* Corresponding author. Present address: Plasma Physics Applications Group, Applied Theoretical and Computa- tional Physics Division, Mail Stop B259, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. Tel.: #1 505 667-6767; fax: #1 505 665-7725; e-mail: murillo@lanl.gov. Physics Reports 302 (1998) 1— 65 Dense plasmas, screened interactions, and atomic ionization Michael S. Murillo!,",*, Jon C. Weisheit# ! Physics Department, Rice University, Houston, TX 77005, USA " Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA # Space Physics & Astronomy Department, Rice University, Houston, TX 77005, USA Received December 1997; editor: R. Slansky Contents 1. Introduction 4 1.1. Plasma preliminaries 4 1.2. The dense plasma environment 6 1.3. Plasma ionization balance 7 1.4. Atomic transitions in dense plasmas 9 2. Plasma density fluctuations 11 2.1. Dynamic structure factor 13 2.2. Plasma susceptibility and dielectric response function 16 2.3. Screening models 18 2.4. Vlasov plasmas with local field corrections 20 3. Static screened coulomb potentials 22 3.1. Classical, multicomponent case 24 3.2. A hybrid potential 28 3.3. Energy level shifts 31 3.4. Total elastic scattering cross section 32 3.5. Number of bound states 33 4. Generalized oscillator strength densities 36 4.1. Definitions 36 4.2. Plane-wave model 37 4.3. Orthogonalized plane-wave model 38 4.4. Numerical partial-wave model 39 5. Ionization rates 42 5.1. Independent electron impact method 43 5.2. Stochastic perturbation method 44 5.3. Plasma impact method 46 6. Numerical study of projectile screening issues 47 6.1. Ionization rates for He‘ (ground state) 48 6.2. Ionization rates for He‘ (excited state) 51 6.3. Ionization rates for Ar‘17 (ground and excited states) 52 7. Numerical study of target screening issues 53 7.1. Non-orthogonality of initial and final states 53 7.2. Bound state level shifts 54 8. Summary and future directions 56 8.1. Important conclusions for ionization rates 56 8.2. Dense plasma issues 57 8.3. Screened interaction issues 57 8.4. Atomic ionization issues 58 Appendix A. List of frequently used symbols 59 Appendix B. Numerical computation of the dielectric responses function 60 Appendix C. Formulary 61 C.1. Plasma parameters 61 C.2. Plasma potentials for ion of charge z 62 References 63 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S0370-1573(97)00017-9
Transcript
*Corresponding author. Present address: Plasma Physics Applications Group, Applied Theoretical and Computa-tional Physics Division, Mail Stop B259, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.Tel.: #1 505 667-6767; fax: #1 505 665-7725; e-mail: [email protected].
Physics Reports 302 (1998) 1—65
Dense plasmas, screened interactions, and atomic ionization
Michael S. Murillo!,",*, Jon C. Weisheit#! Physics Department, Rice University, Houston, TX 77005, USA
" Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA# Space Physics & Astronomy Department, Rice University, Houston, TX 77005, USA
3.1. Classical, multicomponent case 243.2. A hybrid potential 283.3. Energy level shifts 313.4. Total elastic scattering cross section 323.5. Number of bound states 33
4. Generalized oscillator strength densities 364.1. Definitions 364.2. Plane-wave model 374.3. Orthogonalized plane-wave model 384.4. Numerical partial-wave model 39
AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO
Abstract
There now exist many laboratory programs to study non-equilibrium plasmas in which the electron interparticlespacing n~1@3
eis no more than a few Bohr radii. Among these are short-pulse laser heating of solid targets, where
ne&1023 cm~3, and inertial confinement fusion experiments, where n
e'1025 cm~3 can be achieved. Under such
extreme conditions, the plasma environment is expected to have a strong influence on atomic energy levels andtransitions rates. Investigations of atomic ionization in hot, dense plasmas have been motivated by the fact that theinstantaneous degree of ionization is a key parameter for the modeling of these rapidly evolving physical systems.Although various theoretical treatments have been presented in the literature, here we focus on the “random field”approach, because it can readily incorporate (quasi-static) level shifts of the target ion as well as dynamic plasma effects.In this approach, the stochastic perturbation of the target by plasma density fluctuations is described in terms of thedielectric response function. Limiting cases of this description yield the familiar binary cross-sectional model, staticscreening collision models, and the more general dynamical screening models. Screening of the target ion is treated herewith several static screening potentials, and bound state level shifts of these potentials are explored. Atomic oscillatorstrength densities based on these different models are compared in numerical calculations for ionization of He` andAr`17. Finally, we compile a list of atomic/plasma physics issues that merit future investigation. ( 1998 ElsevierScience B.V. All rights reserved.
Fig. 1. (upper panel) Regimes in temperature—density space characteristic of several interesting and important plasmas:pulsar magnetospheres; tokamaks (MCF); ICF experiments; lightning; cores of white dwarf stars, the Sun, and Jupiter;Earth’s ionosphere; non-neutral (pure electron) plasmas; ultra-short-pulse laser plasmas; and electrons in metals.Classical plasmas are left of the dash line denoting ¶
e"1, and weakly coupled plasmas are left of the solid line denoting
Ce"1. (lower left panel) Familiar phase diagram of a simple element (e.g., argon), in terms of the thermodynamic
variables pressure and temperature; both the critical point (CP) and triple point (TP) are marked. (lower right panel) Thesame phase diagram recast in terms of density and temperature. When translated to the upper panel this plot occupiesonly a small rectangular region.
1. Introduction
1.1. Plasma preliminaries
When plasmas are mentioned in non-technical discussions, they often are described with thephrase “the fourth state of matter”, to reinforce the notion that the ionized substances in neonbulbs and lightning bolts differ dramatically from our normal material surroundings. Unfortunate-ly, this phrase is not a particularly good one, because in various circumstances ionized matter canbehave very much like solids, or liquids, or ordinary gases. Better, albeit more technical, definitionsconvey the fact that plasmas are many-body systems, with enough mobile charged particles tocause some collective behavior. Non-neutral (single species) as well as quasi-neutral (electron-ion)plasmas are thereby included.
One encounters a great range of conditions in laboratory and natural plasmas whosephysical properties and behavior are germane to energy, defense, space, and numerous industrialprograms [91]. The upper panel in Fig. 1 marks in temperature—density space the locationsof the dense, hot plasmas generated in inertial fusion experiments; the cool, dilute plasma of
Earth’s ionosphere; the relativistic plasmas in pulsar (neutron star) magnetospheres; the degenerateplasmas formed by electrons in metals; and a few other, well-studied plasma regimes. The lines in thistop panel are associated with two important plasma parameters [43,16]. Coulomb coupling, theratio of the average potential to kinetic energy, for species a is described by the parameter
Ca"2.3]10~7 z2an1@3a /¹a , (1)
where na is the particle density in cm~3, zae is the species charge, and ¹a is the temperature in eV.(In general, various plasma species can have different temperatures.) One can also define aninterspecies coupling parameter, Cab (see, e.g., [49]). The condition C*1 identifies the strongcoupling regime. Fermi degeneracy is measured by the ratio ¶ of the Fermi temperature ¹
Fto the
particle temperature. The degenerate regime, where ¶*1, requires the use of quantum statistics.This criterion usually is relevant only for the electrons, where
¶e"2.4]10~15 n2@3
e/¹
e. (2)
When C@1 one says that the plasma is ideal, and when ¶@1, that the plasma is classical.Further indications of the expected richness of plasma phenomena can be obtained from
consideration of the lower panels of Fig. 1: on the left is a familiar equation-of-state (EOS) diagramfor a simple element like argon; shown are the lines in pressure—temperature space that delineateordinary phase transitions. On the right, this diagram is recast in terms of density and temperature,the state variables of the upper panel. Note first that the parameter ranges in typical EOS plots aremuch smaller than those in the plasma plot, and second that — at a given density — the plasma statecan be achieved by increasing or decreasing the density. (This unusual behavior will be explainedbelow, in Section 1.3.) Moreover in most instances these transitions to the plasma phase occurgradually, as more and more electrons populate positive energy states, in contrast to the abruptchanges that occur when, e.g., a liquid freezes.
This Report is concerned with classical plasmas that are hot and dense, i.e., that have high-energydensity, and specifically with the influence of such an environment on the elementary process ofatomic ionization. Historically, motivation for the study of high-energy-density plasmas first arosein connection with stellar interiors: how were their equations of state, their radiatve opacities, andtheir nuclear reaction rates affected by densities typically exceeding 102 g/cm3 and temperaturesexceeding 103 eV? Similar questions later arose in nuclear weapons research. In recent years,however, there has been a growing interest in such plasmas due to their relevance to shortwavelength (EUV & X-ray) lasers [61,99,75], inertial confinement fusion (ICF) research[69,9,14,36,58], and short pulse X-ray sources [77]. In addition, experiments to study the funda-mental statistical properties of dense plasmas are being carried out with ultra-short-pulse lasers(USP) [89,28,78], compressive shocks [21], and exploding wires [4]. These laboratory plasmas arecharacterized by electron temperatures in the range 101~3 eV, and in many instances the electrondensity exceeds that of a solid. It is expected that even higher energy densities will be created at theproposed National Ignition Facility [67], which should produce plasmas with ¹'10 keV andne'1026 cm~3.Many important statistical properties of real plasmas can be developed from the OCP, the one
component plasma model, in which particles of a single species a are embedded in a homogeneous,neutralizing background whose charge density is !zaena. But, when correlations between differentspecies are important, this scheme must be generalized to two or more components, as described in
a series of papers by Ichimaru and colleagues in the mid-1980s [45]. These models form the basis ofmost of the plasma physics used here.
1.2. The dense plasma environment
As we quantify below, for the purpose of understanding atomic processes in plasmas, key criteriafor the characterization “high density” are a significant overlap of bound state wavefunctions withthose of several plasma particles, or atomic transition energies near that of a plasma collectivemode. Strongly coupled plasmas (C'1), with or without degeneracy effects, obviously are in thehigh-density category, but it will become clear that this true for many weakly coupled plasmas, too.So, how dense is “dense”? We describe here only three simple estimates; more sophisticatedtreatments are possible [81,80].
First, we consider a hydrogenic ion with nuclear charge Z in an excited state having principalquantum number a. (Here, and throughout this paper, in text we refer to ionic charges as z or Z,with the unit “e” being implied.) The “size” of this ion can be estimated by taking the radius of theelectron cloud (for an s state) to be
r.!9
"5SaDrDaT"(15a2/2Z) Bohr , (3)
where 1 Bohr"+2/me2,a0&5.29 nm. The factor of 5 is somewhat arbitrary and is used to
identify not the mean radius SaDrDaT but rather an effective “edge” of the ion. Note that the radius ofthe ion increases as the square of the principal quantum number a. The length r
.!9may be
compared with the mean interionic spacing n~1@3i
to identify those states DaT which are highlyperturbed by neighboring ions.
As an example, for an Al plasma near solid density (typical of short-pulse laser experiments [78],we find from Eq. (3) that all states with a*3 overlap neighboring ions and are therefore stronglyperturbed by them. In fact, as it is not clear which ion most affects electrons in these states, weshould not consider these electrons to be bound to any particular ion; instead they need to beregarded as part of the continuum. This type of ionization, to be contrasted with thermalionization, is called pressure ionization. Pressure ionization is well known in solid state physics[38,16,82,62] because it gives rise to energy bands and conduction electrons in metals; itsappearance has the important consequence for atoms in a high-density plasma environment oflimiting the number of bound states.
Next, we consider the influence of free electrons on the atom, and compare an effectiveinteraction volume (i.e., the volume occupied by a bound electron) with the mean volume occupiedby a plasma electron. Define a density parameter D by the ratio
D"
(4p/3)r3.!9
1/ne
K3]10~22a6n
eZ3
. (4)
The condition D'1 corresponds to having at least one plasma electron within the atomic volume.When this happens, the plasma electron(s) will screen the nucleus and thereby decrease the bindingenergy of the atomic electron. Thus, we expect that atomic electrons are more weakly bound ina dense plasma: bound state energy levels are shifted towards the continuum. Together, pressureionization and energy level shifts are referred to as “continuum lowering” since both phenomenamove bound states closer to the positive energy threshold. In Section 3, we describe this screeningin terms of the plasma’s dielectric response function.
Table 1Principal quantum numbers a of states strongly affected by high-density environments. Shown are various hydrogenicnoble gas ions of nuclear charge Z for plasma densities (in cm~3) characteristic of (low-density) MCF and (high-density)ICF experiments
ne"1015 1024
Z"2 a'17 All10 '39 '118 '53 '136 '75 '2
Table 1 contrasts values of the principal quantum number that correspond to states satisfyingD'1, for plasma densities relevant to magnetic confinement and inertial confinement fusionexperiments (MCF and ICF). Although only Rydberg states, and hence processes like dielectronicrecombination, are affected in MCF experiments, where densities n
e(1015 cm~3, nearly all states
are affected in ICF experiments, where greater-than-solid densities occur.In addition to the effects associated with high number density there are effects associated with
collective behavior at high density. The primary phenomenon is that of electron plasma oscilla-tions. The energy associated with this oscillation,
+ue"3.7]10~11Jn
eeV , (5)
(ne
again is in units of cm~3) provides a third measure of “dense”. At an electron density of1023 cm~3, for example, this corresponds to an energy +u
e"11.7 eV, which is on the order of
atomic transition energies. The collective behavior of the ions at high density leads to a similarresult, but these energies are characteristic of transitions in the Rydberg states which typically arepressure ionized (by the argument given above). Even at modest plasma densities Rydberg statesare broadened into a quasi-continuum [47].
From the scales defined by Eqs. (3)— (5) we conclude that, for the purposes of studying atomicprocesses in plasmas, the term “high density” corresponds to particular combinations of low Z,high a, high n
i, and high n
e.
1.3. Plasma ionization balance
Knowledge of charge state fractions and populations of excited states is important for determin-ing transport properties of, as well as interpreting spectroscopic emission from, hot plasmas. Inthermal equilibrium this knowledge is easily obtained via statistical mechanics, for in suchcircumstances the population n
aof state DaT is related to the population n
bof state DbT by
na/n
b"(g
a/g
b)e~b(Ea~Eb) , (6)
where the E’s and g’s are energies and statistical weights of the states, and b"1/kB¹. Extension of
Table 2A synopsis of the study of ionization balance in plasmas. Early work involved high-density H plasmas in thermodynamicequilibrium, with later studies being of non-thermal, steady-state plasmas at low density. Recently it has become possibleto produce solid density plasmas of moderate nuclear charge Z that evolve on the subpicosecond time scale
System Date Density Temp. Z Thermal Steady state Time dependent
is theionization potential of the charge z ion, and the G’s are atomic partition functions [100]. This is the“Saha—Boltzmann Equation” which, as indicated in Table 2, was originally applied to ionizationbalance in stars. (Note that no reference to underlying ionization/recombination processes need bemade in obtaining the Saha—Boltzmann equation.) This equation gives, for example, the mean ioncharge zN for an element with nuclear charge Z,
zN"+Z
z/0zn
z+Z
z/0nz
, (8)
which is important for obtaining effective Coulomb scattering cross sections used in transportcalculations [43]. The Saha—Boltzmann equation can be used to show that, at fixed temperature,the degree of ionization zN increases as n"+
znz
decreases. And, if one accounts for continuumlowering it is clear that (again at fixed ¹) zN increases as n increases much beyond that of normalsolids. Approximate ionization balance results for equilibrium plasmas also can be obtained fromvarious density functional schemes, such as the Thomas—Fermi and Average-Atom models (see,e.g., [107,24].
It is too bad that none of these straightforward prescriptions apply to most laboratory plasmaexperiments: since true thermal equilibrium cannot be realized on the timescales involved, theseexperiments must be modeled by means of the detailed atomic processes that occur. This procedure,which conceptually is well understood, employs a set of rate equations [87,83,109,3,70] describing thetime evolution of atomic populations due to various gains and losses. In the rest frame ofa homogeneous plasma, the population of state DaT in an ion of charge z, n
z,a(t), obeys the equation
dnz,a
(t)/dt"(formation rate)!(destruction rate) . (9)
Here, “formation”/“destruction” refers to any atomic process that can create/destroy the chargez ion in state DaT. Some of the processes commonly included in Eq. (9) are shown in Table 3. Ofcourse, in principle, there are similar equations coupled to this one for each of the other quantumstates of the same ion, plus all the quantum states of the other atomic species; in practice, the totalnumber of states (and equations) must be limited by some combination of physics arguments andcomputational constraints [111]. In some cases the time evolution is slow enough that the
Table 3The key atomic processes which govern ionization balance in plasmas. Excited states of charge z ions are denoted byz* and photons by c. Note that the resonant capture process cannot occur in a ion without doubly excited states
derivative on the left-hand side can be neglected. As indicated in Table 2, this simplification appliesto such steady-state environments as interstellar HII (H`) regions, tokamaks, and some aspects ofICF experiments. Also indicated are more recent experiments which require the full time depend-ence of Eq. (9). Note that the plasma densities of these recent experiments are quite high.
Traditionally, the electron impact rates in these population equations are obtained by determin-ing a cross section p
ba(�) for the single electron process e~#z[state DaT]Pe~#z[state DbT] and
accounting for the plasma environment by an average involving the flux ne� of free electrons.
