Nuclear Physics and Data for Material Analysis

ICTP Lecture Notes

NUCLEAR PHYSICS AND DATA

FOR MATERIAL ANALYSIS

19 - 30 May 2003

Editor

N. Paver

University of Trieste and INFN, Italy

NUCLEAR PHYSICS AND DATA FOR MATERIAL ANALYSIS First edition

Copyright c 2008 by The Abdus Salam International Centre for Theoretical PhysicsThe Abdus Salam ICTP has the irrevocable and indefinite authorization to reproduce and dissem-inate these Lecture Notes, in printed and/or computer readable form, from each author.

ISBN 92-95003-39-X

Printed in Trieste by The Abdus Salam ICTP Publications & Printing Section

iii

PREFACE

One of the main missions of the Abdus Salam International Centre for

Theoretical Physics in Trieste, Italy, founded in 1964, is to foster the growth

of advanced studies and scientific research in developing countries. To this

end, the Centre organizes a number of schools and workshops in a variety of

physical and mathematical disciplines.

Since unpublished material presented at the meetings might prove to

be of interest also to scientists who did not take part in the schools and

workshops, the Centre has decided to make it available through a publication

series entitled ICTP Lecture Notes. It is hoped that this formally structured

pedagogical material on advanced topics will be helpful to young students

and seasoned researchers alike.

The Centre is grateful to all lecturers and editors who kindly authorize

ICTP to publish their notes in this series.

Comments and suggestions are most welcome and greatly appreciated.

Information regarding this series can be obtained from the Publications Of-

fice or by e-mail to [email protected]. The series is published in-house

and is also made available on-line via the ICTP web site:

http://publications.ictp.it.

Katepalli R. Sreenivasan, Director

Abdus Salam Honorary Professor

vCONTENTS

A.F. Gurbich

Physics of the Interaction of Charged Particles with Nuclei . . . . . . . . . . . . . . 1

A.F. Gurbich

Differential Cross Sections for Elastic Scattering of Protons and Helions

from Light Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

M. Mayer

Rutherford Backscattering Spectrometry (RBS) . . . . . . . . . . . . . . . . . . . . . . . . . 55

M. Mayer

Nuclear Reaction Analysis (NRA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

O. Schwerer

Introduction to IAEA Nuclear Data Services . . . . . . . . . . . . . . . . . . . . . . . . . . 101

vii

Introduction

The Workshop on Nuclear Reaction Data for Science and Technology:

Materials Analysis was held at the Abdus Salam International Centre for

Theoretical Physics in May 19-30, 2003.

It was intended to present an extensive, and up-to-date, overview of the

applications of nuclear data to materials analysis and validation. Dedicated

lectures were given on prompt-charged particle analysis, particle-induced X-

ray spectrometry, nuclear reaction analysis, prompt gamma and neutron ac-

tivation analyses, ion beams analysis. The physics underlying such analyses,

and the determinations of the needed reliable nuclear data, were discussed

thoroughly. The Workshop included also basic lectures on material anal-

ysis, nuclear models, particle-matter interactions, and presentations of the

nuclear data libraries were given. A substantial amount of time was devoted

to practical exercises on simulation and nuclear data retrieval.

We here include lecture notes on: the physics of charged particle interac-

tions with nuclei; differential cross sections of protons and helions on light nu-

clei; nuclear reaction analysis; Rutherford backscattering spectrometry; the

IAEA Nuclear Data Services. For the other lectures, transparencies are avail-

able on-line on the web site http://www.ictp.it/pages/events/calendar.html/

The Workshop was organized by ICTP and IAEA, whose support is

gratefully acknowledged. We are thankful to the speakers for their excellent

presentations, and to the ICTP staff for their invaluable help in running this

activity.

For the Organizers (M. Herman, I. Vickridge)

Nello Paver

March, 2008

Physics of the Interaction of Charged Particles

with Nuclei

A.F. Gurbich

Institute of Physics and Power Engineering,

Obninsk, Russian Federation

Lectures given at the

Workshop on Nuclear Data for Science and Technology:

Materials Analysis

Trieste, 19-30 May 2003

LNS0822001

[email protected]

Abstract

The fundamentals of low energy nuclear reaction physics are consid-ered. The main discussion concerns the physical substance of particle-nuclear interaction phenomena. The corresponding mathematical re-lations are introduced with detailed explanation. All the necessary in-formation to understand the subject is immediately given in the text.Nuclear physics models used for the cross section calculations are de-scribed and how the model parameters are adjusted is discussed. Theaim of the lecture is to give a material scientist an insight into thenuclear physics theory in the limits, which are necessary in order tounderstand the principals of its application in the problem of nucleardata.

