rT, rp, R) are used


as independent variables. These are called generalised nonorthogonaf (or nondiagonal) coordinates: generalised, since R depends strictly on (rT, rf) through R = rT - rf , and nonorthogonal because cross terms appear in the Hamiltonian. An obvious extension of the preceding procedure leads to the result that

1 1 1

2 P T f MT M, - -Vi - - V r r . V, + -V, * V,. (27)

This form will be exploited often in the next sections. Suffice to remark here, that upon dropping terms of order ( m / M T , f ) , i.e., the last two terms in HO(rT, rf , R), the electronic and the internuclear motions decouple completely. This in turn makes the passage to an impact parameter treatment (i.e., a suitable parametrization of R) straightforward.

B. HAMILTONIANS, EIGENFUNCTIONS AND ENERGIES

The full Hamiltonian, H (disregarding the total CM motion), can be written as

H = [Ho + W r d l + [KOp) +

= [Ho + K=(rp)I + [V,(rr) + where the Coulomb interactions are simply

Z f Vp(rf) = -- rT rf

Z T &(rT) = - -

The eigenfunctions belonging to the unperrurbed channel Hamiltonians, H, and Hf, are defined by

(31) H,+,(rT, RT) = W,(rT, RA Hf+krf, Rf) = E+krf, Rf)

and are taken of the form

WrT, R T ) = M r T ) exp(iKi - RT) (32) +/(rf, Rf) = 4/(rP) exdiK,. Rf),

where $,,, is the initial/final bound state of energy E , and E / , respectively. The total energy is then written as

where K, = p iv is, in the CM, the initial momentum of P (= - (initial momentum of (T + e)), and K/ = pfvf is the final momentum of the subsystem


( P + e) (= - (final momentum of T)). The term v is the incident velocity and to order (rn/M,.p), v, = v. Two momentum transfers can now be introduced as

(34) K = a p K j - Ki J = - K j + a,Ki

that satisfy exactly the relation

K i . R , - K , * R , = J - r , + K - r ,

K + J + v = 0.

Making use of (32) and (34) and writing K = K, + K,, and J = J, + J,, in two orthogonal components, one has further

(35)

and to order (rnlM,,p) can be shown to be related by

(36)

C. COULOMB DISTORTED WAVES

The continuum solution for the motion of a particle of mass p and momentum k in a Coulomb potential -Zlr satisfies the differential equation

where the superscript plus or minus corresponds to outgoing-incoming boundary conditions. The outgoing Coulomb wave, normalised to (27~)~6(k - k’) is

(39) $l(r; Z) = exp(ik - r)N(v),F;(iv; 1; +ikr - ik - r) = exp(ik - r)DT(r; 2)

where

(40)

Equation (39) isolates and defines the Coulomb distortion term 0:. The incoming counterparts are obtained from the relations

$;(r; Z) = [$Tk(r; Z ) ] * , (41)

Consideration of the analytic behaviour of the confluent hypergeometric function for large arguments allows the asymptotic forms of (39) and (41) to be extracted; namely,

(42)

G-2 N(v) = eru/T(l - iv), v = - = - k v’

D;(r; Z) = [D Tv(r; Z ) ] * .

- lim Dv2(r; Z ) = DV(r; Z) = exp[Tiv ln(kr T k * r)]. r+m

296 Derrick S . F. Crothers and Louis J . Dubk

These are the well-known long-range Coulomb phases. We will denote hereafter all asymptotic objects by a tilde. Whereas D ,? obey the residual equation

r (43)

one verifies easily that d: satisfies the linear differential equation

(44) Z

iv * V r D :(r; Z ) = - -. r

One of the possible forms of what is known as the continuum distorted waves is

41' = $,@n RT)D +-Jr,; ZP) (45)

4~ = $r(rp, Rp)Dy(rT; Z r ) . (46)

This particular form neglects the distortion caused by the internuclear potential V,, but includes the Coulomb distortion on the electron in its initial and final state. The terms [,+ and

(47)

as can be verified form Eqs. (28)-(29) and Eq. (27). Equipped with this notation, we are now in the position to present the CDW theory.

are exact solutions to order (m/Mr ,p ) of

[(H - V,) - EI41'1 = -(VrT * VrpI4h

III. Time-Dependent Impact Parameter Formalism

A. GENERAL CDW THEORY

The CDW theory is understood most easily within the impact parameter treatment, which indeed was the format originally used by Cheshire (1964) in its inaugural presentation. The Lagrangian for the capture process (1) is given, with Eqs. (16) and (28), by

where r is measured from the nuclear CM. The term - ( 1 / 2 p T P ) V k has been omitted, since the relative motion is described classically. However, the internuclear potential V, has been included. All expressions in this section are valid to order (m/MT,p ) only. One may refer to Section I1 for some of the exact counterparts. In a restricted straight-line impact parameter treatment, we have

R = p + vt p + Z (49)


where v is the impact velocity, t is the time with its origin at the point of closest approach and p is the impact parameter with the property

p - v = 0. (50)

Regarding r T , r, and R as generalised nonorthogonal coordinates, as mentioned in Section 11, i.e., despite the fact that

r T - r p = R, (51)

we may write the Lagrangian of (48) (see Eq. (27)) as

Here the subscripts on r are a reminder that W a f now means keeping r T and r, fixed. Of course, equally well Vr7 means keep rp and t fixed and V,, means keep r T and t fixed. Subscripts on these v operators are omitted for clarity, but typically one should read

(VL)rp,r . (53)

The v * V terms in (52) give rise to the well-known Bates-McCarroll electron translation factors (Bransden, 1972). The essence of the CDW method is to treat the second term on the right of (52), namely - V,, * V,,, as the perturbation. We term this perturbation, the nonorthogonal kinetic energy, since rT and r, are nonorthogonal coordinates, and since at a fixed time, if the electron were confined to the T or P nucleus, it would possess only the quanta1 kinetic energy - 1/2V:, or - 1/2Vfp, respectively. Omitting the perturbation, we have the residual Lagrangian

d d Hcow - i- = H + V,; V,, - i-

dtr dtr (54)

which possesses exact eigenfunctions describing the initial and final states of collision (1) given respectively by

5: = 4, (rT)Ei , -bv(r )D Zv(rp; Z , ) exp ln(vR - v ' t ) ( 5 5 ) 1 (r = 4,(rP)Ef,&)D F(rT; Z , ) exp

with

En," = exp[iu * r - iu2t - k,t ] (57)

298 Derrick S. F. Crorhers and Louis J. Dube'

in the notation of Section 11. Following Cheshire, the explicit logarithmic phases take care of Vp, while exp(-ibv - r) and exp(+iav r) are the explicit Bates-McCarroll electron translation factors, originally derived by the use of Fourier transforms but easily derivable by the partial derivatives of (52) or by a Galilean transformation (Bransden, 1972). The exponents also include associated terms representing the impressed translational kinetic energy. The eigenfunctions 6: , 57 diagonalise the Hamiltonian matrix exactly in all three Cou- lomb interactions Vp, V, and VpT. The D IV functions represent explicitly the continuum distortion correct to order (m/MT,p ) . In the limit r +

lim t: - 4i(rT)Ei,-bv(r) exp ,-.-- .ZP

- I - ln(vr, + v r p ) V

and in the limit r + + 03, we have

z r + i- ln(vr, + v - rT) V

Asymptotic freedom and orthogonalisation follows from (58) and (59), a basic requirement of collision theory (BGS). In addition, 5: and [f satisfy the correct boundary conditions exactly, with special reference to the net logarithmic phases (BGS; Cheshire, 1964; Dollard, 1964; 1971; Crothers, 1982). Equally well, (55) and (56) have no short-range Coulomb singularities, since both the functions 4 and D are regular. If we introduce the total exact time-dependent wave function q: that satisfies the equation

subject to initial and final conditions

lim '4'; - 5: lim 9: - 5: + AAty, (61)

then (suppressing the impact parameter dependence) the transition amplitude, A,', may be taken as the limit for r + + CQ of A,'(r) given by

I - - = I + + -


since t,? and tr are orthogonal at t = - m and where the Dirac bracket notation implies integration over r. It is not difficult to deduce the CDW 1 perturbation expression

since 6: and tr are also orthogonal at t = + m, so that no post-prior discrepancy occurs. Actually (65) comes more readily from the time-reversed amplitude

where 9~ satisfies the equation

subject to

lim 9,- - tr lim 97 - 57 + A j t : . (68)

It may be of interest that historically Cheshire used Eq. (66) as his starting point. Most certainly, (64) and (65) are the standard CDWl expressions that have, in wave form, been so successfully applied at intermediate energies to a wide range of processes (see B: 3). In the wave treatment, the transition amplitude is given immediately by

I - + + P , - - x

T $ ( K , ) = v d p e- ip .KIA ; (PI - (69) I where K = K, + K , , is defined in Eqs. (34), (36) and (37). The total cross section is then obtained equivalently by

One of the principal attractions of the CDWl amplitudes lies in the fact that, upon combining (64) or (65) with (69) and changing dummy variables from R

300 Derrick S. F. Crothers and Louis J . Dube'

and r to r7 and rp, the variables separate, so that the six-dimensional integral factorises into the scalar product of two three-dimensional (easily evaluated) dipole integrals (see B:4).

B. VARIATIONAL MULTISTATE CDW FORMULATJON

I . A Problematic and a Strategy for Its Resolution

It may also be noted that the first term on the right-hand side of (62) is what is called in Burgdorfer and Taulbjerg ( 1 986) the sudace term in the limit t + - =. Precisely because of (61), the exact boundary condition, this surface term does not contribute. On the other hand, close consideration of 5: and 57 shows that

and that

do not satisfy the correct boundary conditions (Crothers, 1982). On the one hand, by comparison with (58) and (59), expressions (71) and (72)

exhibit an incorrect logarithmic phase. On the other hand, the presence of N and N* indicates incorrect normalisation. This failure is not confined to infinite sep- arations, rather it occurs at most values of time and impact parameter (Crothers, 1982). Moreover, the degree of failure increases as v decreases since we have

2nz lim(N(Z/v)12 - -. b-0 V

(73)

By applying the second-order Jacobi variational principle (Crothers, 1982) to the Lagrangian

the worst of the incorrect features may be eliminated in a two-state variational CDW (VCDW) theory. Thus if we use the two-state Ansatz

(75)

f ; = t;((t;It;))-"2 (76)

'P = c , ( t ) f : + c , ( t ) f y

where


are appropriately normalised, we obtain the coupled equations

i ( l - JNg+-I*)C, = ( H , , + + - N g + - H f i - + ) ~ , + ( H d + - - N , + - H , - - ) c , (77)

i ( l - INfi- + 12)Cf = ( H f i - + - Nfi- +H, ,+ + ) c , + ( H I - - - Nfi - + H g + -)c, ( 7 8 )

where typically we define

N g + - = <S,+l8,) = (([TI[?))* = (Nf i -+ )* (79)

Equations (77) and ( 7 8 ) have the advantage that they are Galilean invariant (i.e., do not depend on the choice of coordinate origin) and gauge invariant (i.e., do not depend on V,,, as expected physically). In addition, unitarity is guaranteed as well as detailed balance. Nevertheless, [,+ at t = + w and [,- at t = - w

continue to possess incorrect logarithmic phases. Furthermore, a dichotomy arises as to the ingoing or outgoing status of the distorted waves used. In other words, should a third state (say 3) be ingoing, whether a projectile or a target state, for A,, but then be outgoing for A, or A,,? The first implementation of the program described by Eqs. (77 ) - (78 ) was tackled by Crothers and McCann ( 1 9 8 5 ) , where, anxious to solve the coupled equations exactly for a specimen calculation, they investigated the symmetric resonant collision