Calculations of this kind appear as early as 1912, in Thomson’s study of the collisional ionizationprocess, which even predates the development of quantum mechanics. Since that time, muchprogress has been made in computing accurate cross sections for simple atoms, and full quantumtreatments are now widely available to treat complex atomic targets as well as the indistinguisha-bility of incident and bound electrons. Accurate approximate methods also exist and are describedin the collisional excitation reviews by Bartschat [2], Fritsch and Lin [23], and Burke et al. [8],and in the collisional ionization reviews by Younger [120] and Bottcher [65]. Once the crosssection has been obtained, the rate w
bafor that bound—bound or bound—free process aPb can be
written as
wba"n
ePd3v vF(�)pba
(�),neSp
bavT . (10)
Here F(�) is the velocity distribution of plasma electrons, and we emphasize that pba
refers to a crosssection for a binary (electron—ion) collision. These collisional rates, together with rates for otherimportant processes (e.g., those in Table 3), allow a model to be constructed, from a set of equations(Eq. (9)), which approximately describes the behavior of a nonequilibrium plasma [57].
1.4. Atomic transitions in dense plasmas
One new complication in many of the current laboratory plasma experiments results from thevery short timescales involved. There may be insufficient time for the electron velocity distributionF(�) to relax to its equilibrium, Maxwellian form. Although it can be very difficult to determinewhat F(�) is [116,29], it is straightforward to incorporate any particular result into the rateintegrals of Eq. (10), and this issue, which recently has been explored by Salzmann and Lee [102], isnot considered further. A second new complication results from the high particle densities being
achieved, because the true atomic collisional rates wba
can differ from their low-density values.Quantitative description of these plasmas requires a set of population equations with rates suitablymodified to account for the screening effects at high plasma density.
In recent years there actually has been considerable effort devoted to determining collisionalrates and spectral line shapes for dense plasmas. (Our focus is on collisional transitions; readersinterested in spectral line formation are referred to the monograph by Griem [31] and toproceedings of the conference series on Atomic Processes in Plasmas, and Radiative Properties ofHot Dense Matter, for progress in this important, related field.) However, because the relevantexperiments tend to be “integral”, in the sense that no single phenomenon can be isolated formeasurement, essentially all the collision work has been theoretical. These efforts have employeda wide variety of techniques within various models to address bound—bound excitation, ionization,and three-body recombination. In each case, dense plasma phenomena were typically incorporatedby separating plasma screening effects into two parts: screening of the projectile(s) and screeningwithin the target ion. With this distinction, the models may be divided into two categories, in whichthe projectile screening is treated statically or dynamically.
Calculations of bound—bound excitations within a purely static screening model originated withthe work of Hatton et al. [39]. There, the Born approximation was used with an electron—ioninteraction potential relevant to an ideal plasma. This was later improved upon by Whitten et al.[117] in calculations done with the distorted-wave approximation and close coupling descriptionsusing both ideal and nonideal interaction potentials for hydrogenic ions. Davis and Blaha [12]presented a similar model, based on a finite temperature average atom model, for bound—boundexcitation within the distorted wave approximation that focuses on atomic energy level shifts. Workdirected towards improving the aspherical properties of the potentials used by Whitten et al. (withinthe Born approximation) has been published by Diaz-Valdes and Gutierrez [13,33], and, usinga semiclassical model, by Jung [52,53]. Most recently Jung and Yoon have carried out semiclassicalionization rates for hydrogenic ions in dense plasmas [54]. In each case, the collision cross section (or,equivalently, a collision strength) was computed, from which the rate in Eq. (10) could be obtained.These researchers have found, for example, that the result can in some cases be more sensitive to thecollision physics treatment (e.g., Born versus close coupling) than dense plasma effects.
Dynamical treatments of the plasma—ion interaction are a generalization of the static case andmost often require a more elaborate theoretical approach. Typically, the rate is computed directlyrather than via a binary collision cross section. Advantages of these treatments are that collectiveeffects are included and rates can be obtained for transitions that cannot be described in terms ofbinary collision cross sections. Interestingly, the earliest work incorporating dynamical effectspredates that of the static treatments. In the paper by Vinogradov and Shevel’ko [113] a methodwas proposed in which a bound—bound excitation can be described without recourse to a binarycollision cross section, but rather as transition due to an external random field produced by all ofthe interacting plasma electrons. Weisheit [115] developed a similar model for bound—boundexcitation rates in which the transition is driven by plasma density fluctuations (including electronsand ions). Both have shown how their model reduces to the binary collision model in thelow-density limit. Dynamic screening has also been included in recombination processes as well.Schlanges et al. [103] have introduced a method for obtaining both ionization and recombinationcoefficients within a quantum kinetic approach based on the nonequilibrium Green functionmethod [55]. The static screening approximation was made for their numerical computations,
however. Girardeau and Gutierrez [34,27] treated recombination rates using a second quantiz-ation approach in which recombination energy is transferred purely to a collective mode in theform of a plasmon. Later, Rasolt and Perrot [95] computed three-body recombination rates whichare enhanced by collective behavior. The excitation model of Weisheit [115] has been extended toionization by Murillo and Weisheit [84] and Murillo [85]. Schlanges and Bornath [104,6] haveextended the quantum kinetic approach and have included some nonideal plasma effects.Bound—bound excitations in a relativistic average-atom model incorporating the random fieldapproach have also been computed [121]. Ebeling, Forster, and Podlipchuk have implementeda computational technique in which the time evolution of the ionizing electron’s wave packet iscomputed as it is perturbed by dynamic plasma electrons, these being simulated by a moleculardynamics technique [17].
In this paper, we consider the more general case of dynamic screening in the electron—ioninteraction. A model is presented which, in the spirit of previous approaches, also separates theplasma interaction into a piece that modifies the projectile and a piece that modifies the target.Dynamical screening will be described, as in Murillo and Weisheit [84], in terms of plasma densityfluctuations. In Section 2 these fluctuations are characterized by the dynamic structure factor andthe plasma dielectric response function. Approximations are discussed which recover the lowdensity and static screening cases. Then, in Section 3, screening of the target ion is discussed forvarious plasma conditions. A new potential is introduced which provides a smooth interpolationbetween well-known ideal and non-ideal plasma potentials. Effects on ionic energy levels are thendiscussed. In Section 4 various forms for the oscillator strength of a bound-free transition areconsidered. Then in Section 5 these pieces are put together in a model for calculating atomictransition rates, with numerical results for ionization being presented in Section 6 and Section 7.Our computations treat only hydrogenic ions of nuclear charge Z in various bound states;application of the basic formulae to many-electron targets should be straightforward. Finally,Section 8 summarizes our principal conclusions and offers our opinions on important issues forfuture work in this subject. Frequently used symbols are listed in Appendix A.
2. Plasma density fluctuations
Before addressing collisional atomic processes, we need to discuss static and dynamic structuralproperties of dense plasmas and the functions which describe them. Specifically, the static anddynamic structure factors will be defined and related to other important quantities such as theradial distribution function, the dielectric response function, and the susceptibility. The topicscovered here are a subset of the general subject of linear response theory; see, e.g., the works byKubo [63,64]. This formalism applies equally well to a variety of other condensed systems such asliquid metals [56,41,110], Fermi gases [76], Bose gases [119], supercoiled DNA [19], membranes[123], and viruses [74,71]. However, we will have an electron gas in mind for application to theatomic collision problem which is treated in Section 5. Excellent discussions of the functionsconsidered here can also be found in the texts by Goodstein [30], March and Tosi [124], andHansen and McDonald [37].
A classical ideal gas is characterized by the complete absence of interparticle interactions. Sincethe particles cannot communicate with each other, they behave independently and may be spatially
Fig. 2. OCP radial distribution functions for coupling parameter values of C"0.1,20,100. The structure that appears atlarger C values reflects ordering of the particles.
located anywhere, relative to other particles, with equal probability. In a dense system, however,interparticle correlations lead to a non-random spatial structure. A measure of this structure isprovided by the radial distribution function g(r), which describes the likelihood that there isa particle at r given that there is another particle at the origin. For a completely random system,a classical ideal gas, g(r) is therefore uniform. We may formally define g(r) for a gas of numberdensity n, containing N particles, by
ng(r)"1NT
N+i/1
N+jEi
d[r!(ri!r
j)]U ; (11)
due to interactions, g(r) will have maxima if two particles are likely to be separated by particular r-values and will have minima if particles are unlikely to have particular separations. The averagingdenoted by S2T represents an ensemble average and thus g(r) is a quantity which describes themean static structural properties of the dense system. The definition of Eq. (11) corresponds to anasymptotic (rPR) value of unity for g(r). Fig. 2 shows the OCP g(r) for values of the couplingparameter C"0.1,20,100 [97]. Note that larger C values are reflected in g(r) as larger deviationsfrom the ideal gas result of g(r)"1. The minimum at small r arises from strong Coulomb repulsionwhereas maxima occur at preferential “lattice-like” spacings. As C increases from small to largevalues, the plasma’s structure changes from gas-like to liquid-like to solid-like structure, a trendthat is consistent with our discussion related to Fig. 1.
The radial distribution function is directly measurable in elastic scattering experiments. Forelastic scattering of some probe (e.g. electron, X-ray, neutron) by a many-body target, thedifferential elastic scattering cross section may be written as
where +k is the momentum transferred to the probe, »(k) is the Fourier transform of the effectivetwo-particle interaction energy »(r) between the probe and individual target particles, and
S(k)"1#nPd3r e~*k > r[g(r)!1] . (13)
Once S(k), the static structure factor, has been measured experimentally several thermodynamicproperties, such as the energy
º"
32
n¹#2pn2P=
0
»(r)g(r)r2dr (14)
and the pressure
P"n¹!
2n2p3 P
=
0
d»(r)dr
g(r)r3dr , (15)
can be easily obtained [68]. In both Eq. (14) and (15) the first terms are the ideal gas results and thesecond terms reflect contributions arising from (spherically symmetric) interactions between theparticles, as weighted by the radial distribution function.
2.1. Dynamic structure factor
Since the radial distribution function describes static properties of dense systems, a generaliz-ation is needed for the description of time-dependent phenomena. Such a generalization isprovided by the van Hove correlation function G(r, t) [112,37], defined as
G(r, t)"1NT
N+i/1
N+j/1
d[r!(ri(t)!r
j(0))]U . (16)
In terms of the particle number density n(r, t)"+id(r!r
i(t)), G(r, t) can be written
G(r, t)"Pd3r@Sn(r#r@, t)n(r@, 0)T"Pd3k(2p)3
Sn(k, t)n(!k, 0)Te*k > r . (17)
Physically G(r, t) can be interpreted as the likelihood that there is a particle at r at time t given thatthere was a particle (which may be the same particle) at the origin at time t"0. Thus, G(r, t)contains dynamical information regarding the movement of particles in the system. Later, inSection 5, we will see how the microscopic fluctuating potential produced by these movements canexcite ions in a dense plasma. Often G(r,t) is broken into two pieces,
G(r, t)"1NT
N+i/1
d[r!(ri(t)!r
i(0))]U#
1NT
N+i/1
N+jEi
d[r!(ri(t)!r
j(0))]U ,
where the first term, the “self” term, is the contribution from the particle at the origin being found atr at time t and the second term, the “distinct” term, is the contribution from a different particlebeing found at r at time t. Evidently,
G(r, 0)"d(r)#ng(r) , (18)
which establishes the connection between G(r, t) and g(r).
1For the sake of simplicity, we consider here the classical definition of G(r, t). In the quantal definition, ri(t) and r
j(t) are
non-commuting operators which must be properly ordered. This does not represent a limitation since we never directlyemploy the definition of G(r, t), but rather its Fourier transform S(k, u). For the precise quantal definition, one is referredto the original literature by van Hove [112].
As with g(r), G(r, t) can be measured experimentally, albeit by inelastic scattering experiments.To do so, one considers the generalization of the static structure factor, the dynamic structurefactor (DSF) S(k,u), which is defined as
S(k,u)"NPd3rPdt G(r, t)e~*(k > r~ut) , (19)
where +k is again the momentum transferred and +u is the energy transferred in the collision.1Another dynamic structure factor can be defined in terms of density fluctuations, dn(r)"n(r)!n,rather than of the density itself. This dynamic structure factor satisfies the sum rule
S(k)"1
2pNP=
~=
duS(k, u) . (20)
We need not distinguish between these two definitions of S(k,u) because, as we show below, forinelastic processes (uO0) they are functionally equivalent.
To see how S(k,u), and hence G(r, t), arises in an inelastic scattering experiment consider thepedagogic example of a free electron which is inelastically scattered by a plasma. In the Bornapproximation we may describe this event as an electron in initial momentum state Dp
aT scattering
into final momentum state DpbT while the plasma undergoes a transition from state DAT to state DBT.
If the Coulomb interaction energy of this electron at r and the plasma particles of type a ina volume element at r@ is na(r@)Uea(r!r@)d3r@, the first-order transition rate can be expressed as
wfi"(2p/+)KSBDSp
bDPX
+a
nL a(r@)Uea(r!r@)d3r@DpaTDATK
2d(E
f!E
i) , (21)
where Ef"E
B#E
band E
i"E
A#E
a. Here, nL a(r) is the operator whose diagonal matrix elements
SADnL a(r)DAT give the species density na(r) when the plasma is in state SDAT, and the integrationranges over the plasma volume X. In this paper we consider only the transitions induced byelectron density fluctuations, and therefore we will be concerned only with the term n
eU
ee. (In
Section 8.2 we comment on the neglected terms.)It is possible to write Eq. (21) in a way that is physically more revealing, by decomposing the
Coulomb term Uee
into discrete Fourier modes of wavevector k, and then defining the momentumand energy transferred from the incident electron to the plasma as
+k"pa!p
b,
+u"p2a/2m!p2
b/2m , (22)
respectively. With these manipulations Eq. (21) takes the form
wfi(k,u)"(2p/X2+2)DU
ee(k)D2DSBDnL s(k)DATD2d(u!u
BA) , (23)
where +uBA
"EB!E
Aand where nL †(k)"nL (!k) is the kth Fourier mode of n
For our purposes it is unimportant which particular plasma states DAT and DBT are involved inthe transition aPb. Therefore, we perform a sum over final plasma states and a canonical averageover initial plasma states to obtain the average rate of transitions aPb:
wba"Q~1+
A,B
e~bEAwfi(k,u) , (24)
where Q is the plasma canonical partition function.The mean (electron-induced) transition rate also can be expressed in terms of the DSF S(k,u) as
wba"(1/X2+2)DU
ee(k)D2S(k,u) , (25)
where, from Eq. (23) and (24), it is evident that
S(k,u),2p +A,B
e~bEA
QDSBDnL s(k)DATD2d(u!u
BA) . (26)
To develop some useful relationships that S(k,u) satisfies, we begin by writing out the squaredmatrix element in Eq. (26) and using the integral representation of the Dirac delta function toobtain
S(k,u)"P=
~=
dq e*uq+A,B
e~*uBAqe~bEA
QSADnL (k)DBTSBDnL s(k)DAT. (27)
The time dependence of the electron density fluctuations can now be highlighted by writing thematrix elements in the Heisenberg picture as
where HKpis the Hamiltonian of the plasma, viz. SHK
pDAT"SE
ADAT. Substitution of these quantities
into Eq. (27) gives
S(k,u)"P=
~=
dq e*uq+A,B
e~bEA
QSADnL (k, q)DBTSBDnL s(k, 0)DAT (29)
which, by eliminating the expansion of unity, +BDBTSBD"1, yields the result
S(k,u)"P=
~=
dq e*uqSnL (k, q)nL s(k, 0)T"2pn2d(k)d(u)#P=
~=
dq e*uqSdnL (k, q)dnL s(k, 0)T ; (30)
now, the average S2T,+AQ~1exp(!bE
A)SAD2DAT. In the second step the density has been
separated as n(r)"n#dn(r) and indicates that, for inelastic processes (uO0), we can equivalentlydefine S(k,u) with either the density or its fluctuations. This form, together with Eqs. (17) and (19),establishes connection with G(r, t) and shows that G(r, t) can be measured by inelastic scatteringexperiments. It is easy to see that the DSF is a measure of the amplitude that a density fluctuationof wave vector k created (with the nL s operation) at time zero remains at time q later. Alternatively,one can say that the DSF is the time Fourier transform of the density—density correlation function,
2Many authors use variants of this definition that differ by factors of neand/or 2p.
and thus it represents the spectrum of density fluctuations. Eq. (30), with nL s(k, 0)Pn(!k, 0) andnL (k, 0)Pn(k, 0), is the usual classical definition2 [43], wherein the averaging is taken over N-bodyphase space.
The situation above corresponds to the case in which an incident electron transfers energy andmomentum to the plasma. At finite temperatures the opposite process can occur as well. Since thiscan also be viewed as the incident electron transferring momentum !+k and energy !+u to theplasma, we are led to consider the function S(!k,!u), which can be found easily by writingEq. (26) as
S(k,u)"2p +A,B
e~b(EA `EB~EB)
QDSADnL (k)DBTD2d(u!u
BA)
"2p +A,B
eb+ue~bEB
QDSADnL (k)DBTD2d(u!u
BA) . (31)
The quantity EB!E
Bhas been re-introduced in the exponential and the matrix element has been
rewritten in terms of nL (k). The dummy indices B and A can be switched which allows theidentification
S(k,u)"eb+uS(!k,!u) . (32)
There is an alternate method of arriving at Eq. (32) which provides some physical insight.Consider a thermal equilibrium system in which the state Dp
aT has population n
aand the state SDp
bT
has population nb. From Eq. (6) we know how the populations of these levels are related, and we
known that, in equilibrium, specific transition rates back and forth between the levels are equal, viz.
wba"w
ab. (33)
Eqs. (6) and (25) can be combined with Eq. (33) to yield Eq. (32), which reveals that Eq. (32)embodies the principle of detailed balance for a finite temperature plasma.