Contents

1 Introduction 5

2 Nuclear Forces 6

3 Rutherford Scattering 8

4 Potential Scattering Formalism 9

5 Compound Nucleus Model 15

6 Optical Model 19

7 R-matrix Theory 24

8 Deuteron Induced Reactions 25

9 Conclusion 27

References 29

Charged Particles Interactions with Nuclei 5

1 Introduction

Nuclear physics studying the structure and properties of atomic nuclei dates

back to the discovery of the atomic nucleus in 1911. A tremendous progress

has been achieved since then due to efforts of experimentalists and theoreti-

cians both in understanding the nature of the matter and in application of

the acquired knowledge in different areas. However, until now there is no

comprehensive theory which could describe all the experimentally observed

properties of particle-nucleus interaction from the first principles. This is

a consequence of the complexity of the nucleus as a physical object. Even

in case the nucleon-nucleon interaction force would be known, and it would

be possible to assume that this interaction does not depend on the pres-

ence of other nucleons, the many body problem of nuclear structure could

hardly be resolved. So different models were developed in order to describe

nuclear structure and mechanisms of the nuclear processes. These models

are based on some simplifications and usually each model is suitable only

for the description of a restricted number of nuclear phenomena. Compari-

son of the model calculations with experimental data followed by successive

improvements of the model is a typical approach in nuclear physics.

Ion Beam Analysis (IBA) acquires information about composition and

structure of the sample employing spectroscopy of products of the interaction

of accelerated ions with the nuclei containing in the sample. There are a

number of different IBA methods based on the registration of elastically

scattered particles or the products of nuclear reactions and a reliable source

of nuclear cross section data is needed for all of them except for Rutherford

backscattering for which the cross section can be calculated according to

known formula and for PIXE which is based on atomic rather than on nuclear

physics. When some 30 years ago the first steps were made in IBA this work

was carried out by nuclear physicists. Step by step IBA became more and

more routine and a new generation of scientists came in the field. Now

the present generation belongs mainly to a community of material science

and has, as a rule, no nuclear physics background. So far as a projectile-

nucleus interaction underlies the IBA methods some knowledge in the field is

necessary for material scientists. In addition it appeared that nuclear physics

theory is a powerful tool in the evaluation of the differential cross sections

for IBA [1]. A vice versa process to that made when nuclear models were

developed is now applied to evaluate measured cross sections on the base of

their consistency with nuclear models. The aim of this lecture is to give a

6 A.F. Gurbich

material scientist an insight into the low energy nuclear physics theory in

the limits, which are necessary in order to understand the principals of its

application in the problem of nuclear data.

2 Nuclear Forces

An atomic nucleus is a strongly bound system of nucleons located in a small

domain with a typical size of

R (1.1 1.5) A1/3fm (1fm = 1013cm) . (1)

Nucleons are held together inside nuclei due to nuclear forces. These

forces are strong attractive forces acting only at short distances. They pos-

sess property of saturation, due to which nuclear forces are attributed ex-

change character (exchange forces). Nuclear forces depend on spin, not on

electric charge, and are not central forces. The nature of the nuclear forces

has not yet fully been clarified.

Nuclear forces are said to be strong forces, in the sense that they are

at least 100 times greater than very strong Coulomb forces taken at short

nuclear distances of about 1 fm. The short range of nuclear forces leads to

a strict demarcation of the regions where only long-range Coulomb forces,

or only nuclear forces show up as the latter suppress the Coulomb forces at

short distances

The dependence of the force on the space coordinates is described by

means of the potential. The presence of one of the interacting bodies is

expressed through the potential as a function of the distance from the body

center and the force at the point r, directed from the first body to the second,

is found as a potential derivative with respect to the space coordinates at

this point.

Assuming nucleus is a uniformly charged sphere the electrostatic poten-

tial energy for the projectile-nucleus system can be written as

VC (r) =

{Zze2

r for r RZze2

2R

(3 r2

R2

)for r R (2)

where Z and z are charge numbers of the nucleus and the projectile respec-

tively.