H + + H ( l s ) + H ( 1 s ) + H + , ( 8 1 )

using outgoing waves for both 1s channels. Reasonably good results were obtained for both differential and total cross sections. However, there was a tendency in the 2 to 50 keV range for the total cross section to be too low compared to experiment. A similar trend was noted in the differential cross section at 25 keV, particularly at the more forward scattering angles. Moreover, close examination of typical matrix elements at 25 keV for a not untypical impact parameter of l a.u., revealed that the normalisation element (5,+15,+) is well behaved for t S 0 but not for t > 0. It also revealed that the distortion of distorted waves (5:IH - id/dt,l5,+) is well behaved and appropriately negligible for t 6

0 but not for t > 0. These observations however seemed to indicate a possible strategy for the

resolution of the difficulties encountered; namely, to divide the time plane in two regions where different appropriate sets of functions should be used. In the light of what has just been said and with regards to Eqs. ( 5 8 ) - ( 5 9 ) , the proper choice of expansion functions offers itself immediately. One is to use the set (4 +} for the t S 0 half-plane and the set (5 -} for the t 3 0 half-plane. The application of this strategy has led Crothers ( 1 9 8 7 ) to introduce a “half-way house” phase

302 Derrick S.F. Crothers and Louis J . Dubi

integral VCDW theory based on the factorisation of the S. scattering matrix, into a product of two Mdler scattering matrices (Taylor, 1972)

s = n: n+ (82)

where a, represents the propagation of the initial state from t = --o) to t = 0 - , while a- represents the propagation of the final state from t = + -o) to t = 0 + . Adopting our previous notation, this now means that

- - -

where now we have the boundary conditions

lim 9,- - 8:

Equation (83) may be rewritten as before as

lim 9: - A,:(; + A,'(,. ,+--I !++%

(84)

Once again the 9'T satisfy Eqs. (60) and (67). However, perhaps somewhat perversely we appear to have switched superscripts. The reason is that from the point of view of the relative motion of the heavy particles, 9 r represents incoming waves for t < 0 while for t > 0 9: represents outgoing waves. The theory is then completed by matching Y: and 9; at t = 0 according to some sensible criterion.

2 . The Half-Way House Treatment

Rather than work with two states i and f, we prefer to work with states 1, 2, . . . , n, some of which may be projectile states and some of which may be target states. For t S 0 we define a column vector


where each 4; is an outgoing CDW function like 6: (or 6;) given by Q. (55 ) , but where the hat indicates normalisation as a function of p and t ; that is,

We may then define a symmetrically orthogonalised basis set according to Low- din (1947; 1950):

JI+ = - With the coefficient vector

the total wave function may be written as

Note that the ingoing-wave superscripts refer to the associated heavy-particle motion for t < 0. Applying the Jacobi variational principle, 6%- = 0, where 3 - is given by Eq. (74), we obtain

i c - ( r ) = H++(r)c-(r) - - - where

- H + + = (*+1H - f I * + ) (92)

and where, according to (88),

<*+I*+) = I. (93)

Introducing the evolution matrix u, it follows that for - CQ S t S 0 we have

c - ( t ) = U ( t ) c - ( O ) - - - where u( t ) satisfies the following differential equation

i U ( t ) = H++(t)U(t) - - -

with initial condition

U(0) = I . - - This allows us to write

(94)

(95)

- c - ( -CQ) = - U ( - m ) c - ( 0 ) . - (97)

304 Derrick S.F. Crothers and Louis J . Dube'

From Eq. (95), we also have

- iU+(t) - = Ut(t)[H++(t)lt . - - (98)

However differentiating (93) and invoking Green's third identity, we have

H++(t ) = [H++(t)]', - -

which implies that

(99)

- iUi(r) - = - - U+(t)H + + ( t ) . (100)

h e - and postmultiplying (95) and (100) by U + ( t ) and by U ( t ) , respectively, and subtracting the resulting expressions, we obtain

d -[Ut(r)U(t)] = 0. dt - -

Together with (96), this condition guarantees the preservation of unitarity for all t E [ - to, 01; namely,

- - Ut(t)U(t) = Ut(O)U(O) - - = I . - (102)

It follows then from Eq. (97) that

- c - (0 ) = [U(-w)]tc-(-w) - - (103)

and therefore that the Moller scattering matrix for negative times is

- a+ = [U(-w)]+. - (104)

Similarly for the positive half-plane, t 3 0, we may take

*+ = - - c+T$- . (105)

Following the application of the Jacobi's variational principle we get

i c + ( t ) - = H--(t)c+(t) - - = - - ~ [ H + + ( - t ) ] * ~ c + ( t ) _- (106)

- - ( t ) and H + + ( - t ) is obtained from where the relation between

$ - ( - r , t ) = EJl+*(r, - t ) - _- provided we choose real bound-state wave functions and where g is a constant diagonal n x n matrix whose entries are all real phases. Alternatively and perhaps more simply, we may just choose 8 , rather than v, to be the axis of quan- tisation, in which case (107) is true for complex bound states with g being the identity matrix. Defining

we have, consistent with Eqs. (95) and (96)


i U * ( - t ) = [H++(-t)]*U*(-t), - - - from which it follows immediately that

- c+(+-) = _ _ E[U(--)]*EC+(O). --

- c+(O) = - c-(0)

- at = _ _ E[U(--)]*E. _

( 1 10)

Upon imposing the simple matching condition

(111)

at t = 0, we identify the Mdler scattering matrix for positive times as

(112)

Consistent with the condition ( l l l ) , the S. matrix of (82) can now be written explicitly; viz.,

( 1 13) - s = - - E[U( -w)]*E[U( - - --)It. One verifies easily that from the unitarity of u (Eq. (102)), it follows that

It may further be noted that from Eqs. (90), (105) and the matching condition ( 1 1 1 ) that at t = 0 we have

( V p P - ) = ( c - T $ + I c - T $ + ) = c-+ ($+[$+)c - _ _ - _ - _ _ - - - C - t C -

- - = - - c + t c + (1 15) = c +t($ - I $ -)c + = (c +T$ -1c +T$ -)

= (yl+lP+). - _ - - - - - -

That is to say that the unitarity is preserved at t = 0, despite a local discontinuity in the total wave function due to the change of representation.

In line with R-matrix theory used in nuclear and atomic collisions, in which the projectile is a light particle, we may improve our matching procedure at R =

p, that is at t = 0, by considering the probability density. Essentially this is equivalent to ensuring continuity for both the wave functions *+ and q- and their derivatives at t = 0. For t 0, the current probability density is given by

where

306 Derrick S.F. Crothers and Louis J . DubP

Furthermore, we note that the Hermiticity of 1' + shows that [ c - + h + + c - ] : = C - + h + + c - - - - - - -

and therefore that

C -. J - = - C - + h + + - - -

We then match up at t = 0 with the probability density 9, + (for t 5 0) given by

J + = - c +'E [ h + +( - t)] * EC + (121) - - - -- which implies that both of the following requirements must be satisfied,

[c-(O)]'h++(O)c-(O) - - - = [ c - + ( 0 ) l t ~ [ h + + ( O ) ] * ~ c - - -- '(0) (122)

and

[ c - -(O)]+c - -(O) = [ c - +(O)]'c - +(O). (123)

and a diagon- Since 1' '(0) is Hermitean, there exists a real diagonal matrix alising matrix M so that

M'h++(O)M = D - - - - with

MtM = MM' = I . _ - -- - Setting

f - (O)[ f - (O) l+ = f'(0)["(0)1'. (128)

It therefore suffices to replace ( I I 1 ) by

c+(O) = EM*M+c-(O) or equivalently f + ( O ) = f - ( O ) . (129) - -- - - From Eqs. (1 lo), (129) and (97), the scattering matrix is now given by

where J! is

3. The Continuum and a Rejnemenr of the Matching Procedure

Turning our attention to ionisation, the ingoing and outgoing states of energy E~ = K212 are given by


5; = ( 2 ~ ) - ~ / ~ exp[iK * r,]EK,-bv(r)

[ ZfZTln(vR T v2t) (132) x D,'(r,; Z,)D,',(rf; Z,) exp ki- 1 V

where K' is the velocity of the electron relative to the projectile given by

K' = K - V. (133)

The first two factors in ( 1 32) may be regrouped to read

exp[iK' * r f lEK, ,av exp[ip * Kl (134)

emphasizing the implicitly symmetrical nature of the double continuum eigenfunctions. This reflects in turn the simple physical fact that the electron is simultaneously in the continuum of each nucleus (Belkid, 1978; Crothers and Mc- Cann, 1983). We may now generalise Eq. (90) to the complete expansion

9- = n = I +,'c; + IdK( ;c - (K) = $+,'c;. (135)

Without loss of generality we may assume that

(+n+I5K+) = 0 and (+n'l+m') = a n , (1 36)

since we may replace each 5: of (87) by

5: - 1 W5K+1535K+ (137)

using the well-known Gram-Schmidt orthogonalisation procedure before symmetrical orthonormalisation. In addition, and to good approximation, we may neglect free-free transitions and assume that

(tK+'I5K+) = - K') ( 1 38)

and that

It follows that the formalism developed in the previous paragraphs, starting with Eq. (91) and leading to Eqs. ( 1 13) or (130), carries through provided we interpret the matrices and the column vectors as being infinite and with both discrete and continuous row and column subscripts.

A further advantage of including ionisation in our CDW Ansatz is that we may refine and simplify the matching procedure embraced by Eqs. ( 1 16)-( 131). Ex- panding T* in a complete set of states {+%} as in Eq. ( 1 3 9 , that is

T+ = $ +;c; and 9- = $ +,+c,, (140) " m

308 Derrick S . F. Crothers and Louis J . Dub&

and matching 1I'+ and V - at t = 0 yields the key relation

m

where the overlap matrix element is given by

NIL+ = ($;I$;)? with the obvious property that NIL^'] * = N mn + - .

It follows that at t = 0 we have

= J - . (146)

Equation (143) follows either for the exact 9 + , based on a complete set of $ - , or for an inexact V +, based on a truncated set of $ - according to the Jacobi variational principle. Equation (144) comes from Eqs. (141)-( 142), while Eq. (145) is obtained from the closure relation for each of the complete sets of states {$;} and {$;}. It emerges that the matching condition (141) is sufficient to guarantee conversation of both probability density and current probability density. The connection between the matrix N-+ of Eq. (142) and matrix M of Eqs. ( 124) and (125) is given by

Moreover, the scattering matrix is now


which has the distinct practical advantage that no diagonalisation is required. in contrast, the procedure leading to Eq. (130) involves a diagonalisation off! + ' ( 0 ) for each value of p.

If we approximate $,= by [i and consistently neglect small overlaps, then

cd (0) = c ; ( O M n lr d )I r =o. (149)

We note that these matrix elements are well-defined. For a typical target state, we have

( 150) 1 1; ivr, + i v . r p

1 1; ivr, - iv * rp

and for a typical projectile state we have

If we assume further that the state orbitals concerned are tightly bound, we have a t t = O

(152) V

(153)

x r 1 T i- ,fi ki--; 1; +ipv . ( : ) ( : 1 Assuming yet further that

pv >> 1 (154)

310 Derrick S.F. Crothers and Louis J . Dub6

we obtain from Eq. (42) that at t = 0

V

1 " V V

ZT ln(pv) T i- ln(pv) . (156) 62 = &(rp) exp iuv * r ? i- ZPZT

It follows that the matching condition (141) reduces under these assumptions to multiplying all target based amplitudes by

(157) 2iZpZT 2iZp

( pv) v -u

and all projectile based amplitudes by 2ZpZT 2iZT

( p V ) Y - Y ,

where the v in pv, being a constant, may be safely omitted.

and (59), are given for all t by Interestingly enough, asymptotic CDW wave functions, (*, based on Eqs. (58)

I Z T i4 ln(vR T v2t)

V

I Z T T i- In(vR T v2t) .