2.2. Plasma susceptibility and dielectric response function
We next relate the DSF to the plasma’s susceptibility. The linear susceptibility s(k,u) is definedin terms of the Fourier components of an external potential /
%95(k,u) and the ensemble averaged
electron density fluctuation Sn*/$
(k,u)T it induces, as
s(k,u)"Sn
*/$(k,u)T
!e/%95
(k, u). (34)
Thus, it measures the ability of an external potential to produce density fluctuations. (In theremainder of Section 2.2 all densities will refer to electron density fluctuations and the subscript
“ind” will be omitted.) The density fluctuations n(r, t) can be found by formally including theexternal potential in the equations of motion for the plasma density. Since we are seeking linearresponse functions we apply first-order perturbation theory to write
n(r, t)"!
ie+P
=
~=
dq h(t!q)Pd3r@SAD[nL (r@, q), nL (r, t)]DAT/%95
(r@, q) . (35)
In this expression, which was obtained by allowing the external perturbation to evolve the plasmafrom some initial state DAT, the unit step function h(x) has been included to allow for an integrationover all times q. This form, the fluctuation of a quantity being written in terms of a commutator, isa general and ubiquitous result of many-body theory [94,63].
If the plasma is in thermal equilibrium, we can average over initial states DAT to find the meanthermal density fluctuation Sn(r, t)T. To simplify this expression a complete set of eigenstates isinserted between density operators and, for the first term in the commutator, yields the result,
+A
e~bEA
QSADnL (r@, q)nL (r, t)DAT"
1X2
+k
e*k > (r{~r) +A,B
e~bEA
QDSADnL (k)DBTD2e*uBA(t~q) , (36)
which reveals the translational invariance and stationarity properties of this matrix element.Furthermore, this expression identifies the integrations in Eq. (35) as convolution integrals thatmay be trivially related to the Fourier-transformed quantities Sn(k,u)T and /
%95(k,u). Note the
similarity of the right-hand side of Eq. (36) to Eq. (27). The susceptibility s(k,u) can then beextracted from the defining Eq. (34). Its complete expression is rather lengthy, but we will only needits imaginary part,
Im s(k,u)"!(1/2+X)S(k,u)[1!e~b+u] . (37)
The detailed balance result of Eq. (32) was used to obtain this relation, which is one version of theso-called Fluctuation—Dissipation Theorem [94].
Now that we have obtained S(k, u) in terms of s(k,u) we can proceed to relate S(k,u) to thedielectric response function e(k,u). In this form we will (finally) be prepared to explore the screeningproperties of the plasma. The dielectric response function is defined via the relation
e(k,u)/505
(k,u)"/%95
(k,u) , (38)
where /505
(k,u)"/*/$
(k,u)#/%95
(k,u) is the total potential resulting from the external perturba-tion. These two relations can be combined to give
1e(k,u)
"1#/
*/$(k,u)
/%95
(k, u)"1#A
4pe2k2 Bs(k,u) , (39)
where the (Fourier transform of the) Poisson equation, +2/*/$
"4pen*/$
, and Eq. (34) have beenused. We thus obtain the useful relation
3 In this discussion we are treating e as a dimensionless coupling parameter and not as the fundamental unit of charge.
which can be combined with Eq. (37) to yield the key result,
S(k,u)"!
+k2X2pe2(1!e~b+u)
Im1
e(k,u). (41)
This form for S(k,u) is exact within the context of linear response theory.
2.3. Screening models
In a dense plasma, polarization produces an effective interaction between particles. Variousapproaches exist for obtaining these effective interactions, and in this section we will briefly discussfour of the more common approximations. Three of these approximations will be explorednumerically in Sections 6 and 7, in the context of the atomic transition problem, and will be relatedto the various approaches discuss in Section 1. Many of the experiments discussed in Section 1 arecharacterized by long periods during which the plasma can be described by classical statistics.Therefore, the +P0 limit will be assumed for the remainder of this paper, in which case we may use
S(k,u)"!(Xk2/2pe2bu)Im[1/e(k,u)] . (42)
The utility of this formula is that the dielectric function is frequently easier to compute than theDSF, as well as lending itself to intuitive descriptions.
2.3.1. No screeningThe screening properties of S(k,u) are more easily identified if we express e(k,u) in terms of its
real and imaginary parts, so in the classical limit we put
S(k,u)"Xk2Im e(k, u)
2pe2buDe(k,u)D2. (43)
It is instructive to relate this general result to that of an ideal gas. In a nearly ideal gas the inducedpotential /
*/$(k, u) is vanishingly small due to the weak interactions. The ideal gas result can be
found be taking the limit3 eP0 in Eq. (43) and noting the limits Re e(k,u)P1#O(e2) andIm e(k,u)PO(e2). It follows that we can write the ideal gas limit of Eq. (43) as
S0(k,u),(Xk2/2pe2bu)Im e(k,u) . (44)
Since this result corresponds to a classical non-interacting system, its use is equivalent to many ofthe traditional electron—ion collision treatments in which each plasma electron scatters indepen-dently. These are the approaches reviewed by Younger [120] and Bottcher [65] and discussed herein Section 5.
2.3.2. Static screeningIt is common practice in plasma physics to assume that a charge’s effective potential /
4 It is important to clarify two possible interpretations of having no time dependence. It is easy to show via Fouriertransform theory that the static dielectric function is associated with a time-averaged quantity. Thus, in the presentcontext, time independent refers to a quantity which has previously been time averaged. In contrast, we could considera time-independent system in which all particles remain fixed in some configuration. This situation is described bya frequency integral of the dielectric function.
where m(r) is a screening function. This type of screening can be thought of in terms of the dielectrictheory as follows: a bare charge q produces the “external” potential q/r and gives rise to a “total”potential /
505(r). The screening function m(r) is obviously related to the dielectric response function
via Eq. (39). Since the screening function has no time dependence4 only the static part, e(k, 0), entersthis picture. The familiar Debye theory is an example of this type of screening. We may use the idealgas result, Eq. (44), to write a static screening approximation for S(k,u) as
S45(k,u)"
S0
(k,u)De(k, 0)D2
. (46)
As will become apparent in Section 5, this prescription for the density fluctuations is related to theproblem of electron—ion scattering within a static screening theory. Thus, this is the screeningmodel actually used by Hatton et al. [39], Diaz-Valdez and Gutierrez [13,33], and Jung [52,53].
2.3.3. Dynamic screeningIn general, Coulomb interactions among moving charges involve time dependent screening
functions. If we write our original result as
S(k,u)"S0(k, u)/De(k,u)D2 , (47)
it is clear that Eq. (43) is the time-dependent generalization of the static screening case, Eq. (46).Only when interactions are negligible or time scales are very long are the approximations ofEq. (44) or Eq. (46) appropriate; the screening of particles that can cause ionization usually requiresa dynamical description. This issue will be carefully explored in Section 3.
2.3.4. Dynamic screening: Plasma oscillations onlyIt is a property of (unmagnetized) one component plasmas that a single collective mode can arise.
This collective mode is associated with plasma oscillations. Since collective modes correspond toresonances in a many-body system, it is possible that plasma oscillations play an important role indynamic screening.
Collective modes are determined by the condition
e(k,u)"0 , (48)
which defines a dispersion relationship of the form u"u(k). In principle, one should consider boththe real and imaginary parts of e(k, u) separately. However, the imaginary part for an actual systemcannot be exactly zero, so only the real part need be used to find the collective mode.
To single out the collective mode in S(k,u) we expand Re e(k,u) in a Taylor series,
The first term is zero by the definition of u(k). This expansion can be substituted into Eq. (47) toobtain, to lowest order,
S#0--
(k, u)"S0
(k, u)
CLRe e(k,u)
Lu(k)(u!u(k))D
2# [Im e(k,u)]2
. (50)
Note that this has a Lorentzian form with width (FWHM),
2ck"
2 Im e(k,u)LRe e(k, u)/Lu(k)
, (51)
where ckis the decay rate associated with the oscillation [43]. In the limit that the imaginary term is
very small, this reduces to
S#0--
(k, u)"k2
2e2bu ALRe e(k, u)
Lu B~1
d(u!u(k)) . (52)
In a hot dense plasma the plasma oscillation will dominate the density fluctuations and thisapproximate expression is useful. It is the quantized version (plasmon) that has been used byGirardeau and Gutierrez [27] to study electron—ion recombination.
2.4. Vlasov plasmas with local field corrections
We now turn to the task of obtaining an explicit formula for the dielectric response function.Here we treat the plasma within classical kinetic theory as a one component system. Generaliz-ations to two-component plasmas [11] and to degenerate systems [44] can be found elsewhere.Strong coupling is treated here via local field corrections.
Recall from Eq. (38) that the response function e(k,u) relates an external potential applied to theplasma to the total potential within the plasma, the connection being
/505
(k, u)"/%95
(k,u)#/*/$
(k, u) , (53)
where /*/$
(k,u) is the potential induced in the plasma. The induced potential is related to theinduced number density (fluctuation) via the Poisson equation,
/*/$
(k, u)"!(4pe/k2)n*/$
(k,u) (54)
which yields
e(k,u)"1#(4pe/k2)n*/$
(k,u)//505
(k,u) . (55)
To simplify this expression we must obtain an equation of motion for the number densityfluctuations n
*/$(k,u).
Consider a one component electron plasma described by the phase space density functionF(r, �, t). This function represents the probability of finding an electron at point r with velocity � attime t. The normalization of F(r, �, t) is given by
5This form is often referred to as the STLS (Singwi—Tosi—Land—Sjolander) ansatz.
where n(r, t) now refers to the total electron density. In the absence of any external potential thetotal electron density is assumed to have a uniform, stationary value n
e. This corresponds to an
equilibrium phase-space density of the form F0(�). However, in the presence of a small external
potential, the phase-space density will be driven from its equilibrium value, and we can write
F(r, �, t)"F0(�)#F
1(r, �, t) (57)
where F1(r, �, t) is the fluctuation in the phase-space distribution function. The induced number
density can be written in terms of F1(r, �, t), which in turn gives
e(k,u)"1#4pek2
nePd3vF
1(k, �,u)//
505(k, u) (58)
for the dielectric response function.We now employ kinetic theory to find a suitable equation of motion for F
1(r, �, t). In general, the
phase-space density function F(r, �, t) satisfies the equation
ALLt#� )+r#
em
+r/%95(r, t) )+
�BF(r, �, t)#ne
m+
�)PF(Dr!r@D)F(r, �; r@, �@; t) d3r@d3v@"0 . (59)
Here F(r, �; r@, �@; t) is the two-particle joint probability function and F(Dr!r@D)"!+rUee(Dr!r@D) is
the electron—electron interparticle force. In fact, this is just the first in a set of equations that relateN particle density functions to N#1 particle density functions; altogether these equations areknown as the BBGKY hierarchy [88]. To proceed we must find an appropriate approximationthat truncates the hierarchy and leads to a kinetic equation for F(r, �, t) alone. A reasonable choicefor F(r, �; r@, �@; t), motivated by the exact form in thermal equilibrium5 [105], is
where g(Dr!r@D) is the equilibrium radial distribution function defined in Eq. (11). An integrodif-ferential equation for the single-particle phase-space density F(r, �, t) is obtained when Eq. (60) issubstituted into Eq. (59).
It is useful to define an auxiliary two-particle interaction W(Dr!r@D), according to
F(Dr!r@D)[g(Dr!r@D)!1]"!+rW(Dr!r@D) . (61)
From this formula we see that W(r) is a measure of the degree of spatial correlation, which isimportant in strongly coupled systems. One may obtain g(r) from some separate theory [106,90,37]or, more appropriately, by a self-consistent calculation with the dielectric response function[105]. The self-consistency condition can be seen by noting the relationships between Eqs. (18)and (19), and (41); that is, the g(r) in Eq. (60) should be consistent with the final result for e(k, u)in Eq. (58).
If we then define the induced potential and induced charge density as
/*/$
(r, t)"!
nee PUee
(Dr!r@D)F1(r@, �@, t) d3r@d3v@ , (62)
n*/$
(r, t)"nePF1
(r, �, t) d3v , (63)
we obtain the linearized kinetic equation
ALLt#� )+rBF
1(r, �, t)#
em
+r(/%95(r, t)#/
*/$(r,t)) )+
�F
0(�)
"
1m
+�F
0(�) )+rPW(r!r@)n
*/$(r@, t) d3r@ . (64)
This equation may be solved by Fourier transform techniques to yield
F1(k, �, u)"
1m
k )+�
F0(�)
k ) �!u[!e/
505(k, u)#W(k)n
*/$(k,u)] , (65)
where Eq. (53) has been used. The Fourier transform of the induced density can be gotten byintegrating this equation over velocities as in Eq. (56). Notice that this equation already contains aninduced density term as a result of the local field correction W(k)/(4pe2/k2) [105].
At this point it is straightforward to combine the induced density terms and form the ration*/$
(k,u)//505
(k, u) which can then be substituted into Eq. (55) to obtain the dielectric responsefunction,
e(k,u)"1!4pe2k2
s(k,u)1!W(k)s(k,u)
, (66)
in terms of the susceptibility,
s(k,u)"ne
m Pd3vk )+
�F0(�)
k ) �!u. (67)
If we neglect the local field corrections, we obtain the standard Vlasov dielectric response function,
e0(k,u)"1!
4pnee2
mk2k )P
+�F
0(�)
k ) �!ud3v . (68)
Eq. (68) can be used for obtaining the DSF for ideal (i.e., weakly coupled) plasmas, and in the staticlimit yields the Debye—Huckel screening model. Numerical evaluation of Eq. (68) is discussed inAppendix B.
3. Static screened Coulomb potentials
In the previous section we considered dynamic screening, which is appropriate for treating thescreening of the projectile(s). We now turn to the static screening treatment appropriate for target
ions. Since static screening is often used for both the projectile and the target, we begin this sectionby analyzing the conditions for which the static approximation can also be made for the projectile.Then, in those cases in which the static approximation is valid, the results of this section will applyequally well to the projectile as well as the target without significant modification.
Consider a (classical) test particle of charge q traveling straight through a plasma. For theionization problem we might imagine that this particle impacts a particular target ion with velocity� and impact parameter b. The charge density of this particle can be written as
o%95
(r, t)"qd(r!�t!b) (69)
where the time origin is taken to be at the time of closest approach to the target. Alone, this particleproduces a potential /
%95(r, t) which will polarize the surrounding plasma and yield the total
potential /505
(r, t) of the charge q and its screening cloud. Of course, these potentials are related byEq. (38). If the plasma is weakly coupled we may write the complete dielectric response function interms of the responses of individual species as
[e(k,u)!1]"[ee(k,u)!1]#[e
i(k,u)!1] . (70)
A two-component plasma has been assumed here, but the generalization to many species isobvious. The reason we do not limit this analysis to just the electrons (as in the earlier discussioninvolving S(k,u)) is that atomic transitions are driven by plasma density fluctuations and ionicmotions almost always are too slow to cause ionizations. However, the ionic contribution to testparticle screening may not be ignorable.
The Poisson equation can be used to obtain /%95
(r, t) from Eq. (69) and then, with Eqs. (38) and(70), the total potential can be written as
/505
(k, u)"A4pqk2
e~*k >bBC1
ee(k, k ) �)#e
i(k, k ) �) ! 1D d(u!k ) �) . (71)
The first factor is the (Fourier transformed) potential arising from the bare charge q with impactparameter b and the second factor incorporates the screening from both the ions and the electrons.From this expression it is clear that the potential /
505(r, t) which is experienced at the target depends
sensitively on the velocity of the incident particle. We now discuss some limiting cases of Eq. (71) tofind regimes where static screened Coulomb potentials (SSCP) are valid.
By noting that a species dielectric response function ea(k,u) is close to unity when the angularfrequency u exceeds that species plasma frequency, ua"J4pna(zae)2/ma, various simplificationscan be found. This is a consequence of the fact that the plasma particles a cannot respond effectivelyon time scales shorter than 1/ua. We may therefore define three regimes whereby k ) �: (1) exceedsthe plasma frequencies of both the electrons and ions, (2) is between the electron and ion plasmafrequencies, and (3) is below both plasma frequencies. This is summarized in Table 4. It is importantto recognize that these are strong inequalities. In general, of course, particle velocities in the plasmaare distributed according to velocity distributions F(�) and none of these regimes will be applicableto the entire distribution of any species a. We must therefore be cautious in our choice for thescreening function. We might expect, however, that in some situations a particular part of Fa(�) willdominate and we can safely assume, for that species, just one regime applies to the entire problem.This is the case, for example, when one considers the ionization of a very deeply bound state. Such
Table 4The three time scale regimes of an electron-ion plasma, for a fixed k. In each regime some static potential can be defined
Regime Time scale Description Screening particles
1 k ) �AueAu
iFast particle None
2 ueAk ) �Au
iIntermediate velocity particle Electrons only
3 ueAu
iAk ) � Slow particle Electrons and ions
a process requires impact by a very fast particle and we may conclude that little electron or ionscreening occurs. If, however, for some reason small k values are important (note the 1/k2 factorarising from the Coulomb potential in Eq. (71) this argument can fail; this will be illustrated inmore detail later.