Nuclear forces are also introduced through the potential energy of the

nucleon interaction. The positive potential creates repulsive forces, and the


negative potential creates attractive forces. Therefore, the potential energy

is positive if it corresponds to repulsive forces, and it is negative for attractive

forces. As a result, the potential energy of the point proton interaction with

the nucleus may be presented as is shown in Fig.1

V(r)

r2

T

R r

Figure 1: The nuclear and Coulomb potentials of the nucleus.

The Coulomb repulsion changes abruptly to attraction at the distance of

the radius of action of nuclear forces, i.e. at the boundary of the nucleus R.

The transition from repulsion to attractions proceeds, though rapidly but

continuously, in the region of the space coordinate R. So, to a certain degree

of accuracy the nuclear potential is pictured in the form of a square potential

well which is about 4050 MeV deep.For a charged projectile to reach the range of action of nuclear forces,

it should possess some kinetic energy T sufficient to overcome the Coulomb

potential barrier of height

BC =Zze2

R(3)

which is of the order of 1 MeV even in the interaction of singly charged

particles with the lightest nuclei.

According to quantum mechanics the transparency of the Coulomb bar-

rier is given by the formulae

D e

2

h

r2r1

2(VCT ) dr

(4)

where = MmM+m is reduced mass, r1 =R and r2 is derived from the relation

T = Zze2

r2.

8 A.F. Gurbich

Thus though electric charge of atomic nuclei hinders the initiation of nu-

clear reactions with low energy charged particles the reactions are still feasi-

ble at energies below the potential barrier. These are so-called under-barrier

reactions. The penetrability of Coulomb barrier increases very rapidly as T

approaches BC (Eq. (3)). Therefore, if T does not greatly differ from BC ,

the under-barrier reactions take place with remarkable probability.

3 Rutherford Scattering

If the interaction is solely due to electric forces the differential cross section of

elastic scattering is derived from energy and angular momentum conservation

laws using a concept of impact force. As far as the law of the interaction

(i.e. dependence of the force on the distance) is known it is possible to find

a dependence of the scattering angle on the impact parameter b which is

expressed in a non-relativistic case by the relation

tan =2Zze2

mv2b. (5)

d

b

db

b

+Ze

Figure 2: Scattering of a charged particle by the electric field of the atomic nucleus.

For a single unmoveable nucleus placed on the path of the ion beam of

intensity equal to N particles per square cm in 1 sec the number of the ions


scattered in the angle interval from to +d is dN = 2bdbN where b and

db are derived from eq.(5). The value

d =dN

N= 2b db (6)

is differential cross section which is expressed for the target containing n

nuclei per unit area by the well-known Rutherford formulae

d = n

(Zze2

mv2

)d

4sin4 2. (7)

Distinct of pure Coulomb scattering the cross section cannot be cal-

culated from an algebraic formulae in case of the nuclear interaction. As

far as nuclear forces are acting only at very short distances a classical ap-

proach to the consideration of the scattering process is no longer applicable.

The de Broglie postulates combining the corpuscular and wave properties

of microparticles served as the foundation for the theory of the motion of

projectiles and their interactions with nuclei.

4 Potential Scattering Formalism

In quantum mechanics, the state of a particle is described by the wave func-

tion (x, y, z) which, in the stationary case, depends only on the space coor-

dinates. The specific form of the wave function is determined by solving the

Schroedinger equation including the term expressing the particle interaction

law. The square of the wave function modulus is the distribution of the

probability for the particle to have any space coordinates (x, y, z). The wave

function does not indicate the sequence in which the space coordinates are

occupied with time, as is required when describing the motion of a classical

particle, because this has no meaning for microobjects. For microparticles,

the conception of moving along a trajectory analogous to the trajectory of

the classical particle does not exist. This circumstance is most clearly indi-

cated by one fundamental corollary of the de Broglie postulates known as

the Heisenberg uncertainty principle.

Nuclear scattering is considered below, first for the simplest case of the

projectile with no charge. According to quantum mechanics a particle state

is described by the wave function , which is obtained as a solution of the

wave equation. For the case of elastic scattering of spinless non-identical

10 A.F. Gurbich

particles the wave equation has the form of a Schroedinger equation with a

spherically symmetric potential V(r)

+2m

h2(E V ) = 0 , (8)

where

=2

x2+

2

y2+

z2. (9)

Prior to scattering, the wave function for the particle with a given

momentum p has the form of a plane wave:

= eikz, (10)

where k is a propagation vector

k =p

h=

1

. (11)

Here = /2, where is the de Broglie wavelength.