V

These are just the wave function of the BIB Ansatz. The phase relations between 4: and between ( f are precisely compensated by the exact matching conditions (157) and (158). It follows that using 4: and 5; with continuity at t = 0 suffices, as in now standard BIB theory, and that with this particular choice of gauge essentially there is no distinction between in- and outgoing waves. As reviewed by Bransden (1988), B1B theory is remarkably versatile in the intermediate energy range (see also B: 1.2a). However, the asymptotic development of the Coulomb wave functions in proceeding from Eqs. (55) and (56) to Eqs. (58) and (59) shows that B1B cannot be expected to be accurate for small impact parameters, large angle scattering and therefore charge transfer at high energies. In particular, intermediate continuum coupling is missing in B 1B perturbation theory, for which the post and prior interactions are given by

CONTINUUM DISTORTED WAVE METHODS IN ION-ATOM COLLISIONS 3 1 1

respectively. A further limitation of B 1B theory is that it appears ill-suited to describe ion-

isation. The reason may be seen from Eq. (132). Since, for ionisation, neither orbital is bound, rT cannot be approximated by R and rp cannot be approximated by - R. On the contrary, all three variables r,, rr and R are unbounded. Elec- tron capture to the continuum (ECC), being a subset of ionisation, also seems to lie beyond the B 1B pale.

In case it may be thought that CDW theory is the exception and BIB theory the rule concerning in- and outgoing waves, let us consider the symmetric eikonal (SE) approximation, for which we have

J T i- Z, ln(vrp & v * rp) V

J T i- Z’ ln(vr, T v * rT) . V

In this case a typical nontrivial matching condition is

which is clearly not amenable to a gauge transformation.

4. A Divergence-Free Theory

Following the rather detailed exposition of the previous subsections, where we have examined some of the formal consequences of adopting a theory based upon continuum distorted waves of the type defined by ( 5 5 ) and (56), there remains to consider the “burning” question of the presence or absence of divergences in the formalism.

312 Derrick S .F. Crothers and Louis J . DubP

At the core of the theory lies the evolution matrix defined by (95). By considering the differential equation ( 9 3 , subject to (96), one finds the formal solution

and in particular

on the understanding that time ordering is preserved (Dettmann and Leibfried, 1968), because of course in general H+ + and its time integral do not commute. Clearly expressions (165) and (166) are time-dependent phase integrals. We may therefore refer to the scattering matrix (113) (or (130)), with IJ( -00) given by (166), as a phase-integral VCDW halfway house theory.

A more penetrating method of solving (95) however, rather than the time- ordered phase integral of (165), is to formulate the integral equation

- U(r) = I - i lo dt ’H++( t ’ )U( t ’ )T(r ’ - - - t ) (167)

where 7 is the Heaviside step function that converts the Volterra equation into a Neumann equation and that represents the time-dependent Green function (Mc- Dowell and Coleman, 1970, p. 224). Such a formulation maintains time ordering and does not assume the commutativity of H+ + ( t ) and its time integral. Iteration of Eq. (167) results in the Neumann series

-z

U ( t ) = I + c U“’( t ) - , = I - -

where the rth iterate is given by

where the product is taken left to right as s increases and is understood to be the unit matrix for r = 1. Naturally, matrix multiplication with possibly continuous subscript implies summation over intermediate states including an integration over the continuum. Of course when t = - m, q(tl - t) = 1 , and for instance

CONTINUUM DISTORTED WAVE METHODS IN ION-ATOM COLLISIONS 3 I3

the LIZ, matrix element is given within the second-order VCDW approximation by

r -= UzI = - i d ? , H ~ + ( r , )

170)

where the standard summation convention is understood; that is, a sum and/or integration over sI is taken. In particular taking sI equal to 1, we must consider in the limit t + - m, the matrix element

lim H & + ( ? J , 12- - =

which represents intermediate elastic propagation. Significantly we recall Eq. (58), which implies that

= lirn O(l/Rz) ,-+-=

and therefore that the integrals in Eq. (170) are well defined. We conclude that, in common with the BIB theory but in contrast with strong-

potential Born (SPB) theory (see B: 1.6) and OBK theory, the CDW theory does not contain divergences associated with intermediate elastic propagation (Croth- ers and Dub& 1989). As Eq. (172) emphasises, property (58) is critical. There- fore it is essential to satisfy the correct long-range Coulomb boundary conditions, if divergences are to be avoided. To rephrase, Coulomb interactions may no? be treated perturbarively. Moreover, it may be noted that by definition no off-shell effects arise in an impact parameter treatment, such as we have presented.

By contrast, both OBK and SPB wave theories are obliged to resort to off- shell effects in a futile attempt (in our opinion) to compensate for failure to satisfy the correct asymptotic boundary conditions (Burgdorfer and Taulbjerg, 1986; Macek and Shakeshaft, 1980; Macek and Taulbjerg, 1981; Macek and Alston, 1982; Macek, 1988). In the OBK and SPB impact parameter treatments, there arises the divergent integral (Dewangan and Eichler, 1985) given by

314 Derrick S.F. Crothers and Louis J. Dub6

It may also be noted that incorrect VCDW theory (based on Eqs. (71) and (76)) bears the imputation that

= lim (-%) r - + r

(175)

which leads to the dreaded elastic divergence discussed in Dewangan and Eich- ler (1985). Here is further proof, if any were needed, that the half-way house VCDW theory presented here contains all the necessary ingredients to make it a complete and consistent collision theory. Indeed, it may be that such considerations are critical in sensitive calculations of density matrices (Bransden, 1988; Burgdorfer and DubC, 1984; 1985a; DubC, 1992) that require detailed knowledge of the scattering matrix.

IV. Time-Independent Wave Formalism

A. FULL-HOUSE WAVE THEORY

Although it is quite possible in principle to retain V,,(R) in a full CDW treatment (Crothers, 1982), in practice it is much more convenient to anticipate a semiclassical treatment and remove by gauge transformation the phases ? iZpZr/v In(kR T kZ) in the initial and final states, respectively. The net result is an extra factor in the differential cross section ( pk)l'zpz''' (McCarroll and Salin, 1968). We will henceforth drop the internuclear potential.

Let us now review the by now standard wave version of CDW (Crothers, 1987). We define in the notation of Section 11, the continuum distorted waves

5: = +,(rr) exp[iK, * R - ibv r]D FV(rp; Zp) ,

6; = +//(rp) exp[iKf* R + iav * r]D:(rr; Z r ) .

( 176)

(177)

This comprises a little poetic license, in so far as (176) and (177) are related to ( 5 5 ) and (56) with the time-dependent phases deleted, the heavy-particle plane waves inserted and with the electron translation factors referred to the internuclear centre of mass. Alternatively, with the definitions of Section 11, one easily verifies that, to order (m/M7.p), the heavy-particle plane waves are exp[iK, *

R,] = exp[iK, * R - ibv * rl and exp[iK,. R,] = exp[iK,* R + iav - r].


The transition amplitude for charge exchange from state i to state f is given

(178)

by

T$ = ( 5 r I ( H - El+[*,+)

P,+ = [ I + G + ( H - E)](: .

where the total outgoing initial wave function is given by

(179)

The bracket notation in (178) implies integration over both R and r. We recall the diagonal expression (Eqs. (28) and (16)) of the complete Hamiltonian H :

1 H = - -vz, - -v: + K(rT) + L$(rp) (180)

which in generalised nonorthogonal coordinates R, rT and rp takes the form (see

2P TP 2P

Eq. (27))

As has been emphasised repeatedly, the term nonorthogonal refers to the presence of nonvanishing cross terms in (181), e.g., V,, s V,, whereas the coordinates are generalised, because R is actually rT - r p . As in the impact parameter treatment, the fifth and sixth terms of Eq. (181) give rise to the electron translation factors, while the third term is the CDW perturbation, being the nonorthogonal kinetic energy. The total energy E is conserved (on-shell). The wave functions (176) and (177) are (correct to order (m/MT,p) eigenfunctions of

( 182) (HCDW - E)5i$ = 0

HCDw = H + V,, * V,.

where the unperturbed Hamiltonian is given by

(183)

The exact outgoing Green function G + may be written as

G + = [ E - H + i ~ 1 - I ( 184)

(185) = G Z D W + G,',w(H - H c D w ) ~ +

316 Derrick S . F. Crothers and Louis .I. Dub&

where we have

G&w = [E - HcDw + i e ] - ' ( 1 87)

and where we have used the well-known operator indentity

B-1 E C-1 + C-l(C - B ) B - l . (188)

Substitution of expression (186) into (179) and (178) yields the CDW Neu- mann-Born series

T/i' = ( f r I ( H - Otlf:) + ( 5 i I ( H - E)'G&w(H - E)15:)

+ ( ~ F I ( H - E ) ' G & d H - H c D w ) c b ~ w ( H - E)Ifi+) (189)

+ . . . . Using the alternative operator identity

B - 1 = c - ' + B - ' ( C - B ) C - ' (190)

we may write

G + = G & + G + ( H - HcDw)G&w (191)

= G&w + G + ( H - E)G&w (192)

since both operators acting on a full three-body on-shell CDW function are equivalent to the operation of - V,, - V,.

The transition amplitude may also be written as

T/i' = ( f r ITI f i+> (193)

where T, the transition operator, is given by

T = ( H - E)' + ( H - E ) + G + ( H - E ) . ( 194)

Combining Eqs. (194) and (192), we conclude further that

(H - E) 'G+ = TG&w. (195)

Substitution of Eq. (192) into Eq. (194) gives

T = ( H - E ) t + ( H - E)'G&W(H - E ) (196)

+ ( H - E ) t G + ( H - HCDW)G&w(H - E )

= ( H - E ) t [ l + G&(H - E ) ] (197)

+ TG,',W(H - H C D W G ~ D W ( H - E )

where we have used Eq. (195) in the last step. This is just the well-known Dodd-Greider integral equation (Dodd and Greider, 1966; BGS) for the tran-


sition operator T. Replacing ( H - H,,,) in Eq. (197) by (H - E ) , as in Eq. (192), Eq. (197) may then be solved iteratively to give

m

T = ( H - E ) + C [G,',w(H - E)]". (198)

The fact that V,, connects the electron to nucleus T while V, connects the electron to nucleus P means that the perturbation - V,, * V, ensures a connected kernel of the integral equation. This particular form of the Dodd-Greider formalism is especially transparent due to our use of the generalised nonorthogonal coordinates (see Eq. (181)) and avoids the unnecessary use of nonlocal potentials and operators. To make contact with earlier derivations, our notation may be related to that of Crothers and McCann (1 984) and of Rivarola (1 985) by the following identifications:

n = O

(H - E ) + G c'ow

H - Hc,W -V,; V,

(199)

Furthermore, our wave functions 5: and 5; correspond respectively to o:Qi and my@, of Rivarola (1985), where his operator U; is connected to our transition operator by

This consideration of T; represents the so-called post-interaction (prior wave function) formulation. A similar approach may be taken to T i , the so-called prior-interaction (post-wave function) form (Rivarola, 1985). Due to the symmetrical nature of the CDW approximation, there is no post-prior discrepancy just as in the BIB and the SE theories (Bransden, 1988; Bransden and Dewan- gan, 1988).

We may remark that the first two terms of Eqs. (189) and (198) provided the starting point for the proof of the equivalence of second-order CDW (CDW2) and second-order OBK (OBK2) theories regarding arbitrary charge transfer transitions at high energy (Crothers, 1985b). Also the convergence properties of the CDW series appear quite good (Crothers and McCann, 1984; Crothers, 1987). At high energies both total and differential cross sections converge rapidly at second order, including the notorious so-called interference minimum that lies between the forward and Thomas peaks.