In the remainder of this section we will assume that some SSCP is appropriate for the target ion,i.e. that for it e(k,u) is well approximated by e(k, 0). Unfortunately, there is no single SSCP thatadequately describes every plasma environment. In a dense plasma the free-electron screeningcloud which surrounds the target may be partially degenerate and/or the ions may be stronglycoupled. Derivations of some of the more common plasma potentials are given below to indicatetheir range of validity as the plasma temperature and density vary. Then, we present a hybrid, staticpotential which has applicability over a wide range of the temperature, density parameter space. Thisresult is especially useful in modeling situations where the plasma evolves rapidly through differentregimes. Additional discussion of and references to literature on SSCPs can be found in the reviewdocument by Fujima [25] and the recent articles by Gutierrez et al. [35] and Chabrier [125].
3.1. Classical, multicomponent case
In a very hot plasma the degeneracy parameter ¶eis much less than unity and screening can be
treated within the context of classical statistical mechanics. Furthermore, if the interaction isrelatively weak, the equations describing the potential can be linearized. This leads to the well-known“Debye—Huckel” result, which enjoys perhaps the widest use of any of the plasma potentials.
We begin with the Poisson equation for the potential near a particular impurity ion of charge z.The total potential /, which is produced by the impurity ion, plasma electrons, and other plasmaions, satisifes a Poisson equation of the form
+2/(r)"!4p(!ene(r)#zN en
i(r)#zed(r)) . (72)
Here, the other ions all have been assumed to have the average charge zN , and it is important toremember that this is to be interpreted as an equation for statistically averaged values of thequantities present, in the sense that n(r),Sn(r, t)T. All quantities are therefore spherically symmetricas well as time-independent. The electron number density at a distance r from the target is given by
The denominator of Eq. (73) is the configuration integral, and the potential energy U is a sum overall pairwise interactions, viz.
U" +e:e{
Uee{
# +i:i{
Uii{#+
ei
Uei#+
e
Uez#+
i
Uiz
. (74)
Eq. (73) can also be written as [37]
ne(r)"n
egez(r) , (75)
where gez(r) is the electron-target radial distribution function. This radial distribution function is
similar to the one defined in Eq. (11) except that Eq. (73) measures the likelihood that there areelectrons with separation r from a target ion rather than the likelihood that the electrons have someseparation r from each other. A similar function g
iz(r) describes the ion positions,
ni(r)"n
igiz(r) . (76)
Because it is not easy to evaluate the radial distribution functions exactly, a mean field theorytypically is used to simplify the two-particle terms in the total interaction expression of Eq. (74).This is done by noting that the potential /(r) in Eq. (72) is the total potential from the electrons, theions, and the target. Therefore, if we make the replacement
+e:e{
Uee{
#+ei
Uei#+
e
UezP!e+
e
/(re!r
z) , (77)
and invoke a similar relation for the ions, we obtain the mean field results
ne(r)"n
eebee((r) , n
i(r)"n
ie~bizN e((r) . (78)
(Note that the possibility of a separate ion temperature has been allowed for.) These densityexpressions may now be substituted into Eq. (72) to obtain
+2/(r)"!4pe(!neebee((r)#zN n
ie~bizN e((r)#zd(r)) , (79)
which is known as the Poisson—Boltzmann equation.
3.1.1. Weak couplingSimple analytic solutions can be obtained by linearizing the exponentials, viz.
e~bU(r)+1!bU(r) , (80)
which is valid under the condition of weak coupling (cf. Eq. (1)),
DbU(r)D@1 . (81)
At large r-values the interaction is small and this condition can be satisfied even for lowtemperatures. But, due to the point charge z at r"0, this condition can never be satisfied near theorigin. In any event, wherever these conditions have been satisfied, we obtain the equation
Fig. 3. A comparison of various interactions, in which the static screening factor m(r) is shown for several modelsdiscussed in the text. The electron density and temperature are n
e"1024 cm~3 and ¹
e"15 eV. Only the electron
component of the plasma has been included in computing the interactions.
where
i2D,4pe2(b
ene#b
izN 2n
i) (83)
is the square of the Debye wave vector. In this form Eq. (82) is the modified Helmholtz—Greenfunction equation [1], which is easily solved to obtain the interaction potential
/D(r)"
zer
e~iDr"zer
mD(r) . (84)
The inverse of the Debye wave vector, jD,i~1
D, is the (Debye) screening length of the plasma.
The Debye potential is compared to the bare Coulomb potential of He` in Fig. 3, for an electronplasma with a density of n
e"1024 cm~3 and a temperature of ¹
e"15 eV. The plotted screening
function mD(r) represents the deviation from the bare Coulomb interaction. In this plasma environ-
ment screening is very strong at distances as small as a Bohr radius and significant modifications tomost bound states can be expected.
3.1.2. Partial electron degeneracyFor a plasma with degenerate electrons, ¶
e"b
e¹
F'1, we can compute the useful ratio n
e(r)/n
eby first recalling that the density of non-interacting electrons is given by the standard formula [42]
e/m is the electron thermal wavelength, l is related to the chemical potential
k by l"bek, and f
3@2is the Fermi—Dirac integral. In the spirit of the finite temperature
Thomas—Fermi model [66,18] we take the actual density near an ion to be [101]
ne(r)/n
e"f
3@2(l#eb
e/(r))/f
3@2(l) . (86)
Thus, the mean field / gives rise to an r-dependent energy shift that entices electrons to move toregions where their total energy is lessened. This expression is a quantum mechanical generaliz-ation of Eq. (78) which, for weak interactions, may be expanded about /(r)"0 to give
ne(r)/n
e"1#eb
e/(r) f @
3@2(l)/f
3@2(l) . (87)
This result, which is valid for any degree of degeneracy in a weakly coupled plasma, can becompared with Eqs. (78) and (80) to arrive at a degeneracy corrected Debye length. (Such anapproach has been used by Rose [98] to compute radiative opacities.) For ¶
e&(j3
ene)2@3@1, which
is the classical limit, we have
f3@2
(l)+el (88)
and Eq. (87) reduces to the weak-coupling result of the previous section. But, in the degeneratelimit ¶
eA1 we have
f3@2
(l)+(4/3p)l3@2 (89)
and the density ratio becomes
ne(r)/n
e"1#e/(r)/¹
F. (90)
It follows directly that in this situation we get an interaction whose screening is functionally similarto the Debye—Huckel expression, namely
/(r)"ze e~iTFr/r , (91)
but where the Debye wave vector is replaced by the Thomas—Fermi wave vector iTF
"
J4pe2ne/¹
F[72].
It is not always convenient to compute the necessary functions f3@2
(l) and f @3@2
(l) for an arbitrarydegree of degeneracy. Fortunately, one can define an approximate potential,
/DD
(r)"ze e~iDDr/r , (92)
with
i2DD
"
4pnee2
J¹2e# ¹2
F
(93)
that is trivial to calculate and agrees with results obtained using Eq. (87) to within 5% [7]. (Here,the subscript DD refers to a Debye-like interaction corrected for degeneracy effects.) This approx-imate potential also is shown in Fig. 3, where its screening function m
DD(r)"exp(!i
DDr) is plotted.
The plasma conditions are the same as previously discussed for the classical Debye interaction. It isobvious that the degeneracy correction is significant for these plasma conditions, with the classicalDebye theory overestimating the screening.
6Only strongly coupled ions are considered here since the higher charge of the ions gives rise to a higher Coulombcoupling parameter.
3.1.3. Strongly coupled ionsThe results obtained so far all depend on the assumption of weak coupling, which permits
a linearization of the Poisson equation. If any of the plasma species are strongly coupled thisprocedure is inapplicable and a different approach is necessary.
In a system of strongly coupled ions6 potentials may be computed near the target ion by puttingthe ions in a lattice. The lattice is divided up into cells, much like a solid state (Wigner—Seitz)approach, and the electrons are divided between the cells to give overall charge neutrality to eachcell. In this picture we need to find the electron density in the small cell which surrounds the targetion. That is, in the case of strongly coupled ions we usually consider the situation where theelectrons also interact strongly with the ions.
If we extrapolate Eq. (78) inward toward the target ion we find that the electron density becomesarbitrarily large. This information can be used to estimate how the Fermi energy changes asa function of r. For slowly varying potentials Eq. (78) suggests that the effective Fermi energy maybe estimated as
EF(r)"3.6]10~15[n
eebee((r)]2@3 eV . (94)
Although this extrapolation is not precise, it does indicate that near the ion the effective Fermitemperature ¹
F(r)"2E
F(r)/3 is much greater than e/(r), whence Eq. (90) predicts a nearly uniform
electron density. This model, a cell filled with a uniform electron distribution, is described by thePoisson equation
+2/IS(r)"4pen
e!4pzed(r) (95)
and is known as the ion—sphere model. The radius r4of the cell is determined by the constraint of
charge neutrality,
(4p/3)r34ne"z . (96)
For the reasonable boundary condition of zero potential energy at the ion—sphere radius r4the
solution is
/IS(r)"
zer C1!
r2r
4A3!
r2r24BD"
zer
mIS(r) (97)
for r(r4. Fig. 3 also includes a curve showing the ion—sphere screening function m
IS(r) for the same
He` plasma conditions.
3.2. A hybrid potential
In the previous sections three different potentials have been explored. Two are appropriate forweak coupling and the other, for strong ion coupling. For a particular atomic transition, occurringperhaps under a variety of plasma conditions, it is not clear which potential is “better”. For
bound—bound transitions it is likely that the ion—sphere potential is preferable: bound states arehighly localized and therefore experience the plasma potential when E
F(r) is large. Similarly,
a Debye—Huckel potential, which extends far into the plasma, is probably preferable for free—freetransitions because the states are highly delocalized. These simple arguments suggest that there isno obvious choice when treating bound—free transitions (or bound—bound transitions involvinga Rydberg state). Therefore, in this section we derive yet another screened interaction, one whichsimultaneously has appropriate properties for both bound and free states.
The approach is motivated by the applicability of the ion-sphere picture at small r and theapplicability of the Debye—Huckel picture at large r. Perhaps the simplest way to construct such aninteraction is to match an interior, r(r@, ion—sphere form to an exterior, r'r@, Debye—Huckel format some point r@ (to be determined). In general, r@ will depend on the properties of the surroundingplasma. The split proposed here is qualitatively similar to Region A and Region B of the potentialused by Stewart and Pyatt to find energy level shifts [108], but differs from it in that degeneracyeffects are included. In this section the potential energy U, rather than the potential /, will beconsidered.
The determination of this spherically symmetric hybrid (H) interaction UH(r) for an electron near
a test ion z begins, once again, with the Poisson equation,
1r
d2
dr2[rU
H(r)]"4pe2C+
i
zini(r)!n
e(r)#zd(r)D . (98)
The ion species have various charges ziand densities n
i(r). Near the test ion, r(r@, the electrons will
have their highest concentration and the ions their lowest. This condition is similar to thatexpressed in the ion—sphere model in which n
e(r)+n
eand n
i(r)/n
e+0. Using these approximations
for the densities in Eq. (98) the interaction energy takes the form
U:(r)"c
0/r#c
1!(ze2/2r3
4)r2 . (99)
This is similar to UIS, and as rP0 the interaction is dominated by the point charge z, which gives
c0"!ze2 again. But, the coefficient c
1must be determined by matching to a boundary condition
that differs from the ion—sphere model.Distant from the test ion, r'r@, the weak-coupling approximation is valid and, in the spirit of
the Debye—Huckel approximation, we take the ion densities to be ni[1#b
iziU
;(r)] and the
electron density to be given by Eq. (87). This yields a Poisson equation of the form
This wave vector is a generalization of Eqs. (83) and (93) to include partial electron degeneracy andionic species with differing temperatures. The solution
U;(r)"(c
3/r)e~ir (103)
differs from the usual Debye—Huckel result in that c3
is not determined by a charge z at the origin(the usual boundary condition), but rather by matching to the interior solution, Eq. (99), at r"r@.
To find the three unknowns c1,c3, and r@ we match Eqs. (99) and (103), their derivatives, their
second derivatives, and require r@'0. This yields
c1"
3ze22 i2 r3
4
[[(ir4)3#1]2@3!1] , (104)
r@"1i
[[(ir4)3#1]1@3!1] , (105)
c3"!
3ze2i2 r3
4
r@eir{ . (106)
The full hybrid interaction, using step functions h(x), can thus be written as
UH(r)"U
:(r)h(r@!r)#U
;(r)h(r!r@) ; (107)
this interaction has the correct behavior at small and large r and can safely be used to study bothbound and free states.
It is interesting to compare Eq. (107) with the usual Debye—Huckel interaction for r'r@. In thisregime the interaction can be written as
U;(r)"(!z
%&&e2/r)e~i(r~r{) , (108)
where the effective charge z%&&
, given by
z%&&"z C3
[(ir4)3 # 1]1@3!1
(ir4)3 D (109)
is temperature and density dependent. At moderate to high densities, (ir4)'1 and z
%&&is less than z,
owing to the screening in the region r(r@. This leads to the physical picture of a charge z%&&
ionscreened by a weakly coupled plasma. The screening plasma is guaranteed to be weakly coupledbecause the strongly coupled portion of the plasma is automatically incorporated into the regionr(r@, and therefore into the definition of z
%&&. The shift in the exponential arises from the fact that
the z%&&
“ion” has effective size [email protected], it is r@ which determines the admixture of U
:(r) and U
;(r) in U(r). For very high
temperatures, viz. small i,r@+(ir4)3/3i is very small and almost everywhere U
H(r)"U
;(r). Also, in
this limit the effective charge of Eq. (109) reduces to the bare charge z and we recover theDebye—Huckel expression. Conversely, when the temperature is low, corresponding to large valuesof i,r@+r
4and c
1is essentially independent of i. The first condition indicates that the ion-sphere
form of U:(r) extends to r
4, and the second condition indicates that U
:(r) is independent of the
surrounding ions. Thus, the ion—sphere interaction is recovered in this limit. Another interesting
limit is that of high density and high temperature. In this case UH(r)"U
;(r) results most often
because of the high temperature. But, we can still accomodate ¹F'¹
eas well, which indicates that
this hybrid interaction seems applicable to the study of atomic transitions under wide variety ofplasma conditions.
3.3. Energy level shifts
Screened hydrogenic states can be found via the Schrodinger equation,
!(+2/2m)+2t(r)#U(r)t(r)"Et(r) , (110)
once a particular screened interaction U(r) has been chosen. In general this equation must be solvednumerically for bound and continuum states. But, if the state under consideration is deeply bound,a simple approximation can be made to obtain eigenvalue information. Deeply bound statespredominantly experience their binding interaction at small r, and U can be expanded about thislimit. For each of the interactions discussed so far this limit yields
UDD
(r)P!
ze2r#ze2i
DD, U
IS(r)P!
ze2r#
3ze22r
4
, UH(r)P!
ze2r#c
1, (111)
where iDD
is defined in Eq. (93), r4is defined in Eq. (96), and c
1is defined in Eq. (104). In each case,
the resulting interaction is the bare Coulomb interaction plus a (positive) constant shift *E. Thus,for tightly bound states Eq. (110) can be approximated as
!(+2/2m)+2t(r)!(ze2/r)t(r)"(E!*E)t(r) (112)
which indicates that the hydrogenic wave functions are unchanged but the states have new energyeigenvalues
E"*E!(ze2)2m/2a2+2 . (113)
Note that these uniform level shifts predict smaller ionization energies but do not predict line shifts.Since no large line shifts have been observed experimentally, we do not consider correctionsbeyond the uniform level shifts [46].
In the weak-coupling limit c1Pze2i
DD, and in the strong coupling limit c
1P3ze2/2r
4; thus, the
hybrid case provides a continuous extrapolation of energy level shifts between the two regimes justas it did for the interaction itself. In fact, because c
1is the energy level shift of Stewart and Pyatt
[108] generalized to include degeneracy corrections in i, the interaction UH(r) is consistent with
their shift. For reference, Appendix Cgives numerical formulae for several of the important SSCPresults obtained above.
The ionization potential of the ground a"1 state of He` is shown versus plasma density inFig. 4 for various choices of the energy level shift. The classical Debye potential predicts energylevel shifts at high plasma density which nearly eliminate the state altogether. The degeneracy-corrected Debye potential predicts a smaller shift, indicating a breakdown of the classical Debyepicture. The ion—sphere potential, which is the extreme degeneracy limit, predicts an even smallershift. The hybrid—potential level shift is seen to extrapolate between the Debye and ion—sphereshifts and it predicts the smallest shift at high density. Note the fairly strong disagreement at highdensity. The same information is shown in Fig. 5 for the a"3 state. (Note the different density
Fig. 4. The energy of the ground (a"1) state of He` versus plasma density for various screening models. The classicalDebye screening model (D) predicts the largest changes with the state almost eliminated at n
e"1024 cm~3. Note that the
hybrid model (H) agrees with the Debye model at low densities whereas it agrees with the ion—sphere model (IS) at higherdensities, as expected.
range.) In all models this state is eliminated near ne"1022 cm~3. Because of this, the state does not
exist at densities which require degeneracy corrections, so the degeneracy corrected Debye shiftdoes not differ much from the classical Debye shift. The hybrid shift more closely approximates theDebye shift in this density region for the same reason.