This function is a solution of eq.(8) in case of V(r) = 0, i.e. the equation

of the form

+2m

h2E = 0 (12)

and is normalized to correspond to the flux density equal to the projectiles

velocity.

In the course of scattering the plane wave interacts with the field of

nucleus V(r), that gives rise to a spherical wave divergent from the center of

the interaction. This wave has the form of

f ()eikr

r. (13)

Thus the last stage of the scattering process (after scattering) is depicted

by a superposition of the two waves - plane and spherical ones:

eikz +eikrf ()

r. (14)

Here is a scattering angle (see Fig.3); f() is an amplitude of the di-

vergent wave; the 1/r factor stands for decreasing of the flux in reverse

proportionality to the square of the distance.


The square of the modulus of the scattered wave amplitude is equal to

the differential cross section

d

d= |f ()|2 . (15)

This is easy to prove. By definition the differential cross section d is

equal to the fraction dN/N of the initial particles flux N scattered into the

given solid angle d. Assuming the density of particles in the primary beam

being equal to unity one obtains N = v, where v is the particles velocity. For

dN one obtains (see Fig.3)

r

dS

vdt

v

x

y

z

rd

rsin

d

Figure 3: To the definition of the scattering angles and cross section.

dN =

f () eikr

r

2

v r2sin d d (16)

Taking into account that velocity does not change in the elastic scattering

and that sindd=d one finally obtains that

d =dN

N=|f ()|2 r2v d

r2v= |f ()|2 d . (17)

ord

d= |f ()|2 . (18)

The angular distribution of the scattered particles is defined by the f()

function. For the quantitative analysis of the elastic scattering eq. (8) and

12 A.F. Gurbich

(12) are considered in spherical coordinates. The general solution of these

equations has the form of

=l=0

AlPl (cos)Rkl (r) , (19)

where Rl(r) is a radial wave function; Pl(cos) is Legendre polynomial

(P0=1, P1=cos, P2=(3cos-1)/2,...).

Far from the center of scattering (at large distances r) the radial function

for each of l can be represented in the form of two partial spherical waves

one of which is converging ei(krlpi

2) and the other is divergent ei(krl

pi

2).

For the initial stage depicted by a plane wave both waves have equal

amplitudes and

Rkl (r) = ei(krl pi

2) ei(krl pi2 ) . (20)

So the plane wave expressed through an expansion over Legendre poly-

nomials has the form of

eikz =l=0

(2l + 1) il

2ikrPl (cos)

[ei(krl

pi

2) ei(krl pi2 )

]. (21)

Here each of the spherical waves corresponds to the particles moving with

given orbital momentum l and is characterized by the angular distribution

Pl(cos) (see Fig.4).

Figure 4: Legendre polynomials angular dependence.

Suppose the projectile possesses kinetic momentum p and angular mo-

mentum l. Then from a comparison between classical and quantum mechan-

ical relations for the modulus of the angular momentum~l = p = hl(l + 1) (22)


it follows that

=h

p

l (l + 1) =

l (l + 1) , (23)

i.e. the initial beam behaves as if it were subdivided into cylindrical zones

with radii defined by eq.(20), as shown in Fig.5. A significant difference

between classical and quantum mechanical predictions for the scattering

process is evident: in a classical approach the particle having zero impact

parameter scatters straight in the back direction whereas angular distribu-

tion for the corresponding (l = 0) partial wave is isotropic.

l=1l=0

l=2l=3

Figure 5: The illustration of the initial beam splitting into the partial waves correspond-ing to the angular momenta.

In the process of scattering an additional divergent spherical wave arises.

So the ratio between convergent and divergent waves changes. The change of

the ratio can be formally taken into account by a coefficient at the divergent

wave

Rkl (r) = Slei(krl pi

2) ei(krl pi2 ) . (24)

In the case of the elastic scattering the fluxes for the convergent and

divergent waves should be equal to each other for each of l. This means that

|Sl|2 = 1. So the factor Sl can be written as

Sl = e2il (25)

where l is called a phase shift.