U; = w y t T o : . (200)

318 Derrick S.F. Crothers and Louis J. Dube'

The coordinate representation of CZDW may be given by

where we generalise the notation of (176) to

s,+(p, r,, rT) = +s(rT) exp[ip * R - ibv - r]D Fv(r,; Z,) (202)

and where cs is the energy of the state +s. Three comments are in order. First, the intermediate states include, in prin-

ciple, both target and projectile states, not to mention the double continuum states of Eq. (132). Second, at high energies, GZDW in the second term of Eq. (189) may be approximated by the free Green function, neglecting the distortion factor in (202) and given that (H - E ) and (H - E)' operate to the right and left respectively (Crothers, 1985b). Thirdly, the superscript on the Green funtion is a plus and indicates outgoing waves as required physically and as represented by the + is in the denominator of (201). This refers to the relative motion of the heavy particles, whereas the pluses on the intermediate CDW states could just as easily be minuses. This last point takes us back to the dichotomy already noted in Section 111.

B. HALF-WAY HOUSE WAVE THEORY

How can the preceding matter be resolved? As in Section 111, the answer lies in a close consideration of the boundary conditions and the introduction of a halfway house variant of Eq. (178). Instead of following the exhaustive derivation of Section 111, we will be content here to sketch the basic elements of the formalism and work out some of its implications. Thus on a more intuitive prag- matic basis, we postulate a T-matrix of the form

T; = [6fm + ( t / ' I (H - O * I Y ( + Z ) ~ ~ ) I X (6 i I8n+) Iz=o

(203) The Einstein summation convention is understood in Eq. (203); that is, both m and n are dummy indices to be summed over. The total wave function *,+ satisfies Eq. (179) as before, whereas *; is given by

W,' = [ l + G + ( H - E ) ] [ ; . (204)

X [a,,, + ( t : I (H - E)'I*:Y(-Z))I.

The dichotomy problem is resolved by defining


Note that Z = 0 - R is a time-independent path length and that the superscript plus on the left-hand side corresponds to the + is in the denominator. We may deduce from Eq. (148), taking g = 1, that in the two-state approximation

Sfi = U/i*N;+U,,' + UE*N,+U,,' + U/i*N,+U/it + UE*N,+Ufi' (206) = U*N,+U, , + U,N,+U,, + U , J J + U , + U , N , + U p

This is equivalent to taking {m, n} = { i , f} in Eq. (203). Property (172) and its time-reversed ingoing equivalent are sufficient to guar-

antee the absence of elastic divergences in Eq. (203). Note also that the entire formalism presented in the previous subsection carries through, except that according to Eq. (205) we use ingoing distorted waves for positive Z and outgoing waves for negative 2. Of course, Eq. (203) replaces Eq. (178) and includes a matching matrix at Z = 0, as in the time-dependent formalism of Section 111. Moreover, Eq. (189) has to be generalised since there are now two CDW Neu- mann-Born series, one for positive Z with 5: replaced by 5; and one for negative 2 with 5; replaced by 4,'.

A less ambitious but revealing treatment in the wave formalism is obtained (O'Rourke and Crothers, 1992a), if we start with the simplified Ansatz

(207) @,/ = tI+,r)(-Z) + 5 > r ) ( + Z )

Th+ = (@/I(H - E)+l@.,) (208)

This choice results in a consistent half-way house first-order perturbation theory that agrees with the first and fourth terms of Eq. (206) provided U,, = 1 = UJ and N ; + = 1 = N,- + . The accuracy of these latter approximations may be as- sessed indirectly by including the effect of the term - Vk/2pLrp on r)( ?Z). An important aspect of this version of our theory is that one is able to check the various computer programs because the equivalence of the wave and impact parameter treatments is guaranteed in principle by the convolution theorem (Croth- ers and Holt, 1966). In fact, a recent calculation (O'Rourke and Crothers, 1992b) based on Eq. (208) for the resonant collision H + + H ( 1 s) + H ( 1s) + H + at 400 keV, gives a result identical to that of Brown and Crothers (1991) based on a two-state implementation of the time-dependent approach given by Eq. (148). In addition, O'Rourke and Crothers (1992a) have calculated the second-order amplitude at asymptotical high energies. The amplitude in this case is given by

(209)

the second-order term of which splits into two summed contributions, one part corresponding to elastic scattering for negative 2 and Thomas double scattering for positive 2 and vice versa for the other part.

At lower energies, we cannot expect perturbation theory to provide an accurate, description. We would therefore expect to require a theory based on closeiy

and take

T j + = Th+ + (@/I(H - E)+G&,(H - E)l@,) ,

320 Derrick S. F. Crothers and Louis J. Dubk

coupled equations. For this purpose we use the second-order Jacobi variational principle

(210)

with a linear combination of orthonormalised continuum distorted wave functions (cf. Eqs. (86)-(88) and (107)) given by

8('I':lH - El'€':) = 0

subject to the boundary conditions

exp( iK,R) R

lim %;(R) - a,,, exp(iKi * R) + f;(K,, Ki) R-m

In Eq. (212) K, * Ki is confined to [ - 1, 01 while in Eq. (214) K,, - Ki is confined to [0, + 11. Variation of 9; gives, correct to order ( m / M , , )

cv; + + 2 $ ($XI(VR)rl$:) * v,9: m

= +2M $ ($:IH - eml$;)9: (215)

using Eq. (181) and assuming that $; and 8; differ by a slowly varying amount. In Eqs. (215) and (218) later, parentheses indicate integration over r only. Con- sistent with Eq. (212) we make the semiclassical Ansatz (Bates and Holt, 1966)

m

3; = exp(iK,,Z)c; (216)

subject to the initial condition given by

c"-(m) = 6.i.

Equation (215) gives

2iK,,- ac ; = 2M$ ( $ ; I H - E, - i2- $2 az M az a I ) (218)

X exp[i(K, - K,)Z]ci


where we assume that c ; is sufficiently slowly varying for d2c,/dZ2 to be negligible. If we further assume that all K, may be approximated as Mv, except that the difference of K, and K , is refined to be

2M(E" - E m ) K , - K , = x- E" - E ,

K m + Kn V

and if we absorb the exponential factors into the parentheses, then we recognize the impact parameter time-dependent equations (9 1) of Section 111; namely,

ic,; = H,','c; (220)

where the Einstein convention is again implied. Similar considerations apply to W; of Eq. (204) for positive Z.

V. Conclusions and Future Perspectives

En toutes choses, il faut savoir consi- dkrer la j n .

Jean de Lafontaine

We have presented the continuum distorted wave method in both time-dependent and time-independent form and have clarified its physical interpretation. We have then developed an extension of the CDW Ansatz to a multistate variational close- coupling theory, which removes the normalisation difficulties encountered in the standard CDW approximation. Furthermore, we have shown the new formalism to possess the desirable properties of Galilean invariance, gauge invariance, flux conservation and, above all, absence of divergences. This last point was seen to emerge in a natural and essential way from our use of wave functions that satisfy the correct Coulomb asymptotic boundary conditions. In so doing we have kept the promise made in the Introduction to present a complete, consistent, and anomafy-free formalism.

The implementation of the program detailed in Sections 111 and IV however, is still in its infancy. The main difficulty lies in the expense of calculating the required matrix elements in terms of both human and computer resources. Dif- ferent approximations of varying degrees of severity introduced during the derivation have also to be tested thoroughly for their accuracy. It is fair to say that the few existing calculations, although promising, are still to be taken as indicative rather than definitive. Nevertheless, a close-coupled equations formalism using bases that satisfy all known short- and long-range boundary conditions make the project an attractive proposition, well worth further inquiries.

Before making our final remarks, let us enlarge the scope to highlight some of

322 Derrick S . F. Crothers and Louis .I. Dubk

the recent developments in perturbative methods. Indeed, the last decade has seen the ebullient emergence of many perturbative approaches (see B: 1). Of all those listed in Appendix B however, only the CDWn, the SEn and the BnB series remain when one imposes the requirement that their treatment of Coulomb dis- tortions be symmetric in the initial and final channels. Although one would wish to classify the different approximations according to some reasonable criteria (Dube, 1986; Bransden and Dewangan, 1988), one should bear in mind that in particular, satisfying boundary conditions implies that both the scattering wave functions and the residual perturbation must be consistent with each other: a different distorted wave results in a different perturbation potential. This in turn makes the comparison of the various members of a classification a subtle matter. For example, very little is known on the convergence properties of the CDWn, SEn and BnB series. Work is in progress (Martinez, Rivarola and Dubt, 1992) to clarify this issue. If one relaxes the condition of a symmetrical treatment of both channels, one arrives at “hybrid” approaches, the most commonly used being the CDW,-E, approximation of Crothers and McCann (1983).

Various types of applications have served over the last few years to ascertain the quality of the newly derived approximations. The trend is towards an increasing sophistication going from the calculations of state-to-state transfer cross sections (see B : 3.1 a), ionisation (see B : 3.2), electron capture to the continuum (ECC; see B: 3. lb), and finally densify matrices (see B: 3.3), where the transition amplitudes are tested for size and phase. The necessity to perform such in depth comparisons may be appreciated by the following example. As reviewed by Bransden (1988), the B 1B approximation is quite successful in predicting total cross sections over a wide range of energies and systems. However, a recent study by Dub6 and Mensour (1992) reveals that the B 1B model is incapable of even qualitatively accounting for the experimental density matrix elements of Ashburn et al. (1990). So, even if the size of the transition amplitudes is reproduced correctly by a model, it may fail miserably as to their phases.

In the case of ionisation, as reviewed by Crothers (1992) and Fainstein, Ponce and Rivarola (1991), the CDW,E, model has been remarkably sucessful in describing single ionisation: the model uses (as the acronym suggests) (132) for the final uniform double continuum state and (162) for the eikonal initial state. As for ECC, regarded as a subset of ionisation, discussed in subsection III.B.4, it may be best described by a pure CDW theory in which the initial state is taken from Eq. (55) rather than Eq. (162). Although the ECC monopole term is well reproduced by the CDW approximation, the ECC dipole term (the cusp asymmetry parameter) is less well understood and remains an open question. In this context, it would be advantageous to combine the multistate variational formalism presented here for charge transfer with perturbation methods for ionisation, both within the CDW framework.

Finally, two- and more-electron processes offer a formidable “hunting” ground. As the one-electron amplitudes are reaching an ever greater degree of


confidence (notice that here the phases of the individual amplitudes are of utmost importance), it appears timely to consider extending the existing models to cover those processes. Up to now, such two-electron processes as double capture (Gayet er a f . , 1981; 1991), transfer and ionisation (Dunseath and Crothers, 1991; Gayet and Salin, 1987), and double ionisation (Deb and Crothers, 1990; 1991; Deb et a f . , 1991) have been treated within an independent-electron approximation or an independent-event model (Crothers, 1992; Crothers and McCarroll, 1987) with relative success. Progress is being made, although the strategies adopted by the different groups still contain a level of approximation that has all the flavours of recipes rather than full-grown theories. The correlation of events or particles is a continuing and fascinating issue and now that the experimentalists have thrown down the gauntlet, we can only expect a vital theoretical response in the years to come.

Acknowledgments

DSFC is indebted to Geoffrey Brown (SERC), Narayan Deb (SERC), Kevin Dunseath, David Marshall, Mark McCartney and Francesca O’Rourke (all of them recipients of postgraduate studentships from the Department of Education, Northern Ireland) and is grateful to Jim McCann (Durham) for the kind communication of results prior to publication. LJD wishes to acknowledge John S. Briggs (Freiburg) and Antoine Salin (Bordeaux) for discussions that over the years have helped to shape his appreciation of the subtle nature of ion-atom collisions. This work was supported in part by the Science and Engineering Research Council (United Kingdom) through grant GR/G 06244 and by the Na- tional Sciences and Engineering Research Council of Canada and the Fonds pour la Formation de Chercheurs et 1’Aide a la Recherche (Qutbec).