3.4. Total elastic scattering cross section
It is instructive to compute the elastic scattering cross section for a screened potential. Thisserves both to calibrate the potential with a familiar quantity and to aid in interpreting futurecalculations involving scattering states of this potential. The total elastic cross section can becomputed in terms of the scattering phase shifts d
l(k) as [51]
p505
(k)"4pk2
+l
(2l#1)sin2(dl(k))"+
l
pl(k) , (114)
where +k is the momentum of relative motion. The partial-wave cross sections pl(k) which have
been defined in the second expression are useful for, e.g., quantum transport calculations [122,60].We focus here on the hybrid potential. Partial cross sections are shown in Fig. 6 for the hybrid
potential corresponding to an Ar`17 impurity in a hydrogen plasma typical of ICF experiments.The most notable feature is the very large partial cross section near k"1 for p
3(k). This is
indicative of a low-energy shape resonance which is common to short-ranged potentials. The total
Fig. 5. The energy of the a"3 state of He` versus plasma density for various screening models. In all screening models,continuum lowering eliminates this state near n
e"1022 cm~3. At these lower densities the degeneracy corrected Debye
model (DD) does not deviate appreciably from the classical Debye model (D) and only the classical result is shown.
7Of course, fractional principal quantum numbers do not exist and the next highest integer is implied in each of theexpressions in Eqs. (115) and (116).
cross section from Eq. (114) is shown in Fig. 7 where it is compared with a simpler Born result fora similar Debye potential.
3.5. Number of bound states
It is often useful to assume that the constant energy level shifts of Eq. (179) apply to all boundstates. If this were so, all states within *E of the continuum would be moved into the continuum,leaving a finite number of bound states. In this picture, the uppermost state would have principalquantum number a
.!9given by7
z2/2a2.!9
"*E , (115)
where *E is any of the shifts of Eq. (111) in atomic units. The maximum principal quantumnumbers for the three models are given by
Fig. 6. The partial cross sections pl(k) for the first few l-values. The zeros of the partial-wave cross sections arise from the
presence of the bound states. The large feature in p3(k) near k"1 is a low-energy shape resonance.
Fig. 7. The total elastic scattering cross section p505
(k) associated with the Hybrid potential for the same plasmaconditions as given in Fig. 6. Also shown as a dashed line is the first Born result for a Debye screened potential under thesame conditions.
2must be made for a zero energy s-wave bound state.
Fig. 8. Total number of bound states (neglecting spin degeneracy) of various SSCPs for He` ions in a plasma oftemperature ¹"15 eV. At very high densities virtually all bound states are eliminated, and there is good agreementbetween models; elsewhere, the ion—sphere model predicts far fewer states.
where all quantities in Eq. (116) are in atomic units. The total number of hydrogenic bound states,within this picture, is subsequently given by
N505"
a.!9
+a/1
a2"a.!9
(a.!9
#1)(2a.!9
#1)/6 . (117)
Once the phase shifts of the scattering states are found, we may obtain information regarding thenumber of bound states N
lof a given angular momentum. This is a consequence of Levinson’s
Theorem, which can be simply stated as [51]
Nl"d
l(0)/p . (118)
This theorem is exact for the types of potentials we are considering here8 and may be comparedwith the results predicted by Eq. (117), which represent a relatively crude approximation. Thenumber of bound states for each model discussed is shown in Fig. 8 as a function of plasma density.
For a non-degenerate Debye case, Eq. (116) can be compared with the numerical results ofRogers et al. [96]. For a system in which z/i
D"10, Eq. (116) predicts aDD
.!9"2 whereas the
numerical eigenvalue computations show that the 3s and 3p states are still bound, albeit weakly.Furthermore, the numerical results show that all tightly bound states are approximately shiftedequally. For this case (z/i
D"10), Rogers et al. show that the weakly bound 3s and 3p energies
Having discussed several SSCP for the target ion, we turn to computing oscillator strengths foratomic transitions in such potentials. We focus here on bound—free transitions, in which the energylevel shifts play a larger role than they do for bound—bound transitions. The oscillator strengths arefirst computed in two analytic approximations which incorporate bound state energy level shifts.Then, a partial wave method is described in which the continuum wave function is obtained froma particular SSCP, resulting in an oscillator strength with a more complicated temperature—densitydependence.
4.1. Definitions
The oscillator strength between two discrete levels a and b is defined as [5]
fba"(2m/+2)E
baDSbDxDaTD2 , (119)
where Eba"E
b!E
a. This quantity, which is positive for upward transitions, can be interpreted as
the effective number of classical oscillators participating in the transition [50]. This idea can beextended to give the generalized oscillator strength (GOS),
fba
(k)"(2m/+2)(Eba
/k2)DSbDe*k > rDaTD2 , (120)
which is frequently used in the theory of inelastic collisions [48]. (Here +k is the momentumtransferred in the collision.) Since the states b and a are assumed to be orthogonal, it is easy to seethat Eq. (120) reduces to Eq. (119) in the kP0 limit.
When a transition occurs from a bound to a continuum state the basic definition of the GOS ismodified slightly. This is due to the fact that, for continuum states, we must specify the probabilityof the particle being in range dg of some set of observables g that we are free to choose. That is, wemust specify the oscillator strength for the transition aPg as
The quantity dfa/dg is referred to as the generalized oscillator strength density (GOSD) and the
different choices of g are referred to as “g-scale normalization” [48]. The quantity G(g) is thenumber of continuum states per interval dg. The source of this flexibility can be traced to theorthogonality and completeness relations for continuum states. In general, these conditions can beexpressed as [51]
SgDg@T"F(g)d(g!g@)
PdgG(g)DgTSgD"1 . (122)
In the first expression we are free to choose F(g) and the second expression provides the constraintG(g)"1/F(g). Typically, the set of observables g is taken to be the energy and either the linear orangular momentum. As a specific example, consider the observables to be the energy E of theparticle and the direction of its linear momentum, which points into the solid angle dO. If we
simply wish to integrate our final results over these quantities we would choose the correspondingG to be unity, viz.
PdEdOSDE, OTSE, OD"1 , (123)
which fixes the normalization condition to be
SE, ODE@, O@T"d(E!E@)d(O!O@) . (124)
This particular choice is referred to as “energy-scale normalization”.The information contained in the GOSD is frequently presented as a surface of df
a(k,u)/du
versus u and ln(k2), where +u"Ega is the energy transferred. Plotted in this manner, the surface isreferred to as the “Bethe surface”. The Bethe surface captures all information about an inelasticscattering process, insofar as the collision can be described within the (first) Born approximation.
4.2. Plane-wave model
There is one GOSD which is very simple to use and can be expressed analytically. It isconstructed by treating the initial state of the target as that of an unperturbed hydrogenic systemand the final state as that of a free particle, viz.
SrD1sT"(2Z3@2/J4p)e~Zr
SrDKT"J(K/(2p)3)e*K > r . (125)
Here, the bound state has been taken to be a 1s state of an ion with nuclear charge Z, and thenormalization of the continuum state has been chosen to be on the energy scale with energyE"K2/2. (All quantities will be in atomic units, e"m"+"1, for the remainder of this section.)Thus, the GOSD can be written as
df1s
(k, u)/du"(2u/k2)DSKDe*k > rD1sTD2dO , (126)
where u"E!E1s
. Often we are only interested in the oscillator strength as a function of energyand not the details of the particle’s direction. This is the case when the perturbation causing thetransition is isotropic (on average), rendering the GOSD independent of the specific direction of K.We will assume this to be so and perform an integral over the solid angles in Eq. (126). It is alsobeneficial to write the result in terms of the ionization potential I
1"u!E of the 1s state. (Recall
from the discussion of Section 3.3 that the ionization potential can be shifted due to the presence ofsurrounding high-density plasma.) Together these manipulations give the plane wave GOSD
df1s
(k, u,I1)
du"
16Z5u3pk3
([Z2#(k!J2(u!I1))2]~3![Z2#(k#J2(u!I
1))2]~3) . (127)
Although this result has been obtained for a transition out of the 1s state, it is readily generalized toexcited states. An average over substates for a given level a yields Eq. (127) with the replacementZPZ/a [20,26]. The final form, valid for all principal quantum numbers a, is
Fig. 9. The Bethe surface for the PW GOSD computed for an initial 1s state of He`. The surface has been truncated ata value of one to show smaller details. (Apparent jagged features are an artifact of the shading routine.)
The Bethe surface for a GOSD involving a hydrogenic ground state and plane-wave (PW)continuum states is shown in Fig. 9. Note that there is a narrow ridge which extends out to largeenergy and momentum transfers. This “Bethe ridge” corresponds to classical-like collisions.A second domain, near the origin, is more sensitive to the electronic structure of the initial state,and classically is associated with scattering at large impact parameters.
4.3. Orthogonalized plane-wave model
In our general discussion of quantum mechanical transition rate formulae (Section 2) the initialand final states were assumed to be orthogonal. In the example above this assumption in fact wasviolated: the initial state was an eigenstate of a hydrogenic Hamiltonian and the final state was aneigenstate of a free-particle Hamiltonian. Physically this corresponds to a transition fromsome initial state to a final superposition state which contains the initial state. Such a situationoccurs frequently in studying rearrangment collisions and techniques for handling this issuehave been developed [79]. The method we use here is based on the orthogonalized plane-wave(OPW) approximation [40,10] which has been applied previously to recombination in denseplasmas [34].
The effect of the non-orthogonality of the initial and final states can be exposed by looking ata PW GOSD for small k. In this limit, we can write the GOSD of Eq. (126) as
df1s
(k, u)du
"(2u/k2)DSK D1#ik ) r#2D1sTD2dO . (129)
It is clear that the first term containing the nonzero matrix element SK D1sT will diverge as k~2. Thisbehavior is evident in Fig. 9 where the plotted surface has actually been clipped at a value of 1 toallow the Bethe ridge to be highlighted. This can be remedied simply by subtracting the projectionof the initial state with the final state to form an “improved” final state DK @T,
Fig. 10. The Bethe surface for the OPW GOSD computed for an initial 1s state of He`.
In principle, the final state wave function should be
DK @T"DKT!+blm
DblmTSblmDKT, (131)
where the sum runs over all bound states; this describes the situation in which the bound electronhas no amplitude to be in any bound state following the ionization process. We will not pursue thismore general form in the present paper since there would be a complicated plasma temperature anddensity dependence in the sum, and this would require a treatment of states lying near thecontinuum that is beyond the scope of the OPW [81]. (Our purpose here is to explore thenonorthogonality issue, not to generate a highly accurate GOSD.)
Using the improved state DK @T from Eq. (130) we find the angle-dependent GOSD to be
df1s
(k, u)dudO
"
16uKZ3p2k2C
1[1#(º/Z)2]4
!
32[1#(º/Z)2]2[4#(k/Z)2]2[1#(K/Z)2]2
#
256[1#(K/Z)2]4[4#(k/Z)2]4D. (132)
In this expression K"J2(u!I1) and º"Dk!K D. It is easy to verify that the quantity in square
brackets vanishes as k2 in the limit kP0. Subsequently, an integration over final emissiondirections yields an OPW GOSD with interpretation analogous to that of Eq. (127). The corre-sponding Bethe surface is shown in Fig. 10 for the same conditions as those pertaining to Fig. 9.The effect of removing the non-orthogonality between the initial and final states is quite dramatic:the Bethe ridge is more prominent and the small k divergence clearly has been eliminated.
4.4. Numerical partial-wave model
In the GOSD’s considered above plasma effects could be incorporated only through the (shifted)ionization potential. This allows construction of simple analytic forms for the GOSD that includestatic screening to lowest order. However, plasma effects on the initial and final state wavefunctions have not been included. We may expect that a tightly bound state is well approximated
by a hydrogenic wave function but not that a continuum state is well approximated by a planewave. For electrons which have been ejected from an ion a better choice would seem to have thecontinuum state being an eigenstate of a screened Coulomb potential such as discussed inSection 3. Here we will assume only that the potential has spherical symmetry, which allowsa partial wave analysis. In this representation it is convenient to choose a normalization based onthe energy and angular momentum observables. In choosing an energy-scale normalizationanalogous to Eq. (123) we obtain the conditions
SElmDE@l@m@T"d(E!E@)dll{dmm{
PdE+lm
DElmTSElmD"1 . (133)
There is some practical difficulty in ensuring that these conditions have been met for numericallygenerated wave functions. Let the solution be written as cR
El(r)½
lm(rL ) where R
El(r) is the result
found numerically and c is some constant we must choose to satisfy the normalization criterion. Inthis case, the normalization condition of Eq. (133) reads
DcD2dll{dmm{P
=
0
dr r2REl(r)R
E{l{(r)"d(E!E@)d
ll{dmm{
(134)
which not easy to solve for c. In practice, this problem is handled by normalizing the asymptoticform of the radial wave function, which is presumed to be known. For the short-ranged potentialsconsidered here the asymptotic form involves a spherical Bessel function and has the energy-scalenormalized behavior of J2K/pj
l(Kr). Correct normalization is thus ensured by requiring the exact
solution to have the asymptotic form
limr?=
REl(r)&J(2/pK)
sin(Kr!lp/2#dl(K))
r, (135)
where K"J2E. In what follows, it will be assumed that this procedure has been carried out.In the partial wave representation a sum may be performed over angular momentum substates
to obtain a GOSD pertaining to a transition between energy levels, independent of angularmomentum. This corresponds to the integration over solid angles dO, discussed earlier. Since wemay want to treat excited bound states we also average over initial substates. However, the angularmomentum quantum number l is important for selection rules and so we only average over angularmomentum projections m to obtain
dfa(k,u)du
"
2u/k2
2l#1=+
l{/0
l{+
m{/~l{
l+
m/~l
DSEl@m@De*k > rDalmTD2 . (136)
The GOSD of Eq. (136) represents the average strength of a transition from bound states withquantum numbers a and l to all continuum states with energy E.
Let the initial state DalmT be hydrogenic, SrDalmT"Ral(r)½
lm(rL ), with energy given by the shifted
hydrogenic value as in Eq. (111) and let the final state have the form SrDEl@m@T"REl{
(r)½l{m{
(rL ), withcontinuous energy eigenvalue E. [If I
ais the (shifted) ionization potential then u"I
a#E.] R
El{(r)
is the radial wave function of the chosen screened Coulomb potential. By writing the exponential
Fig. 11. Bethe surface for the Ar`17 Hybrid potential described in the text. Here, the plasma density is 1022 cm~3.
term in a spherical Bessel function expansion and then evaluating the integration over sphericalcoordinates in terms of 3!j symbols, we obtain the intermediate result
dfa(k,u)du
"
8puk2
+l{m{m
(2l@#1)K+LM
iL½*LM
(kK )J2¸#1
]P=
0
dr r2REl{
(r) jL(kr)R
al(r)A
l@ ¸ l
!m@ M mB Al@ ¸ l
0 0 0BK2
. (137)
Eq. (137) can be considerably simplified by writing out the square and performing the sums overm and m@. This eliminates the other M-sum and gives the compact form
dfa(k,u)du
"
2uk2
+l{
+L
(2l@#1)(2¸#1)C P=
0
dr r2REl{
(r)jL(kr)R
al(r)D
2
Al@ ¸ l
0 0 0B2. (138)
Generally this expression represents a fairly extensive computation, due to slow convergenceof the two infinite sums. However, when the initial state has l"0 we obtain a greatly simplifiedGOSD,
dfa(k,u)du
"
2uk2
+L
(2¸#1)CP=
0
dr r2REL
(r)jL(kr)R
a0(r)D
2. (139)
Figs. 11 and 12 show Bethe surfaces corresponding to transitions from the Ar`17 ground state ina Hybrid potential for two hydrogen plasma densities and a temperature of 1 keV. The Ar`17 ionwas chosen because its spectrum is commonly used in ICF plasma diagnostics [118], and becausean unperturbed (hydrogenic) initial state wave-function R
a0(r) simplifies the GOSD calculation.
The continuum wave functions REL
(r) were obtained by numericaly solving the Schrodingerequation in the Hybrid potential. As few as 20 ¸-values could be used for these GOSDs. It is clearfrom the figures that there can be a significant density dependence of the GOSD.
Fig. 12. Bethe surface for the Ar`17 Hybrid potential described in the text. Here, the plasma density is 3]1024 cm~3.
5. Ionization rates
Rates of atomic ionization will now be determined for the simplest case, single-electron (H-like)target ions of nuclear charge Z. Three methods for obtaining these transitions rates will bedescribed. Each derivation begins with a direct calculation of the rate, without recourse to a crosssection, but the derivations represent distinctly different physical pictures. In Section 5.1, which ispedagogical, transition rates are computed within the standard framework of independent plasmaelectrons inelastically scattering from the ion; this is shown to be equivalent to the traditionalbinary cross-sectional approach. Then, Section 5.2 provides a derivation wherein the ionictransition is driven by the time-dependent stochastic field of the surrounding plasma electrons. Thetime-independent picture given in Section 5.3, in which the ion ‘‘impacts” the plasma, turns out tobe equivalent to the stochastic approach of Section 5.2. The models presented in Section 5.2 andSection 5.3 are contrasted to that of Section 5.1, to emphasize those effects which arise from theconsideration of interacting plasma electrons.