Physically the phase shift is explained by the difference of the wave ve-

locity in the presence of the nuclear forces field and outside the nucleus as

is illustrated in Fig.6.

The partial wave after scattering has then the form of

Rkl (r) = ei(krl pi

2+2l) ei(krl pi2 ) . (26)

14 A.F. Gurbich

Figure 6: Formation of the phase shift of the outgoing wave relative to the incident one.

The solution of eq.(8) for the final stage of scattering is

eikz + f ()eikr

r=

l=0

(2l + 1) il

2ikrPl (cos)

[Sle

i(krl pi2) ei(krl pi2 )

]. (27)

The following relation between the scattering amplitude and phases can

be derived

f () =1

2ik

l

(2l + 1)(e2il 1

)Pl (cos) . (28)

Summing up, the differential cross section for elastic scattering is calcu-

lated from eq.(15), the scattering amplitude being expressed through phase

shifts l according to eq.(28). The phase shifts for partial waves are calcu-

lated by resolving Schroedinger equation (8) with assumed potential V(r).

This equation is split into angular and radial ones. The asymptotic general

solution for the radial equation is

Rkl 2

1

rsin

(kr l

2+ l

). (29)

The phase shifts l are defined by the edge conditions. The phase shifts

are functions of k and l but do not depend on the scattering angle.

If the projectile is charged it interacts with combined Coulomb and nu-

clear fields of the target nucleus. The relation for the scattering amplitude

is then

f () = fC () +1

2ik

l=0

(2l + 1) (Sl 1) e2ilPl (cos), (30)


where fC() and l are amplitude and phase shift of the Coulomb scattering

respectively.

The Sl values defined by eq.(25) can be considered as elements of some

diagonal matrix which is called a scattering matrix. In case of pure elastic

scattering phase shifts l are real numbers. However they become complex if

inelastic scattering is also present in the scattering process. This corresponds

to decreasing of the amplitude of the divergent waves i.e. |Sl| < 1.In case a projectile possesses non zero spin all the ideology described

above is retained valid. However, the equations become more complicated

since radial wave equation splits into (2s+1) equation. Suppose projectiles

are protons which spin is 1/2. Then the spin of bombarding particles may

be combined with angular momentum l in two ways to produce the total

angular momentum j=l1/2.The proton elastic scattering differential cross section is obtained in this

case through resolving Schroedinger equations for partial waves as d/d =

|A ()|2 + |B ()|2 , the scattering amplitudes A() and B() being definedby the following relations

A () = fC () +1

2ik

l=1

[(l + 1)S+l + lS

l (2l + 1)]exp (2il)Pl (cos);

B () = 12ikl=0

(S+l Sl

)exp (2il)P

1l (cos),

(31)

where fC() is an amplitude of Coulomb scattering, l are Coulomb phase

shifts, Pl(cos) are Legendre polynomials, P1l (cos) are associated Legendre

polynomials, S+l and S

l are scattering matrix elements for different spin

orientation, k is a wave number.

The above representation of the elastic scattering process produces the

cross section with a smooth dependence on energy. Some rather broad res-

onances called shape (or size) resonances are observed only at energies

when conditions for standing waves to form in the nucleus potential well are

fulfilled (Fig.7). These resonances correspond to the single particle states in

the potential well.

5 Compound Nucleus Model

The mechanism of scattering, considered so far, is called direct or potential

scattering since it proceeds through direct interaction of a single bombarding

16 A.F. Gurbich

3.0 3.5 4.0 4.5 5.0 5.5 6.00

50

100

150

200

250

300

350

400

450

16 O(p,p0)

VR =57.9 MeV VR =58.9 MeV

Cros

s se

ctio

n, m

b/sr

Energy, MeV

Figure 7: A shape (size) resonance and its dependence on the potential well depth VR.

particle with a potential well representing a nucleus. Nuclear interaction

at low energies can proceed also in two stages through the mechanism of a

compound nucleus (Fig.8). The first stage of the interaction is the absorption

of the bombarding particle by the target nucleus and the production of an

intermediate, or compound, nucleus. The compound nucleus is always highly

excited because the absorbed particle brings both its kinetic energy and the

binding energy of the absorbed nucleons into the produced nucleus. The

second stage is the decay of the compound nucleus with the emission of this

or that particle. The original particle may always be such a particle, and

here again the original nucleus is formed. A typical lifetime for a compound

nucleus is 1014 sec that is very long as compared with the time of directinteraction defined as a time (10231021 sec) needed for the bombarding

particle passes through the region occupied by the nucleus potential well.