Appendix A: Recent Reviews of Ion-Atom Scattering

Many reviews and progress reports have appeared in the field of ion-atom scattering in the last 12 years or so. A list of titles of some of these publications follows. The scope of these articles is quite broad, and they should help fill the gaps left in the coverage of the present chapter. The presentations vary greatly in emphasis and in style, some being complementary, some being ”orthogonal” to our exposition.

BelkiC, D., Gayet, R., and Salin, A. (1979). Electron Capture in High-Energy Ion-Atom Collisions. Bransden, B. H. (1988). Charge Transfer and Ionisation in Fast Collisions.

324 Derrick S. F. Crothers and Louis J . Dube'

Bransden, B. H., and Dewangan, D. P. (1988). High Energy Charge Transfer. Briggs, J. S. (1985). The Theory of Electron Capture. Briggs, J. S., and Macek, J. H. (1991). The Theory of Fast Ion-Atom Collisions. Dewangan, D. P. (1988). Semiclassical Treatment of Charge Transfer Collisions. DubC, L. J. (1986). Multiple Scattering Contributions in Electron Capture Theories. Dunseath, K. M. (1990). Transfer and Ionisation Processes in Ion-Atom Collisions. Eichler, J. (1990). Theory of Relativistic Ion-Atom Collisions. Fainstein, P. D., Ponce, V. H., and Rivarola, R. D. (1991). Two-Center Ef- fects in Ionisation by Ion Impact. Jakubassa-Amundsen, D. H. (1989a). Theoretical Models for Atomic Charge Transfer in Ion-Atom Collisions. Janev, R. K. , and Preshnyakov, L. P. (1981). Collision Processes of Multiply Charged Ions with Atoms. Janev, R. K., and Winter, H. P. (1985). State Selective Electron Capture in Atom-Highly Charged Ion Collisions. Shakeshaft, R. (198 1). Atomic Rearrangement Collisions at Asymptotically High Impact Velocities. Shakeshaft, R., and Spruch, L. (1979). Mechanisms for the Capture of a Light Particle (e.g., Charge Transfer) at Asymptotically High Impact Velocities. Taulbjerg, K. (1983). Electron Capture in Ion-Atom Collisions.

Appendix B: Subject-Oriented Index

To facilitate comprehensive referencing and to avoid burdening the text with long lists of authors, we have thought it preferable to present the relevant bibliography in an index form. The index is divided into different subjects, whose coverage should be almost complete-' from the time of the review by BelkiC et al. (1979) to September 199 1.

A panoply of acronyms exist to describe the various theoretical models. Table B.l provides a guide to the most commonly used terminology. We point out synonyms and we have taken the opportunity to state our preferred acronyms as well as suggesting a descriptive naming convention for those theories where the initial and final states are not treated symmetrically.

The subject index Existing Theories should indeed be complete, whereas the other subject list- ings are meant to be indicative of new results and recent advances rather than exhaustive.


TABLE B. 1 ACRONYMS OF EXISTING THEORIES

Suggested Acronyms Synonyms Approximation

OBK OBKn Bn BIB

BnB E En SE SEn CDW CDWn CDW,

CDW,

CDW,-E,

I SPB DSPB

OBKl

CB I TFBA

CDW I

CIS +

PCDW PIA +

CIS - TCDW PIA -

CDW-EIS

First-order Oppenheimer-Brinkman-Kramers (OBK) nth-order OBK nth-order Born Boundary-corrected first Born First-order Born with correct boundary conditions True first Born approximation Boundary-corrected nth Born Eikonal nth-order eikonal Symmetric eikonal nth-order symmetric eikonal Continuum distorted wave nth-order CDW CDW initial state Continuum intermediate state (post form) Projectile CDW Peaked impulse approximation (post form) CDW final state Continuum intermediate state (prior form) Target CDW Peaked impulse approximation (prior form) CDW final-E initial state CDW-eikonal initial state Impulse Strong potential Born Distorted-wave SPB

1. EXISTING THEORIES

1.1. OBKn, Bn

Alston, 1988c; Bates and Mapleton, 1966; Bates, Cook and Smith, 1964; BelkiC and Salin, 1976; Briggs, 1986; Briggs and DubC, 1980; Briggs and Taulbjerg, 1979; Crothers and Todd, 1980; Dewangan, 1980; Drisko, 1955; Dub6 and Briggs, 1981; Golden, McGuire and Omidvar, 1978; Horsdal, Jensen and Niel- sen, 1986; Hsin and Lieber, 1987; Kramer, 1972; Mapleton, 1967; McGuire and Weaver, 1984; McGuire et a l . , 1982; 1983; 1984; 1986; Miraglia et al., 1981; Roy, Saha and Sil, 1980; Shakeshaft, 1974a; 1974b; 1978a; 1980; Shakeshaft and Spruch, 1978; Sil et al., 1979; Simony and McGuire, 1981; Simony, Mc- Guire and Eichler, 1982; Spruch, 1978; Thomas, 1927; Wadhera, Shakeshaft and Macek, 1981; Weaver and McGuire (1985).

(Relativistic version: Decker, 1990; Decker and Eichler, 1991a; 1991b; Deco

326 Derrick S . F. Crorhers and Louis J . Dubt

and Rivarola, 1988a; Humphries and Moiseiwitsch, 1984; 1985a; 1985b; Jaku- bassa-Amundsen, 1990; Jakubassa-Amundsen and Amundsen, 1985; Moisei- witsch (1982; 1985; 1988; 1989).)

1.2 BnB

1.2a B1B. Alston, 1990a; BelkiC, 1988a; BelkiC and Mancev, 1990; BelkiC and Taylor, 1987; BelkiC e ta l . , 1986a; 1986b; 1987; Corchs er al . , 1991; Datta, Crothers and McCarroll, 1990; Decker and Eichler, 1989b; 1989c; Deco, Hanssen and Rivarola, 1986b; Dewangan and Chakraborty, 1989; Dewangan and Eichler, 1985; 1986; 1987a; 1987b; 1989; Dub6 er a l . , 1990; Dunseath, Crothers and Ishihara, 1988; Grozdanov and Krstic, 1988; Saini and Kulander, 1988; Toshima and Ishihara, 1989.

(Relativistic version: Eichler, 1987.) 1.26 BIB and Beyond. BelkiC, 1988b; 1991a; BelkiC and Taylor, 1989;

Decker and Eichler, 1989a; 1989c; Dewangan and Bransden, 1988; Dewangan and Eichler, 1987b.

(Relativistic version: Toshima and Eichler, 1990.)

1.3. E' , SE'

Anholt and Eichler, 1985; Chan and Eichler, 1979a; 1979b; 1979c; Chan and Lieber, 1984; Crothers and Todd, 1980; Deco and Rivarola, 1985; Deco er al . , 1984; 1986a; 1986c; Dewangan, 1975; 1977; 1982; Dub6 and Eichler, 1985; Eichler, 1981; Eichler and Chan, 1979; Eichler and Narumi, 1980; Eichler, Tsuji and Ishihara, 1981; Fainstein and Rivarola, 1987; Ferrante and Fiordilino, 1980; Gien, 1984; Glauber, 1959; Ho et al . , 1981a; 1981b; 1981c; 1982; Ishihara and Tsuji, 1982; Kobayashi, Toshima and Ishihara, 1985; Maidagan and Rivarola, 1984; Mittleman and Quong, 1968; Sinha, Tripathi and Sil, 1986; Wilets and Wallace, 1968.

(Relativistic version: Eichler, 1985; Moseiwitsch, 1986; 1987a; 1987b.)

1.4. I ' , CDW= (CIS', PCDW, P I A + ) , COW,= (CIS-, TCDW, P I A - )

Amundsen and Jakubassa, 1980; Banyard and Shirtcliffe, 1984; BelkiC, 1977; 1978; Bransden and Cheshire, 1963; Briggs, 1977; 1980; Briggs er al . , 1982a; Cheshire, 1963; Coleman, 1969; Coleman and McDowell, 1964; 1965; Coleman and Trelease, 1968; Crothers and Dunseath, 1987; 1990; Crothers and Todd, 1980; Deb, 1988; Deb and Crothers, 1989b; DubC, 1983b; 1984; Dunseath er al . , 1988; Ghosh et al . , 1984; 1987; Gravielle and Miraglia, 1988; Jakubassa- Amundsen, 1981a; Jakubassa-Amundsen and Amundsen, 1980a; 1981; Koc- bach, 1980; Kocbach and Taulbjerg, 1985; Macek and Dong, 1988; Macek and Taulbjerg, 1989b; Mandal, Datta and Mukherjee, 1983; 1984; McCann, 1992;


McDowell, 1961; Miraglia, 1982; 1984; Miraglia and Macek, 1991; Nagy, Ma- cek and Miraglia, 1991; Pradhan, 1957; Saha, Datta and Mukherjee, 1985.

(Relativistic version: McCann, 1985 .)

1.5. CDW + ModijicationsIExtensions

Bachau, Deco and Salin, 1988; Banyard and Shirtcliffe, 1979; BelkiC, 1991b; BelkiC and Gayet, 1977a; 1977b; BelkiC and Janev, 1973; BelkiC and McCarroll, 1977; BelkiC and Salin, 1978; BelkiC, Gayet and Salin, 1981; 1983; 1984; Brown and Crothers, 1991; Burgdorfer and Taulbjerg, 1986; Cheshire, 1964; Crothers, 1981; 1982; 1983; 1985a; 1985b; 1987; Crothers and DubC, 1989; Crothers and McCann, 1982; 1983; 1984; 1985; 1987; Crothers and Todd, 1980; Datta et af., 1982; Dodd, 1970; Dodd and Greider, 1966; DubC, 1984; Fainstein, Ponce and Rivarola, 1987; 1988a; 1988b; 1989; 1990; 1991; Gayet, 1972; Greider and Dodd, 1966; Martinez and Rivarola, 1990; Martinez et uf., 1988; McCann and Crothers, 1987; McCarroll and Salin, 1967a; 1967b; 1968; Miraglia, 1983; Mukherjee and Sil, 1980; Rivarola, 1984; Rivarola and Fainstein, 1987; Rivarola and Salin, 1984; Rivarola et af., 1980; 1984; Saha, Datta and Mukherjee, 1987; Salin, 1970; Shakeshaft, 1973; Shirtcliffe and Baynard, 1980.

(Relativistic version: Deco and Rivarola, 1986; 1987a; 1987b; 1988b.)

I .6. SPB + ModijicationsIExtensions

Alston, 1982; 1983; 1988a; 1988b; 1989a; 1989b; 1990b; 1991; Amundsen and Jakubassa-Amundsen, 1984a; 1984b; Barrachina, Garibotti and Miraglia, 1985; Briggs, Macek, and Taulbjerg, 1982b; Burgdorfer and Taulbjerg, 1986; Deb, Sil and McGuire, 1985; 1987a; 1987b; Dewangan and Eichler, 1985; Dub& 1983a; 1983b; 1984; Freire and Montenegro, 1987; Gorriz, Briggs and Alston, 1983; Hsin and Lieber, 1987; Hsin et al., 1986; Jakubassa-Amundsen, 1984; Kocbach and Taulbjerg, 1985; Macek, 1985; 1988; Macek and Alston, 1982; Macek and Dong, 1988; Macek and Freed, 1985; Macek and Shakeshaft, 1980; Macek and Taulbjerg, 1981; 1989a; 1989b; Marxer and Briggs, 1989; McGuire, 1983; 1985; McGuire and Sil, 1983; 1986; McGuire and Weaver, 1986; McGuire et al., 1985; 1987; Sil and McGuire, 1985; Taulbjerg, 1990; Taulbjerg and Briggs, 1983; Taulbjerg et al., 1990; Ward and Macek, 1991.