We begin by recalling that time-independent scattering theory is based on a Hamiltonian of theform
HK "HKX#HK
Y#»K , (140)
where HKX
and HKY
are the Hamiltonians of the (possibly composite) subsystems undergoing thecollision and »K is the interaction between them. Since the full Hamiltonian is used, transitions areinduced by allowing the interaction »K to be adiabatically turned on, as »K ect, with the limit cP 0taken at the end of the calculation. If the interaction is sufficiently weak, a first-order transition ratecan be computed from the Golden Rule,
w"(2p/+)DSy@DSx@D»K DxTDyTD2 d(E@!E) . (141)
Eigenstates of HKX
and HKYhave been used to describe the asymptotic initial and final (primed) states
It is useful to start with the simple approach to atomic ionization rates presented in this section,because it connects the traditional (binary) cross-sectional picture to the stochastic model of thenext section. Here, all interactions among plasma electrons are neglected, and plasma ions areignored entirely. Thus, the three parts of the Hamiltonian of Eq. (140) are:
HK!50.
"
p2
2m!
Ze2r
,
HK1-!4.!
"+e
p2e
2m,
»K "+e
Uee
(r, re) . (142)
The perturbation »K is the sum of Coulomb interactions between the plasma electrons at re,
with momenta pe, and the bound atomic electron at r, with momentum p. Let the atomic electron
be in state DaT, and the N free electrons be in plane wave states DpeT, e" 1,2,N. Since only
»K contains two-particle interactions, the initial and final composite states may be written asproducts,
DtiT"DaTDp
1T2Dp
NT,
DtfT"Da@TDp@
1T2Dp@
NT. (143)
Here the Hartree factorization has been made (electron exchange is neglected).With the initial and final states defined by Eq. (143), the Golden Rule can be written as
wfi"
2p+ KSp@
ND2Sp@
1DSa@D+
e
Uee
(r,re)DaTDp
1T2Dp
NTK
2d(E
f!E
i) . (144)
It is possible to simplify this transition rate formula by expanding the two-particle interaction interms of its Fourier components Uk "4pe2/k2. This leads to the expression
wfi"
2pX2+ K +k UkSa@De*k >rDaT+
e
Sp@eDe~*k > r
eDpeT <e{Ee
Sp@e{Dp
eTK
2d(E
f!E
i) . (145)
It is a consequence of using first-order perturbation theory with this two-particle interaction thatonly two electrons simultaneously undergo transitions; and, since we are considering an atomictransition, only one plasma electron is involved.
Of course, we do not know the initial state of the plasma at the level of detail required to evaluatethe above rate, nor do we care in which final state the plasma is left. We do know, presumably, theplasma’s statistical properties and we can therefore compute the more relevant mean quantity forthe atomic states DaT and Da@T:
which involves a sum over final states and an average over initial plasma states. The F( pe)’s are the
probabilities associated with finding electron e in state DpeT, as given by, e.g., a Maxwellian
distribution. This averaging reduces Eq. (145) to the simpler form
wa{a
"
2pN+X2
+pe
F( pe)+
p{e
+k
DUkSa@De*k > rDaTD2DSp@eDe~*k > r
eDpeTD2 d(E
f!E
i) . (147)
From this formula it is clear that wa{a
is an average over the rate due to a single electron impactingthe ion.
The matrix element involving plane wave states reduces to a Kronecker delta function times theplasma volume X, so the sum over p@
ecan be immediately effected. This results in the yet simpler
expression
wa{a
"
2pN+X
+pe
F( pe)+
k
DUkD2DSa@De*k > rDaTD2 d(Ea{!E
a#( p
e!+k)2/2m!p2
e/2m) . (148)
The relationship between this rate and the standard, binary cross-sectional rate can easily beworked out. To get a form suitable for making this connection, we first multiply and divide by theparticle flux v/X within the average over p
e. This average next is written as an average over electron
velocities �"pe/m to yield
wa{a
"
2pN+X
+�
F(�)vX
Xv
+k
DUkD2DSa@De*k > rDaTD2 d(Ea{a
#(+k)2/2m!� ) +k) . (149)
Now, we identify the electron density ne"N/X, and denote the velocity averaging by angle
brackets. We may then trivially rewrite the preceeding expression in the familiar form
wa{a
"neSvp
a{a(v)T (150)
where the cross section is defined as
pa{a
(v),2p+v
+k
DUkD2DSa@De*k > rDaTD2 d(Ea{a
#(+k)2/2m!� ) +k) . (151)
This (Born) cross-section is a sum involving the square of an atomic form factor times a Coulombterm Uk, together with the constraints of energy and momentum conservation.
This form for the transition rate includes the physics of the plasma in two ways: the factorne
represents the mean density of plasma electrons, and statistical properties of the plasma areincluded via the momentum distribution function F( p). As reviewed in Section 1, there arenumerous publications in the literature describing transition rate calculations in this framework,but also incorporating additional plasma effects. Most of these use some model of static screening inthe interaction »K ; a few use more sophisticated cross-section prescriptions, too. However, as weshow in the next section, all such treatments are of limited validity.
5.2. Stochastic perturbation method
The physical picture of the previous section was one of plasma electrons interacting indepen-dently with the target, and their cumulative effect was simply a density factor multiplying
a transition rate for a single (average) electron. At very high plasma densities this picture isinadequate due to the increasingly strong interactions between the plasma electrons themselves.Any given electron’s trajectory will be modified by the presence of the other electrons and thebackground ions. One approach to including this physics is to construct basis states not from theproducts of free particle states Dp
iT (Eq. (143) but, rather, to use eigenstates of
HK!50.
#HK @1-!4.!
"
p2
2m!
ze2r#+
e
p2e
2m# +
e:e{
Uee{
#+ei
Uei
. (152)
(Note that free electron—target interactions are still being neglected.) In this picture no singleplasma electron makes a transition independently of the other plasma electrons, a fact whichintroduces a complicated density dependence. Moreover, since it is no longer possible to write theplasma state as a product of one-electron states, in the traditional cross-section method this wouldraise the problem of defining the flux. Fortunately, this problem does not arise when rates arecomputed directly, as we now show.
The stochastic approach extends the above notions, and views the target ion as an atomic systemsurrounded by a gas of dynamic electrons interacting with each other and the background ions.This approach is explicitly time dependent as the random motions of particles within the gasproduce a stochastic field at the position of the ion. On average, the electrons and ions will tend toscreen the target nucleus, and dynamical effects, due predominantly to the lighter electrons, are notlikely to be highly modified by the presence of the target. The stochastic approach can be describedwith the atomic Hamiltonian
HK @!50.
"!(+2/2m)+2#»0(r)#»
1(r,t)"H
0#»
1(r, t) , (153)
where all coordinates now refer the bound electron. In this Hamiltonian »0(r) represents the mean
(spherical) interaction produced by the nucleus and the quasi-static screening from the plasmaelectrons and ions, and »
1(r, t) represents the time-dependent portion of the plasma—target interac-
tion associated with the stochastic motion of the electrons. In this section, we will assume that theelectrons are weakly coupled. Of course, the fluctuations in the potential arise from fluctuations inthe plasma (electron) density n(r, t). With this we can write the time-dependent part of Eq. (153) forsome particular realization of the plasma as
»1(r, t)"e2Pd3r@
n(r@, t)Dr!r@D
. (154)
It will be assumed that the states associated with H0
can be found by methods such as discussed inSection 3. Here, the transition rate due to the perturbation given by Eq. (154) will be sought, anda statistical average involving plasma states will be taken at the end of the calculation.
As before we begin with the transition rate determined by first order perturbation theory. Due tothe persistent time dependence of the interaction, the first-order rate for DaTP Da@T must be written as
w"limc?0
e4+2
ddt KP
t
~=
dqSa@DPd3r@n(r@,q)Dr!r@D
DaTe*(ua{a~*c)qK2
(155)
which is just a generalization of Eq. (141). This can be rewritten (as in the previous section) in termsof the atomic form factor, by Fourier transforming the interaction »
written in terms of the Fourier modes of the density fluctuations, n(k, q), of one particular plasmarealization as
w" limc?0
1X2+2
ddt K +k UkSa@De*k >rDaTP
t
~=
dq e*(ua{a~*c)qn(k,q)K2
. (156)
In this form we can see distinctly the difference in this approach. The plasma interaction ofEq. (145), essentially a momentum transfer factor, has been replaced here by a complicated plasmatime evolution. Simplification can be achieved by expanding the square and then performing theaverage over plasma states. This yields the mean rate
wa{a
" limc?0
1X2+2
+k
DUkSa@De*k > rDaTD2P=
~=
dq e*(ua{a~*c)qSn(k, q)n(!k, 0)T . (157)
From Section 2.1 we can identify the time integration of the density—density correlation function asthe dynamic structure factor. Employing that result we are able to write the rate as
wa{a
"
1X2+2
+k
DUkD2DSa@De*k > rDaTD2S(k,u) , (158)
where u,ua{a!ic and the cP0 limit is understood.
Eq. (158) has a very intuitive interpretation due to the three main pieces into which the problemhas factorized. Beginning from the right, we have the power spectrum of density fluctuations, whichwas covered in detail in Section 2; S(k,u) contains the ‘‘physics” of the plasma electrons. Themiddle term is the atomic form factor of Section 4, which depends only on the properties of theatomic states involved in the transition; recall, though, that in the stochastic model this factor doesinclude some plasma effects, due to the static screening in »
0. However, it is independent of the
dynamic properties of the plasma. The third term is just the Fourier-transformed Coulombinteraction, which connects the other two factors. Earlier, we noted that the wave vector k in thestructure factor corresponds to the spatial Fourier modes associated with inhomogeneities, and thefrequency u, to temporal Fourier modes associated with oscillations. In the present context wehave additional information. From the atomic form factor we see that +k also has an interpretationas the momentum transferred to the target ion. Thus, short-wavelength density fluctuationscorrespond to the plasma’s ability to transfer a large amount of momentum to the ion. We also see,from u,u
a{a!ic, that plasma fluctuation frequencies near resonance are needed to drive the
transition.
5.3. Plasma impact method
In this section, the ionization rate will be derived a third time by appealing to a somewhatunusual picture. Recall that the main reason for developing the time-dependent stochastic modelwas that the scattering event could not be pictured as a simple binary collision with independentplasma electrons impacting the target ion. Having abandoned the need for a flux (and a crosssection), we may now turn the problem around and view the event as the bound atomic electron‘‘impacting” the plasma. That is, we let the atomic electron undergo inelastic scattering by theplasma. It will be seen that this is formally equivalent to the stochastic approach, above.
In the traditional picture we would take, say, state DxT to be a plasma electron impacting the ionwhich is initially in state DyT. Now, we take the opposite viewpoint and let DxT"DaT be the atomicelectron ‘‘impacting” the entire plasma, which is initially in state DyT"DAT. Since the reference hereis not to a flux, but rather to asymptotic states of individual subsystems, this picture is perfectlyvalid too.
For the atomic transition DaTP Da@T, Eq. (141) takes the form
wa{a
"
2p+
+A,A{
e~bEA
Q KSA@DSa@DPnL (r@)Uee(r!r@)d3r@DaTDATK
2d(E@!E) , (159)
where nL (r@) is a density operator, Uee" e2/Dr!r@D is the Coulomb interaction, and an average over
initial plasma states DAT and a sum over final plasma states DA@T has been made. For comparisonwith earlier results, we Fourier transform the interaction to obtain an expression similar toEq. (21),
The matrix element can be squared to yield a result which contains a double integral over theFourier transform variables. If the second Fourier transform variable is q and the plasma isassumed to be translationally invariant, the result will contain the product
The delta function dkq arises from the fact that translationally invariant states are momentumeigenstates. Using this information, the transition rate then reduces to
wa{a
"
2pX2+
+k
DUkD2DSa@De*k > rDaTD2 +A{,A
e~bEA
QDSA@DnL (k)DATD2d(E@!E)
"
1X2+2
+k
DUkD2DSa@De*k >rDaTD2S(k,u) , (162)
which is identical with Eq. (158). It is Eqs. (26) and (30) that connect the time-dependent approachof the previous section to this time-independent approach.
6. Numerical study of projectile screening issues
In Section 5.2 the plasma’s perturbation of the atomic system was broken into pieces whichrepresent separately static target screening and dynamic projectile screening. Quantitatively, thispicture led to the Hamiltonian of Eq. (153) (repeated here in atomic units),
HK @!50.
"!12+2#»
0(r)#»
1(r, t) . (163)
From the discussions of Section 3 and Section 2, we know that both »0(r) and »
1(r,t) contain effects
of high plasma density. In this section, the high-density effects of »1(r, t) will be illustrated, with
The interaction between two particles in vacuo is often quite different than the complicatedeffective interaction arising from the presence of other particles in dense systems. It was remarkedin Section 5.1 that this might be taken into account in a static screening model by replacing thebare Coulomb interaction between two particles with a screened interaction, such as
»(r1, r
2)" 1/Dr
1!r
2D e~i4@r1~r
2@ (164)
where i4is some screening parameter. Thus, a collision is not just an electron impacting the ion but
in fact is an electron with its associated screening cloud impacting an ion. This static screeningpicture suggests that collisional transition rates will be reduced at high-density as the screeningbecomes more efficient. Numerical results based on the collisional model of Section 5.2 will bepresented here to examine this prediction quantitatively.
It is instructive to consider the collisional ionization process for various initial (bound) states.Our model system, chosen purely for illustrative purposes, of He` ions in a 15 eV plasma isconsidered here for a wide range of plasma densities, and for both the a"1 and a"3 initial states.The ionization rate is determined for these two cases to directly examine projectile screeningproperties of the plasma at high density. In particular, each of the various screening approxima-tions of Section 2.3 is used to compute the rate coefficient as a function of density, after the dielectricresponse function of Section 2.4 was employed to obtain e(k,u) in each case. At the highest densitiespresented here, this scheme breaks down and details of the results become suspect. Nevertheless, thesecalculations do provide considerable insight into issues regarding projectile screening.
6.1. Ionization rates for He` (ground state)
Fig. 13 shows the results of our numerical calculation of the total ionization rate for He` fromthe a"1 state, in a 15 eV plasma. The PW GOSD of Section 4.2 has been used here both to keepthe atomic physics as simple as possible and to use a GOSD form that also is valid for the excitedstate calculation of Section 6.2. Results are shown for each of the three screening approximations andare plotted relative to the no screening case. In this way, the screening models are easily compared.
In the no screening (NS) case the dynamic structure factor has been approximated as that of anideal gas, (e.g. Eq. (44))
S(k,u)PS0(k, u) , (165)
which implies that each plasma electron impacting the ion does so independently of the otherelectrons. Since the electrons are independent, there can be no high density information containedwithin this picture and the ionization rate coefficient, that is, w/n
e, is density independent.
Therefore, the flat curve shown in the figure represents the actual NS density dependence and is notmerely an artifact of plotting a ratio; our computed NS He`(a"1) ionization rate coefficient is1.4] 10~9 cm3/s.
The static screening (SS) result shows behavior consistent with predictions based on interactionsof the form of Eq. (164). In fact, within the context of static screening and the dielectric responsefunction of Section 2.3.2, the functional form of Eq. (164) is exact. This can be seen by writing theinteraction as
Fig. 13. The total ionization rate of He` from the a"1 state versus plasma density, for both dynamic (DS) and static(SS) screening models. The plasma temperature is 15 eV and the atomic transition is treated via the PW GOSD.Specifically, the ratio of the rate for each screening model to the rate for the no screening case is shown (left ordinate)versus the logarithm of plasma density. Also shown (dashed line, right ordinate) is I
1!+u
e, the difference in the
ionization potential and the plasmon energy.
which has, with i4"i
D, Eq. (164) as its Fourier inverse. Thus, at high densities, with interactions
weakened by screening, the ionization rate is reduced. This reduction also can be exploredgraphically by plotting the screening function 1/De(k,u)D2 versus k and u. This is shown inFig. 14 for the SS screening function, and clearly illustrates both the screening behavior at smallk and the absence of any u dependence. Note, however, that this static screening does not becomeimportant until the plasma has reached a density of about 1022 cm~3. It is easy to explain thisbehavior by noting that, within the stochastic model, the wave vector for the plasma density modek also corresponds to the momentum transferred to the atomic system. From Eq. (166) we see thatmodes with wave vectors less than i
Dare strongly screened, while those above i
Dare essentially
unchanged. Thus, screening reduces the contribution from small momentum transfers k(iD. We
may use the relation
k2.*/
/2"z2/2a2 , (167)
in which the ionization energy has been equated to the classical energy k2.*/
/2, to estimate theminimum momentum k
.*/required to ionize the electron from level a. The ionization process is not
affected by screening until the density increases enough that the momentum iD
Fig. 14. The screening function 1/De(k,u)D2 in the static screening approximation. In this approximation the modes belowkD
are screened and there is no u dependence. This was computed at a density of 1022 cm~3 and a temperature of 15 eV.Atomic units are used.
we estimate the onset of screening effects by the condition k.*/
" 5iD
(see discussion followingEq. (3)), a threshold density of
n.*/
+ 8 ] 1020(Z2¹/a2) cm~3 (168)
is predicted by Eq. (167), with ¹ in eV. For the case considered in this section, n.*/
is about5] 1022 cm~3, which is in good agreement with the computational result.