For the case of light nuclei the compound nucleus has discrete energy

levels as shown in Fig.9 and so the cross section of the elastic scattering

through this mechanism has a resonance structure. Because of the relatively

long lifetime and due to the uncertainty relation (written in energy-time

coordinates it is E t h) the widths of the compound nucleus levelsare rather small. So are the widths of the resonances observed in the cross

section.

One of the ways to take resonance scattering into account is to add Breit-


Projectile

Shap

e elas

tic

Scatt

ering

Absorbtion

CompoundNucleus

Elastic

Scatt

ering

via Co

mpou

nd

Nucle

us

CompoundNucleus Decay

Figure 8: The particle-nucleus interaction channels.

Wigner resonance terms to the diagonal elements of the scattering matrix:

Sl = exp(2il

) [exp

(2l

)+ exp (2ip)

ip

E0 E 12 i

], (32)

where l + i

l is the off-resonance nuclear phase shift describing the elastic

scattering of particles of energy E from spin zero nuclei. The quantities E0, ,

and p are the energy, total width and partial elastic width, respectively. The

subscript l is the relative angular momentum of the proton and the target

in units of h. The plus and minus signs in superscripts refer to summing of

orbital and spin momenta with different mutual orientation. The quantity

p is a resonance phase shift.

Because of the interference between potential and resonance scattering

the excitation function has a typical structure with resonances pictured as

dips and bumps rather than as Breit-Wigner functions (Fig.10).

In case of the nuclei of middle and heavy mass the energy level den-

sity is high and the width of resonances exceeds the distance D between

them, D at a relatively low excitation energy (see Fig.9). Then acontinuous background produced by the scattering via compound nucleus

with overlapped levels is observed in the scattering yield. This background

can be evaluated in the framework of a statistical model. It is assumed in

this model that the compound nucleus decay is independent from the way

the compound nucleus was created (yet all the conservation laws - energy,

momentum, parity etc. - naturally are fulfilled). If the number of the over-

lapped levels is great enough it becomes possible to depict the properties of

18 A.F. Gurbich

E1

E2

E3

E i

E i +1

(E)

D <

E

D (E)

E

Figure 9: A scheme of nuclear levels of light (left) and heavy (right) nuclei.

the compound nucleus by averaging over excited states. Due to the averag-

ing the quantum mechanical effects disappear and the semiclassical approach

using statistical physics methods becomes possible. Computer codes based

on the Hauser-Feshbach formalism [2] are widely in use for such calculations

(see e.g. [3]). The input data needed are level-density parameters and trans-

mission coefficients. The level-density parameters can be found e.g. in a

Reference Input Parameter Library (RIPL) [4] produced in the result of the

recent IAEA coordinated research project. The transmission coefficients are

calculated using the optical model discussed below.

Even at high excitation energy when compound nucleus levels are over-

lapped, there are still some sharp resonances superimposed on a continuous

background. These resonances are caused by the population in the com-

pound nucleus (Z,N) of so-called isobaric analogue states which have a rather

simple structure, because they look alike low-lying states in the nucleus hav-

ing Z-1 proton and N+1 neutron; a proton takes on the role of a neutron and

vice versa. Because of the independence of nuclear forces on electric charge

these nuclei are similar, with states being displaced due to the difference in

the Coulomb energy between (Z,N) and (Z-1,N+1) nuclei.

In the intermediate case when D the so-called Ericson fluctuations ofthe cross sections are observed [5]. These fluctuations are uncorrelated over

the energy, angle and reaction channel. Thus only statistical properties of

the fluctuations not detailed structure of the cross section can be calculated.


0

200

400

600

800

1000

Potential Scattering

Cros

s se

ctio

n, m

b/sr

0

200

400

600

800

P1/2

S 1/2 Breit-Wigner Resonances

0.5 1.0 1.5 2.00

200

400

600

800Interference of Potential and Resonance Scattering

Energy, MeV

Figure 10: A pattern of the excitation function in case of resonance scattering.

A comprehensive review of compound nuclear reactions can be found

elsewhere [6].