2. ASYMPTOTIC RESULTS AND STUDIES

2.1. Theory

Briggs, 1986; Briggs and Dubt, 1980; Briggs and Taulbjerg, 1979; Crothers, 1985a; 1985b; Dewangan, 1982; DubC, 1983b; 1984; DubC and Briggs, 1981;

328 Derrick S.F. Crothers and Louis J . Dubk

D U E and Eichler, 1985; Karnokov, 1982; McGuire and Sil, 1986; McGuire et al., 1986; Rivarola, 1984; Rivarola and Miraglia, 1982; Rivarola et al., 1984; Shakeshaft, 1978a; 1978b; Shakeshaft and Spruch, 1978; 1979; Spruch, 1978; Spruch and Shakeshaft, 1979; 1984; Toshima, lshihara and Eichler, 1987.

2.2. Experiment

Breinig et al., 1983; Horsdal et al., 1986; Horsdal-Pedersen, Cocke and Stockli, 1983a; Vogt et al., 1986.

3. SOME RECENT APPLICATIONS

3.1, Charge Transfer

3.la Total and State to State Cross Sections. BelkiC, 1991b; Bruch et al., 1982; Burgdorfer and Dub6, 1985b; Chetioui et al., 1983; Cline, Westerveld and Risley, 1991; Dub6 and Burgdorfer, 1985; Dub6 et af . , 1985; Hippler et al., 1987a; 1988a; 1988b; Horsdal et al., 1986; Horsdal-Pedersen et al., 1983b; Hvelplund et al., 1983; Jolly er al., 1984; Knize et al., 1982; 1984; Knudsen et af . , 1981; O’Rourke and Crothers, 1992b; Rodbro et al., 1979; Schwab et al., 1987; Vogt et al., 1986.

3. lb Electron Capture to the Continuum. Andersen et al., 1984; 1986; Bar- rachina, 1990; Barrachina and Garibotti, 1983; Bernardi etal., 1989; Breinig et al., 1982; Crothers and McCann, 1987; Dub6 and Saiin, 1987; Focke et al., 1983; Garibotti and Miraglia, 1980; 1981a; 1981b; Groeneveld et al., 1984; Hvelplund et al., 1983; Jakubassa-Amundsen, 1981b; 1983; 1988; 1989b; Knud- sen, Andersen and Jensen, 1986; Moiseiwitsch, 1991; O’Rourke and Crothers, 1992a; Ponce and Meckbach, 1981; Skulartz, Hagmann and Schmidt-Boecking, 1988.

3.2. tonisation

Andersen et al., 1984; BelkiC, 1978; Bernardi etal., 1990; Breinig et al., 1982; Crothers and McCann, 1983; Deb et al., 1991; Dunseath and Crothers, 1991; Fainstein and Rivarola, 1987; Fainstein et al., 1987; 1988a; 1988b; 1989; 1990; 1991; Martinez et al., 1988; Miraglia, 1983; Miraglia and Macek, 1991; Mc- Cann and Crothers, 1987; McCarroll and Salin, 1978; Rivarola and Fainstein, 1987.

(Relativistic version: Deco, Fainstein and Rivarola, 1988.)

3.3. Coherence and Density Matrix Studies

Ashburn et al., 1989; 1990; Burgdorfer, 1979; Burgdorfer and Dub6 1984; 1985a; DeSerio et al., 1988; Dub6 1992; Dub6 and Mensour, 1992; Havener et


al., 1982; 1984; 1986; Hippler e ta l . , 1986; 1987b; 1989; 1991a; 1991b; Knize, Lundeen and Pipkin, 1982; 1984; Westerveld et al . , 1987.

4. INTEGRALS AND PROGRAMS

Barut and Kleinert, 1967; Barut and Wilson, 1989; BelkiC, 1981; 1983; 1984a; 1984b; BelkiC and Lazur, 1984; BelkiC e f a l . , 1981; 1983; 1984; Crothers, 1981; Dalitz, 1951; Datta, 1985; Deb, 1988; Deb and Crothers, 1989a; DubC, 1984; Holt and Driessen, 1981; Lewis, 1956; McDowell and Coleman, 1970; Nord- sieck, 1954; Sil, Crees and Seaton, 1984.

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Index

A Absorption law, 220 Abundance anomaly, 268 Alignment, 47, 56 Angular distribution of fragment ions,

Anisotropic scattering, 264-267 Applications

110- 112, 129- 132

charge transfer, 328 coherence, density matrix, 328 ionisation, 328

Associative ionization, 176 Asymptotic freedom, orthogonal, 298 Asymptotic theory, 327

Atomic beam, laser-cooled, 142, 181 Atomic coherence, 217, 225

experiment, 328

ground-state, 225, 234, 245-247, 251, 253 interlevel, 225

Autoionization, in hydrogen, 114- 118

B Barrier, centrifugal, I5 1 Bennett peak, 222 Black resonance, 246 Bloch equations, 209, 216, 223, 225 BnB, 326 Bound, S-matrix,unitary, 150 Bound states

in associative ionization, 180 collision spectroscopy, 176 effect on threshold law, 151

Boundary-corrected Born, BIB, 288 Broadband laser, 255 Buffer gas, 261-267

atomic, 261-262 molecular, 262-267

C Catalysis laser, 175 CDW2,OBK2,317 CDWn, 327 Cell coating, 203, 239, 242, 243 Chemical loss, 236, 241 Chemically peculiar stars, 268 Cheshire, 296 Close-coupling (CC) method

coupled equation, 7 exchange potential, 7 helium (He), 14, 18-20, 23-27, 30, 32 Li+, 34-36

Coherent population trapping in LID, 240,

Cold molecules, I76 Collisional

spectroscopy, 176 theory, complete, consistent, 3 14

244-255

anomaly-free, 321

broadening, 218, 228 cross-section, 202

Collisions, 200

quenching, 264-266 relative difference of, 202, 244, 261-263, 261, 277

fine structure, 167, 171, 223, 225, 227, 228, 230-232, 262

hyperfine, 153, 224 inelastic, 220, 264

with forward preference, 264 with backward preference, 266

ionizing, 144, 155, 174, 176 isotope effects, 156 kernel, 203, 209, 226, 264

Keilson-Storer kernel, 214, 216, 223, 226, 227

339

340 INDEX

Collisions (continued) models, 212, 214

phase-interrupting, 216, 225, 228, 232 polarized atoms, 144 type 1, in absence of light, 143 type 11, in a light field, 143, 158 unpolarized atoms, 144 velocity-changing, 209. 216, 218, 221, 223,

Strong, 212, 214, 216, 221, 230, 251

226, 228, 230-232, 250-253 Complete set, 308 Composition of mixed beams, 48-60.63, 64 Concentration gradient, 234-236 Condon point, 169, 171, 178 Continuum

distorted wave, CDW, 288 double, 307, 322 electron capture to, 3 I I monopole, dipole, 322

generalized nonorthogonal, 294

function, 4 variables, I82

'boundary condition, 288 explosions, I2 1 - I29 interaction, non-perturbative, 3 13

helium (He), 15, 23, 26-27, 41 hydrogen (H), 1 1

Cross section, 4, 6, 145 differential (DCS)

helium (He), 14-16, 20-33, 37-41 Li', 33-40 06+, 33-40 Si lz+ , 33, 39-40

by electrons, 105-106, 131 by fast electrons, 105- 139 by photons, 106, 119, 121

argon, 67, 68-70, 71, 82, 90 barium, 67, 82, 92-94 helium, 67.68-70.74-77, 87-89.94-95 hydrogen, 67, 77-82 krypton, 67.90 magnesium, 67 neon, 67, 82, 89, 90 sodium, 67, 72-73,91, 96-98

Coordinate, 290

Correlation

Coulomb

Coupled-channel-optical (CCO)method, I 1 - 12

for dissociative ionization

species

strontium, 67, 84 thallium, 67 xenon, 67. 90

apparent level excitation, 84-91 differential, 91 -98 electron-impact excitation, 84-91 ionization, 73-84 line excitation, 84-91 total, 66-73

types of

D Detailed balance, 47, 65 Detection of excited atoms

ionization, 65 optical, 6 1 secondary electron emission, 64 superelastic scattering, 66 thermal, 61

Deuterium, 268 Dichotomy, full-house CDW, 314 Differential cross section, see Cross section Diffusion

coefficient, 186, 21 I , 226, 227, 230. 235,

collisional. 186 laser cooling, effect on, 185 tensor, 211-212 wave packet, 185

241, 270

Disconnected diagrams, 289 Distorted-wave (DW) method, 12- 14, 289

distortion potential, 13-20.22, 30 helium (He), 20-22, 24, 26-27, 30 Li+, 33-37, 39-40

Silz+, 33. 37, 39-40 unitarized (UDW), 34

0". 33, 37, 39-40

Distorted-wave polarized-orbital (DWPO), method, 21, 23, 25-27, 30

Drift velocity, 202, 210, 227, 241, 243, 274 with broadband excitation, 255-261 influence of buffer gas, 204, 261 -267 for Na, 244-267 techniques for measuring, 234-243

Doppler limit, 219

E ELSEt, 326 Eikonal-Born series (EBS) method, 21,

26-27, 30

INDEX 34 1

Elastic collisions, 143, 146 intermediate divergence, 289

absence, 3 I 1 Electron

capture, 287

translation factor, 297 Energy distributions

of fragment ions

by fast ions, 106, 107, 134

H2andD2, 113-118 N2, 120-121 02, 118-120

of fragment ion pairs H* and Dz, 121 - 125 Nz, 125-126

of multiply-charged fragments, 126-129

Energy loss spectrometry, 136 Equation, Volterra, Neumann, 312 Evolution

of intensity, 235, 270 of particle density, 235, 270

Exchange potential, 7 Excitation of atoms, methods

charge exchange, 53 discharge, 49 electron beam, 5 I laser, 56

F Fine structure transistions, 167, 171 First-order form of many-body theory

(FOMBT), 20, 23-24,26-27, 29, 37, 39-40

FM laser, 257-261 Formalism, time-dependent

impact parameter, 296 wave, 314

Franck-Condon principle, 107, 109. 121, 123,

Fraunhofer absorption lines, 268 Full-house CDW, 314

168

G Gas-kinetic effects of light, 208-213 General CDW theory, 296 Glauber approximation, 30 Gradient velocity, 212 Gravitational redshift, 274, 277

Green's function free, target Coulomb, 289 time dependent, 3 I2

H Halfway house CDW, 302

wave, 318 closely-coupled equations, 319

multistate variational, 321 first-order perturbation, 319 semiclassical, 320

Hamiltonian, eigenfunction, energy, 294

Helium-like ions Li', 33-40

Si'*+, 33, 39-40 06+, 33, 39-40

Hermitean, 306 Hydrogen collisons, 142, 154 Hydrogen isotopes, 268, 275 Hyperfine pumping, 203, 206, 227, 232, 244,

251, 255, 258-260, 272-274 non-velocity-selective, 222 velocity-selective, 222

excited-state, 234, 247-253, 261 ground-state, 203, 272

Hyperfine splitting, 203, 222, 255

Hyperfine structure, 153, 178

I 1' . CDW,,,, CIS', PCDW, TCDW, PIA',

326 Impact parameter, 147 Integral

computer programs, 329 equation, Dodd-Gseider, 316

Interaction, long range dipole-dipole, 158 retarded, 162 sodium potentials, 178 van der Waals, 152

Internuclear potential, 314 Invariance

Galilean, gauge, 301 Ion-atom collision, 288 Ionization

associative, 144, 176 by fast ions, 106, 121-126 Penning, 144, 174

Isotope separation, 204, 232, 275-277

342

K

INDEX

Kinetic energy, nonorthogonal, 297

L Lamp laser, 255 Laser cooling, effect of long-range collisions

collisional diffusion, 181 deterministic analysis, 187 numerical simulations, 186