The dynamic screening (DS) case plotted in Fig. 13 shows unexpected behavior, based on theprevious analysis of the SS case. Whereas the SS case showed a decrease in the rate at high densitythe DS case shows just the opposite, an increase. Recall that the DS model is related to the SSmodel by the generalization e(k, 0)Pe(k,u) and thus it contains the SS model. However, we do notsee any evidence for diminution via screening of the interaction in the DS case. We may understandthis new behavior by comparing the energy transferred I
1to the plasmon energy +u
e. The quantity
I1!+u
e"2!1.365] 10~12Jn
e, (169)
which also is plotted in Fig. 13, indicates that the ionization potential exceeds the plasmon energyat all densities considered here. Thus, the ionization process is only sensitive to density fluctuationswhose characteristic frequencies exceed u
e. A high-frequency expansion of e(k,u),
e(k,u)+1!u2e/u2!u4
e/u4k2/i2
D!2 , (170)
clearly shows that the static screening, which arises from the iD
dependence, is a higher order andtherefore a less important effect. In this regime, the relevant plasma fluctuations occur at frequen-cies which cannot be screened on a time scale of order u~1
e. The expression that replaces Eq. (166)
is
Uk/e(k,u)+(4p/k2)/(1!u2
e/u2!2) , (171)
which indicates an enhanced interaction, in agreement with the calculated result.Note further that this expansion predicts a divergence in the interaction in the very high-density
regime where the plasmon energy becomes comparable to the ionization energy. This expectationis illustrated in Fig. 15 where the dynamic screening factor 1/De(k,u)D2 is shown. It is clear from the
Fig. 15. The full (dynamic) screening function 1/De(k,u)D2 is shown versus k and u. The screening function is about unityover most of the surface indicating that, to a good approximation, no screening occurs. A plasma oscillation peak doesappear which serves to modestly enhance the ionization rate. Below the plasma frequency the surface dips below unityreflecting static screening behavior. The plasma conditions are identical to those of previous figure.
plot that this screening factor is unity for most values of k and u. There is a plasma oscillation peakwhich enhances the interaction above the NS case and at higher densities this peak becomes morepronounced. This indicates, perhaps surprisingly, that the DS case is more similar to the NS casethan to the SS case!
6.2. Ionization rates for He` (excited state)
The total ionization rate has also been computed for the He`a"3 state, using the PW GOSD.Results analogous to those in the previous section are presented in Fig. 16. The NS and SS curvesare in qualitative agreement with the a" 1 calculations. The static screening now, however, muchmore severely inhibits the ionization rate at high plasma densities. Furthermore, the screeningbegins to become important at a density of &1021 cm~3, in agreement with Eq. (168).
The a"3, DS case is not even in qualitative agreement with the previous, a" 1, example. Theanalysis of the previous section suggests that there would be a divergence of the ionization rate asthe plasmon energy approaches the ionization potential. It is easy to see from the graph of I
3!+u
ein the figure that this condition is actually realized for the a" 3 state, but evidently there is nocorresponding divergence in the ionization rate. Upon closer inspection of the interaction in thisregime we find the screened interaction takes the form
KU
ke(k,u)K
2"
16p2/k4
[Re e(k,u)]2#[Im e(k,u)]2+
16p2/k4
(1!u2e/u2)2#[Im e(k,u)]2
. (172)
The divergence has been avoided by the damping of the plasma oscillation. Although the dampingis often ignored, it plays an important role in circumventing divergences near the plasmon energy.
Not only is a divergence not present but the DS ionization rate is in fact dramatically reduced atdensities where I
3!+u
e(0. In this regime fluctuations at frequencies high enough to ionize the
target now exist below the plasma frequency. These fluctuations are screened and we have
Fig. 16. As in Fig. 13, for the total ionization rate of He` from the a" 3 state, again in a 15 eV plasma. Also shown(dashed line, right ordinate) is I
3!+u
e, the difference in the ionization potential and the plasmon energy.
a situation entirely analogous to the SS case. To be more exact we must remember that thesecalculations are of the total ionization rate and we have to consider all possible bound—freetransitions. With a temperature of 15 eV and a density of n
e"1024 cm~3 more than half of the
transitions take place as a result of fluctuations below the plasma frequency. Furthermore, it is inthe lower-energy region where the GOSD takes its largest values. Thus, at high plasma densities,the total ionization of this excited state is dominated by fluctuations which are screened.
6.3. Ionization rates for Ar`17 (ground and excited states)
Ionization of Ar`17 represents an important case due to its use as an ICF diagnostic [118].Results are shown in Fig. 17 for the bound states a"1,3,5 at a temperature of 650 eV. Bothprojectile screening and Hybrid SSCP level shifts are included in these calculations. It is clear thatthe tightly bound ground state is almost completely unaffected under these conditions (cf. Eq. (3).)However, the ionization rates for the excited states are greatly affected by high plasma density. Fora"3 the enhancement in the rate is dominated by (quasi-static) shifting of the level, except at veryhigh density where the dynamic and static projectile screening regimes are entered and theenhancement is diminished. The level shifts greatly enhance the a"5 ionization rate until the stateis lost (via continuum lowering) just below n
e"1024cm~3. For this case, projectile screening effects
at high density cannot be realized before the state is lost. Because each state’s ionization ratechanges in a different way, it does not seem likely that one can produce a simple prescription for theoverall effect of high density on ionization balance in non-equilibrium plasmas.
Fig. 17. The total ionization rate for Ar`17 from the ground and excited states a"1,3,5. The results were obtained withHybrid level shifts and dynamic screening. Plotted is the ratio of the rate to the rate Rate
0computed with no level shifts
or screening. The result for a"5 is shown up to the density for which that bound state is lost via continuum lowering.
7. Numerical study of target screening issues
High plasma densities also affect atomic transition rates through quasistatic perturbations of thetarget. In the Hamiltonian of Eq. (163), »
0(r) contains the quasistatic effects. In the previous
section, we employed a PW GOSD, which assumed that the bound state was unperturbed and thecontinuum state was a free-particle state. Here we take up issues associated with the modificationsof the atomic system due to the departure of »
0(r) from a pure Coulomb term representing the
bound electron’s interaction with the atomic nucleus.As discussed in Section 3, free charges in a plasma tend to screen out charges of the opposite sign,
on average. The local potential around an ion, for example, differs from that of the bare ion becausefree electrons are attracted and other ions repelled. This screening modifies the eigenstates of theatomic system and, therefore, the GOSD for the transition. In Section 7.1, ionization calculationswith the OPW GOSD of Section 4.3 are compared with those of the PW GOSD to address theorthogonality issue. Ionization rates are then calculated in Section 7.2 with bound state energyeigenvalues that reflect the quasistatic screening. Level shifts are estimated as in Section 3.3.
7.1. Non-orthogonality of initial and final states
The total ionization rate as a function of electron density has been computed for He` (a"1)with the OPW and PW GOSD for a fixed plasma temperature of 15 eV. These results are shown
Fig. 18. The He` a"1 ionization rate for the different screening models for both the OPW and PW GOSD is plotted inratio with the PW, NS case. Results with the OPW GOSD are shown as solid lines and those of the corresponding PWGOSD are shown as dotted lines.
together in Fig. 18. The solid lines are the OPW ionization rates normalized to the no screeningPW result. With the OPW GOSD, the ionization rates are nearly a factor of two less than the PWGOSD results for each screening model. This change is of the same order as that caused by thevarious projectile screening models, with nearly an order of magnitude spread in the rates at highdensity. Thus, a seemingly innocuous change in the GOSD leads to a substantial modification ofthe predicted ionization rate. A comparison of Fig. 10 with Fig. 9 reveals that, in this case, thereduction in the rate is due to the elimination of the small k divergence in the PW GOSD.
In addition to an overall reduction in the ionization rate at all densities, there is also a slightchange in the behavior of the different screening approximations. In particular, the rates computedwith SS or DS do not differ from the NS case as much in the OPW calculation as they do in the PWcalculation. Recall from comments related to Fig. 15 that much of the difference between thescreening models arises in the small k regime. This is exactly where the PW approximationoverestimates the GOSD, and so the (PW) screening effects we found earlier were actuallysomewhat exaggerated.
7.2. Bound state level shifts
The He` (a"1) ionization rate has also been computed with corrected binding energiesfor the ground state, using the hybrid level shifts of Eq. (179). These results, together with the PW
Fig. 19. The He` a"1 ionization rate with bound state energy level corrections is shown for the various screeningmodels and the OPW GOSD. Specifically, the rate is shown in ratio with the NS case without level corrections. Thedotted lines are the same as those the previous figure.
results of Section 6.1, are plotted in Fig. 19. It is clear from the figure that level shifts play avery important role in determining the ionization rate. At high plasma densities the levelapproaches the continuum and ionization becomes progressively easier. Now, in each screeningmodel we explored, the ionization rate is enhanced at high plasma density by a substantialfactor.
In the SS case, there is a competition at high density between the screening of the projectile andthe target screening. The shift in the energy level dominates at lower densities, producing first anincreased ionization rate as the density rises. At the highest density shown, however, screening ofthe interaction is quite strong and the ionization rate subsequently drops to become about equal toits low-density value.
The DS and NS cases are very similar: as the plasma density increases, the collective modecontained in the DS case serves to enhance the ionization rate slightly over the NS case. This effectwas also seen in Fig. 13. But, new behavior now appears at the highest densities, where the twoapproximations yield rates that are nearly equal. The DS result decreases as a function of plasmadensity just as with the SS case, because the shift in the energy of the level has put the state into theregime where some of the transitions are statically screened: in essence, the excited state behavior ofFig. 16 is now being seen, albeit weakly, for the shifted ground state.
We have investigated transitions driven by stochastic perturbations for the special case of atomiccollisional ionization. The most useful number associated with this process is the total ionizationrate from a single bound state to all continuum states. Atomic form factors for bound—freetransitions were computed in various approximations that reflect properties of the average plasmainteraction. Properties of the plasma electron fluctuations were investigated within three screeningmodels which correspond to no screening, static screening, and dynamic screening. Conclusionsbased on our numerical computations are:
f A naive screening model that replaces the bare Coulomb interaction with a static screenedpotential is almost always a poor approximation. This is because most atomic ionizationsinvolve energies that exceed +u
eand are not screened.
f When the transition energy is near +ue
there is an enhancement in the ionization rate.Transitions below this energy, if there are any, are essentially statically screened and those abovethis energy are only weakly screened. Thus, for the total collisional ionization rate we mustconsider different screening properties for the various transitions. All of this information iscontained in the dynamic screening model. Only in cases where all important transitions areeither above or below +u
ecan a simplification be made.
f The generalized oscillator strength density (GOSD) must be carefully chosen. It has beenshown that merely orthogonalizing the initial and final states produced a factor of two changein the ionization rate, which indicates a strong sensitivity to this effect. Also, deviations fromthe no screening case are exaggerated with a GOSD having non-orthogonal initial and finalstates.
f A new static screened Coulomb potential was developed for this problem. The usual Debyepicture was deemed invalid for treating the initial bound state, and the complementary,ion-sphere potential was deemed invalid for treating the continuum states. A hybrid potentialderived from these two limiting models was constructed; it has good small-r and large-r behaviorand is applicable over wide ranges of temperature and density.
f At high densities, modest changes in the bound state energies can produce large changesin the ionization rate. These changes are generally more significant than those associated withdynamic screening. Furthermore, level shifts can bring a state with ionization potential I
aA+u
e,
which originally was in the ‘‘no screening regime”, into a regime described only by dynamicscreening.
The results presented here can be extended in many directions. Perhaps the most importantextension would be to consider the inverse process of three-body recombination. Then, theproblem of ionization kinetics and the importance of density corrections could be ascertained forseveral cases of experimental interest.
There also remains much that can be done to improve the underlying physics for both of theseprocesses. These physics issues can be partitioned into the three areas of plasma physics, atomicstructure, and the description of the interaction between plasma and atom. We end this Report withan annotated ‘‘shopping list”.
1. Plasma degeneracy. There are experiments in which the plasma may spend sometime as a degenerate electron gas. This is the case, for example, in laser produced plasmasat early and late times. In fact, at low laser intensities the electrons may never leavethe degenerate regime. The stochastic model can be easily extended to the degenerate regimeby simply using the finite temperature Lindhard dielectric response function (DRF),which is the appropriate generalization of the Vlasov DRF used here. References are given inAppendix B.
2. Strong coupling. Strong coupling among the plasma electrons appears under conditions similarto those which result in degeneracy: high density and/or low temperature. Strong coupling isalso possible in certain non-degenerate experimental regimes we have considered. Relevantextentions of the model used here are well known; the theory is based on treating correlationsmore carefully in the underlying kinetic equation for the phase space distribution function. Thisgives rise to the so-called ‘‘local field corrections” in the DRF and, hence, the dynamic structurefactor. An account of this procedure has been outlined in Section 2.4.
3. Ions. The ions in a plasma are more likely to be strongly coupled than the electrons since, ina two-component plasma, one has C
z"z5@3C
e. The quasistatic potentials described in this
Report are invalid under strong ion coupling conditions. Also, ions have been neglected entirelyin obtaining the plasma density fluctuations (see Section 8.4 below).
4. Double counting. The common problem of double counting [15] plasma perturbations on theatom has not been addressed in a systematic way. Some double counting is inevitable in thestochastic model as a result of treating the quasistatic and fluctuating perturbations withseparate interaction terms »
0(r) and »
1(r, t). This issue is closely related to the previous issue of
treating the ions self-consistently.5. Non-thermal distributions. It was stated in Section 1 that most plasmas are not in thermal
equilibrium. It was useful, however, to use a thermal distribution for the plasma electrons toobtain an analytic form for the DRF. In some experiments where electron—ion temperaturedifferences are important this may still be a good approximation whereas in others, such as somelaser-produced plasmas, it is known not to be [102,29,73]. The form of the non-equilibriumdistributions is likely to be strongly dependent upon the specific time history of the experimentand simple descriptions may not be possible.
8.3. Screened interaction issues
1. Many-electron ions. Our work here has been restricted to hydrogenic ions which, in some cases,happen to be the most important charge state. Often, however, other charge states are key. Forexample, some X-ray lasers are based on Ne-like ions, and future ICF X-ray diagnostics arelikely to be based on a many-electron charge state of Xe. Not only is the basic atomic structurecomplicated by many bound electrons, but the collision problem itself is further complicated bythe appearance of resonances and the likelihood of several important inelastic scatteringchannels. These are well known, but still challenging issues even for binary electron—ioncollisions. How stochastic perturbations are influenced by correlations among bound electronsis an unexplored topic.
2. Higher-order bound state corrections. The plasma-induced modifications of the bound stateshave only been treated as energy level shifts obtained from spherically symmetric quasistaticpotentials. This lowest-order correction can be extended, and since a numerical method fortreating the GOSD has been detailed, this problem has in principle been solved. A code whichcalculates the exact bound state energy and wave function is all that is needed to complementthe existing codes used for obtaining accurate GOSDs. Such an improvement definitely isneeded to obtain accurate results for weakly bound states. It is also known that the (hydrogenic)angular momentum degeneracy is lifted by any screened potential. This fact was ignored inobtaining the PW and OPW GOSDs.
3. Nonspherical quasistatic potentials. It has been assumed that all the quasistatic potentialshave spherical symmetry. But, recall that the stochastic model treats the ions as staticand the electrons as having both static and dynamic aspects; that is, the ions do not causetransitions. If the ions do not move significantly during the ionization process, then it isimpossible that a discrete number of them can produce a completely spherical potential.This issue is important in line broadening theory, and results from that area might beborrowed to explore the importance of a non-spherical potential on the ionization problem[93]. This issue also has been explored by Perrot [92] in a study of external electric fieldeffects on inelastic cross sections and in a microfield stochastic model (MSM) developed byMurillo [86] to treat the perturbations of slowly moving ions on electron impact processes. Insuch descriptions it is likely that level shifts in agreement with spectroscopy [31] can beobtained.
8.4. Atomic ionization issues
1. Beyond the Born approximation. The range of applicability of the Golden Rule has not beenexplored in this work. For plasmas which are cool, the slower free electrons may not be ina regime in which the (first) Born approximation is valid. For transitions between states withsimilar energies, screening relaxes such a constraint because the interaction is weakened bystatic screening. However, as we have seen, this typically is not the case. In fact, the interactionoften becomes stronger at high densities due to collective effects. The collisional ionizationprocess is particularly complicated because one has to consider both slow electrons in thedistribution as well as transitions which are modified and possibly enhanced by dynamicscreening effects.
2. Coulomb three-body problem. The final state of the ionization process contains (at least) twocharged particles in the field of an ion and thus represents a Coulomb three-body problem. Thisfact has been ignored entirely in the stochastic model whose plasma fluctuations are not affectedby the presence of the ion. As the Coulomb three-body problem is insoluble, only modestimprovements can be made. Some improvements may, however, be essential for obtainingresults in quantitative agreement with experiments.
3. Potential and exchange scattering. The quantum statistical effects of the (identical) electrons havenot been accounted for here. These effects appear in both the initial state and the final state. Inthe initial state this effect arises from the indistinguishability of the plasma electron(s) and thebound state electron(s) [120]. In the final state one often refers to the ‘‘exchange scattering”between the plasma and ionized continuum electrons. A careful study of these effects in
traditional cross-section calculations, but with plasma and ion parameters of interest, would beuseful for assessing the importance of these complications.
4. Distorted waves for plasma electrons. Our description of plasma density fluctuations has beenbased on the one-component plasma model. In this model the electrons are assumed tointeraction with each other and a uniform, positively charged background. Polarization of theplasma due to the presence of a high-z ion requires one to consider dynamic structure factors fortwo-component, electron—ion plasmas. In the traditional (cross section) approach, such polar-ization is accounted for by treating the plasma electron within, for example, the distorted-waveapproximation.