6 Optical Model

If nuclear reactions contribute to the total cross section along with elastic

scattering this should be taken into account. Though some progress has

been achieved in microscopic theory of nuclear reactions it is practical to

apply a phenomenological approach consisting in consideration of the pro-

jectile interaction with the nucleus as a whole, the nucleus being represented

by an appropriate potential. The potential parameters are found through

fitting theoretical calculations to the available experimental data. To make

this approach more physical the potential shape is derived from the known

features of the nucleon-nucleon interaction and from distributions of matter

and charge in the nucleus.

In the so-called optical model [7] nucleus is represented by means of a

complex potential. The interaction of the projectile with the nucleus is then

reduced to de-Broglies wave refraction and absorption by an opaque sphere.

The name of the model originates from the formal analogy with the light

plane wave passing through a semitransparent sphere.

Also refraction and absorption of the light is described by a complex

20 A.F. Gurbich

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40

2

4

6

8

10

12

14

14N(p,p0)

Experiment Calculations with RIPL OP

d

/d

R

E, MeV

Figure 11: Comparision of the optical model calculations using Reference Input Param-eters Library [4] with experimental data for proton scattering from nitrogen.

index

n = nr + ia

the complex potential of the form

U = V + iW (33)

is used to take into account scattering and absorption of the projectile by the

nucleus. The real part of the potential is responsible for scattering due to

the direct mechanism whereas the imaginary part stands for the absorption.

It is implied in the model that the compound nucleus formed in the result

of the absorption of the incident particle is excited to such an extent that

its energy levels are completely overlapped.

The standard form of the optical potential is as follows:

U (r) = UC (r) + UR (r) + iU1 (r) + Uso (r) , (34)

where UC is the Coulomb potential defined by eq.(2),

UR (r) = VRfR (r) (35)

U1 (r) = 4aiWDdfi (r)

dr(36)


Uso =

(h

mpic

)2Vso

1

r

dfsodr

l s (37)

fR (r) =

[1 + exp

(r Rxax

)]1(38)

Rx = rxA1/3 . (39)

The potential terms represent, in sequence, the real central volume po-

tential of the depth VR, the imaginary central surface potential of the depth

WD (volume absorption is negligible at low energies), and the surface spin-

orbit potential of the depth Vso, while fx(r) is a Saxon-Woods formfactor,

Rx is a half value radius, ax is a diffusivity parameter, A is a target mass

number, m is a -meson mass, c is light velocity, l and s denote angular

momentum and spin operators respectively.

1.0 1.5 2.0 2.5 3.00

50

100

150

200

250

300

28Si(p,po)

c.m.

=165.5o WD =0.0 MeV WD =0.1 MeV

Cros

s se

ctio

n, m

b/sr

Energy, MeV

Figure 12: The dependence of the calculated cross sections on the imaginary potentialin case of low energy proton scattering from silicon.

The optical model does not take into account specific features of a par-

ticular nucleus. Thus resonances (except for single particle ones) are not

reproduced in the framework of this model. Resonances are specific features

of a particular compound nucleus whereas the optical model describes nu-

clear matter as a whole and so the optical potential parameters have only

22 A.F. Gurbich

a general trend on mass number and energy. Due to more than 30 years

of application of the optical model the general features of phenomenological

optical potential parameters are well established. So-called global sets of

parameters obtained through the optimization procedure that was based on

a wide collection of experimental data were developed [8] - [10]. A Reference

Input Parameter Library [4] also contains the recommended optical model

parameters. Generally the results obtained with optical model for scattering

of nucleons for nuclei with mass number A>30 are quite reliable in the en-

ergy range of 10


as illustrated in Fig.(12). As far as the imaginary part of the potential is

equal to zero the cross section is represented by S-matrix formalism rather

than by the optical model.

1p1p1/21p3/2

1s1/21s

Figure 13: Splitting of the levels due to spin-orbital interaction.

It is interesting to note that the differential cross section at higher en-

ergies is insensitive to the spin-orbit potential (38) and it influences only

polarization data. In the region of separated resonances such is not the case.

Because of the spin-orbit interaction the energy levels split with respect to

the total angular momentum, as shown in Fig.13.

1.0 1.5 2.0 2.5 3.00

50

100

150

200

250

300

28 Si(p,p0) 1p3/2

1p1/2

c.m.