Laser frequency fluctuations, 253-255 Laser-induced fluorescence, 61 Level degeneracy, 221, 226 LID function, 217, 219 Light-induced

diffusive pulling, 204, 212, 267, 278 drift, 199, 208, 274, 278

experiments, dynamic, 237-243

experiments, steady-state, 235 - 236 in astrophysics, 206, 244, 267-277 of electrons, 203 of molecules, 203, 220, 221, 234, 278 with resonance lamp, 277 with tailored excitation, 207, 244, 255-261 with white light, 206, 268-277

heat flux, 280 kinetic effects, 203, 208, 278, 279

macroscopic description, 210 microscopic description, 209

optically thin regime, 239

pressure anisotropy, 280 viscous flow, 204, 279

Line emission by stellar core, 274 Local classical path approximation, 181 Long-range molecule, 178

M Matching, 305 Matrix, 312

evolution, 314 Maxwell’s demon, 201,202 Metastable atoms

detection, 60-66 production, 48-60

multilevel model, 213, 220 analytic model, 221 numerical model, 222 rate-equation model, 203, 225, 240, 241,

251, 254, 258, 261

Models for light-induced drift, 203, 2 I3

two-level model, 203, 214, 227 analytic model, 216 random-walk model, 215

Modulation index, 257 Molecular buffer gas, 204 Momentum

conjugate, 290 correlation, 186 transfer, in dissociative ionization, I 1 I ,

112 Monte Carlo simulation, 188 Multichannel quantum defect theory, 148, 151 Multicharged ions, collisions with molecules,

Multifrequency excitation, 232, 255-261 Multistate CDW theory, variational, 300

117, 132, 133, 136

N Nonabsorbing state, 246-251 Noniterative integral equation method, 14

0 OBKn, 325 Off-shell effect, 313 On-shell three body CDW, 315, 316 Operator identity, 316 Oppenheimer-Brinkman-Kramers

approximation, 132 Optical Blach equation,

with diffusion, 175 operator form, 162 for trap loss, 169

manipulation of collisions, I77 molasses, 177 piston, 202, 237-239 potential, 7, 10, I I , 19, 27-28 shutter, 240 trap, 177

Optical

Orientation, 47, 56 Oriented molecules, 129- 132 Orthogonalisation, Gram-Schmidt, 307 Oscillator strength, 19-20, 27

P Phase, long-range, Coulomb, 296 Photoassociation spectropscopy, 175 Piston velocity, 237 Polarization, 22 1

operator, 6 potential, 22

Post-prior discrepancy, CDW, BIB, SE, 317 Potential energy curves, for HI, 108 Predissociation, I 1 I , 118, 121 Pressure shift, 156

INDEX 343

Probability density, current, 305 excitation-survival, 168 survival, 165

Processes, two-electron, 322 double capture, 322 double ionisation, 322 independent-electron, -event, 322 transfer ionisation, 322

Projection operator, 6

Q Quantum reflection, 157 Quasistatic distribution, 159, 167, 169,

Quenching collisions, 204, 264 190

R r-centroid approximation, 109- 110, 125 R-matrix(RM) method, 8- I 1

continuum basis orbital, 9, 14- 15 helium (He), 14-20, 23-24, 37-41 hydrogen (H), 11, 41 intermediate energy, 10, 11, 41 Li + , 34- 39 pseudo-orbital, 9, 14- 15, 19-20 pseudo-resonance, 11, 15, 28-29 pseudo-state, 9- I 1, 15

Rabi frequency, 216, 247-249 Radiation pressure, 200, 267 Radiative escape, 164, 167 Raman resonance, 245-251, 254 Rate coefficient, 145

associative ionization, 178 low temperature limit, 150 trap loss, alkali species, 167, 172 trap loss, He metastable, 174

Rate equations, 203, 217, 225 Rearrangement, formal theory, heavy-particle

collision, 288 Reference potential, 148 Reflection approximation, 109- 110, 125 Rubidium, 204-205, 232, 239, 244, 262-263,

267, 275

S S-wave, 147, 150 Saturated hydrocarbons, 266 Saturation parameter, 218, 219 Scattering

ion-atom review, 323 large-angle, 226- 23 1

length elastic, 149, 154 inelastic, 149, 155

matrix, Moller, 302 quantum

channel state, 145 opacity function, 146 Schrbdinger equation, matrix, 146

small-angle, 226-231 theory

differential cross section, 4, 6 electron- atom(ion) 3

Second-order potential method, 22 Series, Neumann, 312

CDW Neumann-Born, 316 convergence, 3 17

Short-lived excited atoms, 56-60 Sideband generation, 26 I SPB, 327 Spontaneous emission

effect on collision, 17 1 one-atom, 186 retarded, 171 two-atom, 162, 194

State-dependent interaction, 200, 208 Stellar atmosphere, 267 Strong-potential Born, SPB, 289 Sudden approximation, 227, 262 Superelastic

collisions, 264 scattering, 47, 65

Surface scattering, 156 Surface-light-induced drift, 204, 278 Symmetric orthonormalisation, 303

T T-matrix

coupled-channel optical method, 12 distorted-wave method, 13

Doppler cooling, 142, 158 recoil limit, 142, 159

Lambda-type, 221, 245, 249-250, 272-274 V-type, 221

Threshold laws modification by light, 174 onset of, 151 Wigner, 143, 147

ordering, 312 reversal, 299

Temperature

Three-level systems, 221, 232, 250-251

Time

344 INDEX

Total cross section, see Cross section Transit relaxation, 225, 228 Transition

amplitude, 3 15 matrix, 5 see also T-matrix

alkali atom, 164, 166, 171 helium 3S, 174 metastable rare gas, 174

Trap loss

Two electron excitation processes, 106, 112,

Two step excitation processes, 112, 123, 124 123, 124

U Uncoupling condition, 222, 234 Unitary, 304

V Variational CDW, 300 Variational principle, second order Jacobi, 300 Velocity-selective

excitation, rates for, 199, 208, 226, 232,

heating-cooling, 279 255, 258, 268, 273-275, 277

WXYZ Wall adsorption, 202, 237, 239 Wave, Coulomb, distorted, 295 Wave packet, spread of, 162, 164 Wavelength, de Broglie, 142, 147 White-light-induced drift, 268 - 277

in three-level atoms, 272-274 WKB approximation, 148, 165

Contents of Previous Volumes

Volume 1

Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A . T. Amos

Electron Affinities of Atoms and Molecules, B . L. Moiseiwirsch

Atomic Rearrangement Collisions, B. H. Bransden

The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi

The Study of Intermolecular Potentials with Mo- lecular Beams at Thermal Energies, H. Pauly and J . P. Toennies

High-Intensity and High-Energy Molecular Beams, J . B. Anderson, R. P. Andres, and J . B. Fenn

Volume 2

The Calculation of van der Waals Interactions,

Thermal Diffusion in Gases, E. A . Mason, R. J .

Spectroscopy in the Vacuum Ultraviolet,

A . Dalgarno and W. D. Davison

Munn. and Francis J . Smith

W. R. S . Garton

The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson

The Theory of Electron-Atom Collisions, R. Peterkop and K Veldre

Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J . de Heer

Mass Spectrometry of Free Radicals, S. N. Foner

Volume 3

The Quanta1 Calculation of Photoionization Cross Sections, A. L. Stewart

Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt

Optical Pumping Methods in Atomic Spectros- copy, B. Budick

Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf

Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney

Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder

Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J . Wood

345

346 CONTENTS OF PREVIOUS VOLUMES

Volume 4

H. S . W. Massey-A Sixtieth Birthday Tribute, E. H. S. Burhop

Electronic Eigenenergies of the Hydrogen MO- lecular Ion, D. R. Bares and R. H. G. Reid

Applications of Quantum Theory to the Vis- cosity of Dilute Gases, R. A . Buckingham and E. Gal

Positrons and Positronium in Gases, P. A. Fraser

Classical Theory of Atomic Scattering, A. Burgess andl. C. Percival

Born Expansions, A. R. Holt and B . L. Moiseiwitsch

Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke

Relativistic Inner Shell Ionizations, C. B. 0. Mohr

Recent Measurements on Charge Transfer, J. B. Hasted

Measurements of Electron Excitation Functions, D. W. 0. Heddle and R. G. W. Keesing

Some New Experimental Methods in Collision Physics, R. F. Stebbings

Atomic Collision Processes in Gaseous Nebulae, M . J . Searon

Collisions in the Ionosphere, A. Dalgarno

The Direct Study of Ionization in Space, R. L. F. Boyd

Volume 5

Flowing Afterglow Measurements of Ion- Neutral Reactions, E. E. Ferguson, F. C. Fehsenfeld. and A. L. Schmeltekopf

Experiments with Merging Beams, Roy H. Neynaber

Radiofrequency Spectroscopy of Stored Ions 11: Spectroscopy, H. G. Dehmelt

The Spectra of Molecular Solids, 0. Schnepp

The Meaning of Collision Broadening of Spec- tral Lines: The Classical Oscillator Analog, A. Ben-Reuven

The Calculation of Atomic Transition Probabili- ties, R. J. S. Crossley

Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations sAsSI~ps C. D. H. Chisholm, A. Dalgarno, and F. R. Innes

Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Volume 6

Dissociative Recombination, J. N. Bardsley and M. A. Biondi

Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman

The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa

The Diffusion of Atoms and Molecules, E. A. Mason and T. R. Marrero

Theory and Application of Sturmian Functions, Manuel Rotenberg

Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston

Volume 7

Physics of the Hydrogen Master, C. Audoin, J. P. Schermann. and P. Grivet

Molecular Wave Functions: Calculation and Use in Atomic and Molecular Processes, J. C. Browne

Localized Molecular Orbitals, Hare1 Weinstein, Ruben Pauncz, and Maurice Cohen

General Theory of Spin-Coupled Wave Func- tions for Atoms and Molecules, J. Gerratt

Diabatic States of Molecules-Quasi-Stationary Electronic States, Thomas F. O’Malley

Selection Rules within Atomic Shells, B . R. Judd

Green’s Function Technique in Atomic and Mo- lecular Physics, Gy. Csanak, H. S. Taylor. and Robert Yaris

CONTENTS OF PREVIOUS VOLUMES 347

A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Na- than Wiser and A . J . Greenfield

A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress, Jr.

Volume 11 Volume 8

Interstellar Molecules: Their Formation and De- struction, D. McNally

Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Sys- tems, James C. Keck

Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y. Chen and Augus- tine C. Chen

Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. 1. Schoen

The Auger Effect, E. H. S. Burhop and W. N . Asaad

Volume 9

Correlation in Excited States of Atoms, A. W. Weiss

The Calculation of Electron-Atom Excitation Cross Sections, M. R. H. Rudge

Collision-Induced Transitions between Rota- tional Levels, Takeshi Oka

The Differential Cross Section of Low-Energy Electron-Atom Collisions, D. Andrick

Molecular Beam Electric Resonance Spectros- copy, Jens C. Zorn and Thomas C. English

Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy

Volume 10

Relativistic Effects in the Many-Electron Atom, Lloyd Armstrong, Jr. and Serge Feneuille

The First Born Approximation, K. L. Bell and A . E. Kingston

Photoelectron Spectroscopy, W. C. Price

Dye Lasers in Atomic Spectroscopy, W. Lunge. J . Luther, and A. Steudel

Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett

The Theory of Collisions between Charged Par- ticles and Highly Excited Atoms, 1. C. Per- cival and D. Richards

Electron Impact Excitation of Positive Ions, M. 1. Seaton

The R-Matrix Theory of Atomic Process, P. G. Burke and W. D. Robb

Role of Energy in Reactive Molecular Scat- tering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine

Inner Shell Ionization by Incident Nuclei, Jo- hannes M. Hansteen

Stark Broadening, Hans R. Griem

Chemluminescence in Gases, M. F. Golde and B. A . Thrush

Volume 12

Nonadiabatic Transitions between Ionic and Co- valent States, R. K. Janev

Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R. -J. Champeau

Topics on Multiphoton Processes in Atoms, P. Lambropoulos

Optical Pumping of Molecules, M. Broyer, G. Goudedard, J . C. Lehmann, and J . Vigue'