Acknowledgements
Much of this work was performed with National Science Foundation support through grantsPHY-9321329 and PHY-9024397 to Rice University. Some of this work was performed under theauspices of the United States Department of Energy through support of the Theoretical Division ofLos Alamos National Laboratory. We would like to thank Dr. D.P. Kilcrease for a careful readingof the manuscript with accompanying helpful comments. The Aspen Center for Physics providedus with a stimulating environment during August 1995, when a preliminary draft of this Report wascreated during the workshop on Elementary Processes in Astrophysical Dense Matter.
Appendix A. List of frequently used symbols
Notationa, b principal quantum number, atomic state labelA, B plasma state labelE energyf oscillator strengthF phase-space distribution functionF electron—electron forceg statistical weightg radial distribution functionG continuum density of statesG atomic partition function+ Planck’s constantIz
ionization potential of a charge z ionk plasma spatial Fourier modeK continuum electron wave vectorl angular momentum quantum numberm,ma massnL electron density operatorna particle species densityN number of electrons
q, qa charge (of species a)Q plasma canonical partition functionr positionr4
ion—sphere radius¹,¹a temperature (of species a)¹
FFermi temperature
v, � velocity» general interaction (energy)wba
transition rate (aPb)z, za ion charge (of species a)Z nuclear chargea generic species labelb inverse temperature, energy unitsCa coupling parameter (of species a)D density parametere dielectric response functiong continuum state observablesi inverse screening lengthiD
Debye wave vectoriTF
Thomas—Fermi wave vectorje
thermal DeBroglie wavelengthk chemical potentialm screening functionp scattering cross section¶ degeneracy parameter/ electric potentialU Coulomb interaction energyt wave functionW effective electron—electron interactionu plasma temporal Fourier modeua plasma frequency of species aX plasma volume
Appendix B. Numerical computation of the dielectric response function
Having derived the dielectric response function, Eq. (68), we now obtain a form suitable fornumerical computations. As all directions are equivalent in a homogeneous, unmagnetized plasma,k can be oriented as k"kzL . In addition, to be consistent with the use of the Vlasov equation, wemust choose F
0(�) to be the equilibrium ideal gas distribution. This distribution is the familiar
It is convenient to shift to dimensionless variables K and W defined by
K"k/kD, W"u/u
e, (B.2)
where kD
is the Debye wave number and ue
is the electron plasma frequency. In terms of thesequantities we have
e(K,W)"1#(1/K2Jp)P=
~=
x exp(!x2)/(x!W/J2K) dx (B.3)
for the dielectric response function. The integral can be evaluated with the Dirac identity,
limg?0
1(x!a)$ig
"PC1
x!aDGipd(x!a) , (B.4)
and the result can be written in terms of the error function complement with complex argument,
¼(z)"e~z2 erfc(!iz) (B.5)
which has been well studied [22]. After considerable manipulation, one finally obtains
e(K,W)"1#1
K2!S
p2W
K3Im[¼(W/J2K)]#iS
p2W
K3Re[¼(W/J2K)]. (B.6)
The problem of evaluating plasma density fluctuations through S(k,u) therefore has been reducedto evaluating Eq. (B.5). Efficient routines are available to compute ¼(z) [114]. Calculation of theresponse function for a finite-temperature degenerate plasma has also been investigated [32,59].
Appendix C. Formulary
Here, we collect numerical expressions for several of the important quantities discussed in thetext. In this formulary, species number densities na are in cm~3, temperatures ¹ and inversetemperatures are in eV and eV~1, respectively; masses ma and charges za are in units of the electronmass and charge; lengths r are in Bohr radii (a
[1] G. Arfken, Mathematical Methods for Physicists, Academic Press, Inc., San Diego, 1985.[2] K. Bartschat, Phys. Rep. 180 (1989) 1.[3] D.R. Bates, A.E. Kingston, R.W.P. McWhirter, Proc. Roy. Soc. A 267 (1962) 297; Proc. Roy. Soc. A 270 (1962) 155.[4] J.F. Benage Jr., W.R. Shanahan, in: H.M. Van Horn, S. Ichimaru (Eds.), Strongly Coupled Plasma Physics, vol.
187, University of Rochester Press, 1993.[5] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Plenum, New York, 1977.[6] Th. Bornath, M. Schlanges, Physica A 196 (1993) 427.[7] H. Brysk, P.M. Campbell, P. Hammerling, Plasma Physics 17 (1975) 473.[8] P.G. Burke, V.M. Burke, K.M. Dunseath, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 5341.[9] M.D. Cable et al., Phys. Rev. Lett. 73 (17) (1994) 2316.
[10] J. Callaway, Phys. Rev. 97 (4) (1955) 933.[11] R. Cauble, D.B. Boercker, Phys. Rev. A 28 (2) (1983) 944.[12] J. Davis, M. Blaha, J. Quant. Spectrosc. Radiat. Transfer 27 (3) (1982) 307.[13] J. Diaz-Valdes, F.A. Gutierrez, Phys. Lett. A 193 (1994) 154.[14] T.R. Dittrich et al., Phys. Rev. Lett. 73 (17) (1994) 2324.[15] J.W. Dufty, Phys. Rev. 187 (1969) 305.[16] W. Ebeling et al., Thermophysical Properities of Hot Dense Plasmas, B.G. Teubner, Stuttgart, 1991.[17] W. Ebeling, A. Forster, V.Yu. Podlipchuk, Phys. Lett. A 218 (1996) 297.[18] R.P. Feynman, N. Metropolis, E. Teller, Phys. Rev. 75 (1949) 1561.[19] D.M. Fishman, G.D. Patterson, Biopolymers 38 (1996) 535.[20] V. Fock, Z. Physik 98 (1935) 145.[21] V. Fortov, V. Ternovoi, S. Kvitov, I. Lomonosov, in: H.M. Van Horn, S. Ichimaru (Eds.), Strongly Coupled
Plasma Physics, vol. 177, University of Rochester Press, 1993.[22] B.D. Fried, S.D. Conte, The Plasma Dispersion Function: The Hilbert Transform of the Gaussian, Academic
Press, New York, 1961.[23] W. Fritsch, C.D. Lin, Phys. Rep. 202 (1991) 1.[24] P. Fromy, C. Deutch, G. Maynard, Phys. Plasmas 3 (1996) 714.[25] K. Fujima, Atomic Models for Hot Dense Plasmas, Inst. Plasma Phys., Nagoya Univ. Report dIPPJ-AM-57,
1988.[26] J.D. Garcia, Phys. Rev. 159 (1) (1967) 39.[27] M.D. Girardeau, F.A. Gutierrez, Phys. Rev. A 38 (3) (1988) 1624.[28] J. Glanz, Science 262 (1993) 1379.[29] T.E. Glover, J.K. Crane, M.D. Perry, R.W. Lee, R.W. Falcone, Phys. Rev. Lett. 75 (1995) 445.[30] D.L. Goodstein, States of Matter, Dover Publications, New York, 1985.[31] H.R. Griem, Principles of Plasma Spectroscopy, Cambridge University Press, Cambridge, UK, 1997.[32] C. Gouedard, C. Deutsch, J. Math. Phys. 19 (1) (1978) 32.[33] F.A. Gutierrez, J. DiazValdes, J. Phys. B: At. Mol. Opt. Phys. 27 (1994) 593.[34] F.A. Gutierrez, M.D. Girardeau, Phys. Rev. A 42 (2) (1990) 936.
[35] F.A. Gutierrez, H. Jonin, E. Cormier, J. Quant. Spectrosc. Radiat. Transfer 51 (1994) 665.[36] B.A. Hammel et al., Phys. Rev. Lett. 70 (9) (1993) 1263.[37] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic Press, San Diego, 1990.[38] W.A. Harrison, Solid State Theory, McGraw-Hill, New York, 1970.[39] G.J. Hatton, N.F. Lane, J.C. Weisheit, J. Phys. B: At. Mol. Phys. 14 (1981) 4879.[40] C. Herring, Phys. Rev. 57 (1940) 1969.[41] J.P. Hill et al., Phys. Rev. Lett. 77 (1996) 3665.[42] K. Huang, Statistical Mechanics, Wiley, New York, 1987.[43] S. Ichimaru, Statistical Plasma Physics, vol. I: Basic Principles, Frontiers in Physics, vol. 87, Addison-Wesley, New
York, 1992.[44] S. Ichimaru, Statistical Plasma Physics, vol. II: Condensed Plasmas, Frontiers in Physics, vol. 88, Addison-Wesley,
New York, 1994.[45] S. Ichimaru, S. Mitake, S. Tanaka, X.-Z. Yan, Phys. Rev. A 32 (1985) 1768; plus references in X.-Z. Yan,
S. Ichimaru, Phys. Rev. A 34 (1986) 2173.[46] C.A. Iglesias, J. Quant. Spectrosc. Radiat. Transfer 54 (1/2) (1995) 181.[47] D.R. Inglis, E. Teller, Astrophys. J. 90 (1939) 439.[48] M. Inokuti, Rev. Mod. Phys. 43 (3) (1971) 297.[49] N. Itoh, H. Totsuji, S. Ichimaru, H.E. DeWitt, Astrophys. J. 234 (1979) 1079.[50] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975.[51] C.J. Joachain, Quantum Collision Theory, Elsevier, New York, 1987.[52] Y.D. Jung, Phys. Plasmas 2 (1) (1995) 332.[53] Y.D. Jung, Phys. Plasmas 2 (5) (1995) 1775.[54] Y.D. Jung, Y.S. Yoon, J. Phys. B 29 (1996) 3549.[55] L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Green’s Function Methods in Equilibirum and
Nonequilibrium Problems, Addison-Wesley, New York, 1989.[56] S. Kambayashi, G. Nowotny, J. Chihara, G. Kahl, J. Non-Crystalline Solids 205—207, Part II, 914, 1996.[57] T. Kawachi, T. Fujimoto, Phys. Rev. E 55 (1997) 1836.[58] C.J. Keane et al., Phys. of Fluids B 5 (9) (1993) 3328.[59] F.C. Khanna, H.R. Glyde, Can. J. Phys. 54 (1976) 648.[60] H. Kitamura, S. Ichimaru, Phys. Rev. E 51 (6) (1995) 6004.[61] A. Klisnick, et al., in: W.L. Rowan (Ed.), Atomic Processes in Plasmas, vol. 201, AIP Press, New York, 1995.[62] D. Saumon, G. Chabrier, Phys. Rev. A 46 (4) (1992) 2084.[63] R. Kubo, J. Phys. Soc. Japan 12 (1957) 570&1203.[64] M. Toda, R. Kubo, N. Saito, N. Hashitsume, Statistical Physics, Springer, New York, 1991.[65] C. Bottcher, in: D. Bates (Ed.), Advances in Atomic and Molecular Physics, Vol. 20, Academic Press, New York,
1985, p. 241.[66] R. Latter, Phys. Rev. 99 (6) (1955) 1854.[67] R.W. Lee, R. Petrasso, R.W. Falcone, Science on High-Energy Lasers: From today to the NIF, Lawrence
Livermore National Laboratory Report dUCRL-ID-119170 (1995).[68] R.L. Liboff, Kinetic Theory: Classical, Quantum, and Relativistic Descriptions, Prentice-Hall, Englewood Cliffs,
NJ, 1990.[69] J. Lindl, Phys. Plasmas 2 (1995) 3933.[70] R.W.P. McWhirter, in: H. Huddlestone, S. Leonard (Eds.), Plasma Diagnostic Techniques, vol. 201, Academic
Press, New York, 1965.[71] T. Maeda, Macromolecules 24 (1991) 2740.[72] G. Mahan, Many-Particle Physics, Plenum Press, New York, 1990.[73] R.C. Mancini, et al. in: W.L. Rowan (Ed.), Atomic Processes in Plasmas, vol. 322, AIP Press, New York, 1995,
p. 161.[74] C. Martin et al., Journal de Physique II 5 (1995) 697.[75] D.L. Matthews et al., Phys. Rev. Lett. 54 (2) (1985) 106.[76] F. Mazzanti, A. Polls, J. Boronat, Phys. Lett. A 220 (1996) 251.
[77] H.M. Milchberg et al., Phys. Rev. Lett. 67 (19) (1991) 2654.[78] H.M. Milchberg et al., Phys. Rev. Lett. 61 (1988) 2364.[79] M.H. Mittleman, Phys. Rev. 122 (6) (1961) 1930.[80] R.M. More, J. Quant. Spectrosc. Radiat. Transfer 27 (1982) 345.[81] R.M. More, in: D. Bates, B. Bederson (Eds.), Advances in Atomic and Molecular Physics, vol. 305, Academic Press,
New York, 1985.[82] N.F. Mott, Metal-Insulator Transitions, Taylor & Francis, London, 1974.[83] J.A.M. van der Mullen, Phys. Rep. 191 (1990) 109.[84] M.S. Murillo, J.C. Weisheit, in: H. Van Horn, S. Ichimaru (Eds.), Strongly Coupled Plasma Physics, vol. 233,
University of Rochester Press, 1993.[85] M.S. Murillo, in: W. Goldstein (Ed.), Atomic Processes in Plasmas, vol. 301, AIP Press, AIP CP, New York, 1996.[86] M.S. Murillo, Proc. 1997 Internat. Conf. on Strongly Coupled Coulomb Systems, 1997, to appear.[87] J.K. Nash, J.M. Salter, W.G. Eme, R.W. Lee, J. Quant. Spectrosc. Radiat. Transfer 54 (1995) 283.[88] D.R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983.[89] A. Ng et al., Phys. Rev. Lett. 72 (21) (1994) 3351.[90] K.C. Ng, J. Chem. Phys. 61 (7) (1974) 2680.[91] National Research Council (U.S.), Plasma Science: From Fundamental Research to Technical Applications,
National Academy Press, Washington DC, 1995.[92] F. Perrot, Physica Scripta 37 (1988) 351.[93] E.L. Pollock, J.C. Weisheit, in: F. Rostas (ed.), Spectral Line Shapes, Vol. 3, Walter de Gruyter & Co., New York, 1985.[94] D. Pines, P. Nozieres, The Theory of Quantum Liquids, Addison-Wesley, New York, 1966.[95] M. Rasolt, F. Perrot, Phys. Rev. Lett. 62 (19) (1989) 2273.[96] F.J. Rogers, H.C. GraboskeJr, D.J. Harwood, Phys. Rev. A 1 (6) (1970) 1577.[97] F.J. Rogers, private communication; see also F.J. Rogers, D.A. Young, H.E. DeWitt, M. Ross, Phys. Rev. A 28 2990 (1983).[98] S.J. Rose, Ap. J., 453 (1995) 45.[99] M.D. Rosen et al., Phys. Rev. Lett. 54 (2) (1985) 100.
[100] G.B. Rybicki, A.P. Lightman, Radiative Processes in Astrophysics, Wiley, New York, 1979.[101] E.E. Salpeter, Australian J. Phys. 7 (1954) 377.[102] D. Salzmann, Y.T. Lee, J. Quant. Spectrosc. Radiat. Transfer 54 (1995) 339.[103] M. Schlanges, Th. Bornath, D. Kremp, Phys. Rev. A 38 (4) (1988) 2174.[104] M. Schlanges, Th. Bornath, Physica A 192 (1993) 262.[105] K.S. Singwi, M.P. Tosi, R.H. Land, A. Sjolander, Phys. Rev. 176 (2) (1968) 589.[106] J.F. Springer, M.A. Pokrant, F.A. Stevens Jr, J. Chem. Phys. 58 (11) (1973) 4863.[107] L. Spruch, Revs. Mod. Phys. 63 (1991) 151.[108] J.C. Stewart, K.D. Pyatt, Astrophys. J. 144 (3) (1966) 1203.[109] S.R. Stone, J.C. Weisheit, J. Quant. Spectrosc. and Radiat. Transfer 75 (1986) 67.[110] K. Tsuji et al., J. of Non-Crystalline Solids 205—207, Part I, 295 (1996).[111] J.A.M. van der Mullen, Phys. Rep. 191 (1990) 109.[112] L. vanHove, Phys. Rev. 95 (1954) 249.[113] A.V. Vinogradov, V.P. Shevel’ko, Sov. Phys. JETP 44 (3) (1976) 542.[114] W. Gautschi, SIAM J. Numer. Anal. 7 (1) (1970) 187.[115] J.C. Weisheit, Advances in Atomic and Molecular Physics 25, Academic Press, New York, 1988, p. 101.[116] K.G. Whitney, P.E. Pulsifer, Phys. Rev. E 47 (1993) 1968.[117] B.L. Whitten, N.F. Lane, J.C. Weisheit, Phys. Rev. A 29 (2) (1984) 945.[118] N.C. Woolsey et al., Phys. Rev. E 53 (1996) 6396.[119] W.-C. Wu, A. Griffen, Phys. Rev. A 54 (1996) 4204.[120] S.M. Younger, in: T.D. Mark, G.H. Dunn (Eds.), Electron Impact Ionization, Springer, Wien, 1985.[121] J.K. Yuan, Y.S. Sun, S.T. Zheng, J. Phys. B 29 (1996) 153.[122] J.K. Yuan, Y.S. Sun, S.T. Zheng, Phys. Rev. E 53 (1) (1996) 1059.[123] A.G. Zilman, R. Granek, Phys. Rev. Lett. 77 (1996) 4788.[124] N.H. March, M.P. Tosi, Atomic Dynamics in Liquids, Dover, New York, 1976.[125] J.G. Chabrier, J. Phys. France 51 (1990) 1607.