Highly Ionized Ions, Ivan A. Sellin

Time-of-Flight Scattering Spectroscopy, Wil-

Ion Chemistry in the D Region, George C. Reid

helm Raith

Volume 13

Atomic and Molecular Polarizabilities-A Re- view of Recent Advances, Thomas M. Miller and Benjamin Bederson

Study of Collisions by Laser Spectroscopy, Paul R. Berman

Collision Experiments with Laser-Excited At- oms in Crossed Beams, 1. V. Hertel and W. Stoll


Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J . Peter Toennies

Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet

Microwave Transitions of Interstellar Atoms and Molecules, W. B. Somerville

Volume 14

Resonances in Electron Atom and Molecule Scattering, D. E. Golden

The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster. Michael J . Jamieson. and Ronald F. Stewart

(e, 2e) Collisions, Erich Weigold and Ian E. McCarthy

Forbidden Transitions in One- and Two-Electron Atoms, Richard Marrus and Peter J . Mohr

Semiclassical Effects in Heavy-Particle Colli- sions, M. S. Child

Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin

Quasi-Molecular Interference Effects in Ion- Atom Collisions, S. V. Bobashev

Rydberg Atoms, S. A. Edelstein and T. F. Gallagher

UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree

Volume 15

Negative Ions, H. S. W. Massey

Atomic Physics from Atmospheric and Astro- physical Studies, A . Dalgarno

Collisions of Highly Excited Atoms, R. F. Srebbings

Theoretical Aspects of Positron Collisions in Gases, J . W. Humberston

Experimental Aspects of Positron Collisions in Gases, T. C . Grrjirh

Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein

Ion-Atom Charge Transfer Collisions at Low

Aspects of Recombination, D. R. Bares

The Theory of Fast Heavy Particle Collisions, B. H. Bransden

Atomic Collision Processes in Controlled Ther- monuclear Fusion Research, H. B. Gilbody

Inner-Shell Ionization, E. H. S. Burhop

Excitation of Atoms by Electron Impact,

Coherence and Correlation in Atomic Colli-

Theory of Low Energy Electron-Molecule Col-

Energies, J . B . Hasted

D . W. 0. Heddle

sions, H. Kleinpoppen

lisions. P. G. Burke

Volume 16

Atomic Hartree-Fock Theory, M. Cohen and R. P. McEachran

Experiments and Model Calculations to Deter- mine Interatomic Potentials, R. Diiren

Sources of Polarized Electrons, R. J . Celorta and D . T. Pierce

Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain

Spectroscopy of Laser-Produced Plasmas, M. H. Key and R. J . Hutcheon

Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch

Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N. Forfson and L. Wilets

Volume 17

Collqctive Effects in Photoionization of Atoms,

Nonadiabatic Charge Transfer, D. S. F. Crothers

Atomic Rydberg States, Serge Feneuille and Pierre Jacquinor

Superfluorescence, M. F. H. Schuurmans, Q. H. F. Vrehen. D. Polder, and H. M. Gibbs

Applications of Resonance Ionization Spectros- copy in Atomic and Molecular Physics, M. G.

M. Ya. Amusia

CONTENTS OF PREVIOUS VOLUMES 349

Payne, C. H. Chen, G. S. Hurst, and G. W. Foltz

Inner-Shell Vacancy Production in Ion-Atom Collisions, C. D. Lin and Patrick Richard

Atomic Processes in the Sun, P. L. Dufton and A. E. Kingston

Volume 18

Theory of Electron-Atom Scattering in a Radia- tion Field, Leonard Rosenberg

Positron-Gas Scattering Experiments, Talbert S. Siein and Walter E. Kauppila

Nonresonant Multiphoton Ionization of Atoms, J . Morellee, D. Normand, and G. Petite

Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards

Recent Computational Developments in the Use of Complex Scaling in Resonance Phe- nomena, B . R. Junker

Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, A'. Anderson and S. E . Nielsen

Model Potentials in Atomic Structure, A. Hibbert

Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D . W. Norcross and L. A . Collins

Quantum Electrodynamic Effects in Few- Electron Atomic Systems, G. W. F. Drake

The Reduced Potential Curve Method for Di- atomic Molecules and Its Applications, F. JenE

The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson

Vibrational and Rotational Excitation in Mo- lecular Collisions, Manfred Faubel

Spin Polarization of Atomic and Molecular Pho- toelectrons, N. A. Cherepkov

Volume 20

Ion-Ion Recombination in an Ambient Gas,

Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T. D.

Mark and A. W. Castleman, Jr.

Nuclear Reaction Effects on Atomic Inner-Shell Ionization, W. E. Meyerhof and J.-F. Chemin

Numerical Calculations on Electron-Impact Ion- ization, Christopher Bottcher

Electron and Ion Mobilities, Gordon R. Free- man and David A. Armstrong

On the Problem of Extreme UV and X-Ray La- sers, I. I. Sobel'man and A. V. Vinogradov

Radiative Properties of Rydberg States in Reso- nant Cavities, S. Haroche and J . M. Raimond

Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction-Rydberg Mole- cules, J . A . C. Gallas. G. Leuchs, H. Walrher, and H. Figger

D. R. Bares

Volume 19 Volume 21

Electron Capture in Collisions of Hydrogen At- oms with Fully Stripped Ions, B . H. Bransden and R. K . Janev

Interactions of Simple Ion-Atom Systems, J . T. Park

High-Resolution Spectroscopy of Stored Ions, D. J . Wineland, Wayne M. Itano, and R. S . Van Dyck, Jr.

Spin-Dependent Phenomena in Inelastic Electron-Atom Collisions, K . Blum and H . Kleinpoppen

Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O'Brien. Pierre Meystre, and Her- bert Walther

Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen

Theory of Dielectronic Recombination, Yukap Hahn

Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Pro- cesses, Shih-l Chu


Scattering in Strong Magnetic Fields, M. R. C. McDowell and M. Zarcone

Pressure Ionization, Resonances, and the Conti- nuity of Bound and Free States, R. M. More

Volume 22

Positronium-Its Formation and Interaction with Simple Systems, J . W. Humberston

Experimental Aspects of Positron and Positro- nium Physics, T. C. Grijirh

Doubly Excited States, Including New Classifi- cation Schemes, C. D. Lin

Measurements of Charge Transfer and Ioniza- tion in Collisions Involving Hydrogen Atoms, H. B. Gilbody

Electron-Ion and Ion-Ion Collisions with Inter- secting Beams, K. Dolder and B. Peart

Electron Capture by Simple Ions, Edward Pol- lack and Yukap Hahn

Relativistic Heavy-Ion- Atom Collisions, R. An- holt and Harvey Could

Continued-Fraction Methods in Atomic Physics, S. Swain

Volume 23

Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal

Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney

Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Po- tential, D. E. Williams and Ji-Min Yan

Transition Arrays in the Spectra of Ionized At- oms, J . Bauche, C . Bauche-Arnoult, and M. Klapisch

Photoionization and Collisional Ionization of Excited Atoms Using Synchrotron and Laser Radiation, F. J . Wuilleumier, D. L. Ederer. and J . L. Picque

Volume 24

The Selected Ion Flow Tube (SIFT): Studies of Ion-Neutral Reactions, D. Smith and N. G . Adams

Near-Threshold Electron-Molecule Scattering,

Angular Correlation in Multiphoton Ionization of Atoms, S. J. Smith and G . Leuchs

Optical Pumping and Spin Exchange in Gas Cells, R. J. Knize, 2. Wu, and W. Happer

Correlations in Electron- Atom Scattering, A. Crowe

Michael A. Morrison

Volume 25

Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor

Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane

Alexander Dalgarno: Contributions to Aer- onomy, Michael B. McElroy

Alexander Dalgarno: Contributions to Astro- physics, David A. Williams

Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson

Flow Tube Studies of Ion-Molecule Reactions, Eldon Ferguson

Differential Scattering in He-He and He+-He Collisions at KeV Energies, R. F. Stebbings

Atomic Excitation in Dense Plasmas, Jon C. Weisheit

Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu

Model-Potential Methods, G . h u g h l i n and G. A. Victor

2-Expansion Methods, M . Cohen

Schwinger Variational Methods, Deborah Kay

Fine-Structure Transitions in Proton-Ion Colli-

Watson

sions, R. H. G. Reid

CONTENTS OF PREVIOUS VOLUMES 35 1

Electron Impact Excitation, R. J . W. Henry and A. E. Kingston

Recent Advances in the Numerical Calcula- tion of Ionization Amplitudes, Christopher Bottcher

The Numerical Solution of the Equations of Mo- lecular Scattering. A. C. Allison

High Energy Charge Transfer, B . H. Bransden and D. P. Dewangan

Relativistic Random-Phase Approximation, W. R. Johnson

Relativistic Sturmian and Finite Basis Set Meth- ods in Atomic Physics, G. W. F. Drake and S. P. Goldman

Dissociation Dynamics of Polyatomic Mole- cules, T. Uzer

Photodissociation Processes in Diatomic Mole- cules of Astrophysical Interest, Kare P. Kirby and Ewine F. van Dishoeck

The Abundances and Excitation of Interstellar Molecules, John H. Black

Volume 26

Comparisons of Positrons and Electron Scatter- ing by Gases, Walter E. Kauppila and Talbert S. Stein

Electron Capture at Relativistic Energies, B. L.

The Low-Energy, Heavy Particle Collisions- A Close-Coupling Treatment, Mineo Kimura and Neal F. Lane

Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis

Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Franpise Mas- nou-Sweeuws, and Annick Giusti-Suzor

On the p Decay of '*'Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Larry Spruch

Progress in Low Pressure Mercury-Rare Gas Discharge Research, J . Maya and R. Lagushenko

Moiseiwifsch

Volume 27

Negative Ions: Structure and Spectra, David R.

Electron Polarization Phenomena in Electron- Atom Collisions, Joachim Kessler

Electron-Atom Scattering, I . E. McCarfhy and E. Weigold

Electron-Atom Ionization, I . E. McCarthy and E. Weigold

Role of Autoionizing States in Multiphoton Ion- ization of Complex Atoms, V. 1. Lengyel and M. I . Haysak

Multiphoton Ionization of Atomic Hydrogen Us- ing Perturbation Theory, E. Karule

Bates

Volume 28

The Theory of Fast Ion-Atom Collisions, J . S.

Some Recent Developments in the Fundamental

Briggs and J . H. Macek

Theory of Light, Peter W. Milonni and Surendra Singh

Zaheer and M. Suhail Zubairy Squeezed States of the Radiation Field, Khalid

Cavity Quantum Electrodynamics, E. A. Hinds

Volume 29

Studies of Electron Excitation of Rare-Gas At- oms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. W. Anderson

Cross Sections for Direct Multiphoton Ionization of Atoms, M. V. Ammosov, N. B. Delone, M. Yu. Ivanov, 1. I . Bondar, and A . V. Masalov

Collision-Induced Coherences in Optical Phys- ics, G. S. Aganval

Muon-Catalyzed Fusion, Johann Rafelski and Helga E. Rafelski

Cooperative Effects in Atomic Physics, J . P. Connerade

Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Mo- lecular Collisions, J. H. McGuire


Volume 30

Differential Cross Sections for Excitation of He-

Theory of Collisions Between Laser Cooled Atoms, P. S. Julienne, A . M . Smith, and K. Burnett

lium Atoms and Heliumlike Ions by Elec- tron Impact, Shinobu Nakazaki

Cross-Section Measurements for Electron Im- pact on Excited Atomic Species, S . Trajmar and J . C . Nickel

Light-lnduced

Continuum Distorted Wave Methods in lon- Atom Collisions, Derrick s, F, Crorhers and Louis J ,

E , R, Eliel

The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J. Latimer

ISBN O-L2-003830-7

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Advances in Atomic, Molecular, and Optical Physics, Volume 30

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