=165.5 V

so=4.8 MeV

Vso

=5.0 MeV

Cros

s se

ctio

n, m

b/sr

Energy, MeV

Figure 14: Dependence of the distance between split resonances on the spin-orbitalpotential..

As a result the distance between split resonances strongly depends on

the spin-orbit potential (see Fig.14).

24 A.F. Gurbich

7 R-matrix Theory

The R-matrix theory describes a nuclear reaction proceeding via formation of

intermediate states of the compound nucleus in terms of some set of states

that can be associated with those of the compound nucleus. If the wave-

function and its derivative are known at the boundary of the nucleus it can

be found everywhere outside the nucleus. The idea of the R-matrix approach

is that the scattering matrix is expressed through R-matrix which is defined

to connect values l with its derivative at the nucleus boundary

l(a) = Rla

(dldr

)r=a

. (40)

Matrix elements Rl are expressed as

Rl =

2l,E E

(41)

where

l, =

(h

2ma

)1/2l,(a) . (42)

Functions l, correspond to actual states E of the nucleus. Quantities

l,, called reduced width amplitudes, are connected with the energy width

of real states (2l, l,). It is shown in the theory that the cross-sectioncan be written in terms of the R-matrix. The differential cross section for the

scattering of charged particles is a sum of three terms which correspond to

pure Coulomb scattering, resonance and reaction scattering and interference

between Coulomb and resonance scattering. Potential scattering is taken

into account through the so-called hard sphere scattering phase shift. The

R-matrix theory is a formal one in the sense that it uses expansion of the

wave function over eigenvalues regardless of the nature of the states.

Application of the R-matrix theory is quite simple in the case of a

zero spin target nuclei, especially if the distances between compound nu-

cleus levels are great enough and level interference effects are so negligi-

ble. The parameters of nucleus levels needed for the calculations can be

taken from Ajzenberg-Seloves compilation for A


1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

50

55

60

65

70

75

80

85

90

95

27Al(p,p0)27Al

= 1 keV

Ipitarget = 5/2

+

J pi =2 -

l=1 s=2 s=3

d

/d

, m

b/sr

Energy, keV

Figure 15: Effects of the channel spin on a resonance curve shape.

of the analysis in this case significantly increases. For a target of spin Itand a projectile of spin Ip the two spins are coupled to form a channel spin

s. This channel spin is then combined with the relative orbital momentum

l to form the spin of the compound nucleus state. For proton scattering

there are two values of the entrance channel spin s = It 1/2 and for elasticscattering both s and l mixing are possible. With such mixing the measured

elastic scattering cross section includes reaction terms with s 6= s and l 6= lwhere primed values correspond to the exit channel. The spin of the channel

remarkably influences the shape of a resonance curve, as shown in Fig.15.

A comprehensive review of R-matrix theory is presented by Lane and

Thomas [13]. A practical description of the analysis of resonance excitation

functions for proton elastic scattering is given in Ref.[14].

8 Deuteron Induced Reactions

Reactions due to deuterons possess some specific features. The deuteron

consists of one proton and one neutron, its binding energy is 2.22 MeV, or

about l MeV per nucleon, which is much less than 8 MeV, the mean binding

energy of the nucleon in most nuclei. In addition, the mean distance at

26 A.F. Gurbich

which nucleons are spaced from each other in the deuteron composition is

relatively long ( 4 fm ). The particle possessing such properties proves tobe able to interact with nuclei not only with the production of a compound

nucleus but by direct interaction. If in the deuteron nucleus collision, a

compound nucleus is produced with the capture of both nucleons, then its

excitation energy appears to be very high due to the great difference in the

binding energies of two nucleons in the nucleus and in the deuteron, i.e.,

about 14 MeV. Therefore, all the reactions due to deuterons (d, p), (d,n),

(d,), are exoenergic and have high yields. Apart from the production of

the compound nucleus, the reactions (d,p) and (d,n) can proceed in some

other way. Because of the weak binding of nucleons in the deuteron and of

the relatively great distance between them, the deuteron-nucleus interaction

may result in the absorption of only one nucleon, while the other nucleon

will remain beyond the boundaries of the nucleus and continue its motion

predominantly in the direction of the initial flight. If the kinetic energy of

the deuteron is lower than the height of the potential barrier of the nucleus,

then the yield of the (d,p)-reaction turns out to be comparable with the

yield of the (d, n)-reaction for light and intermediate nuclei, and for heavy

nuclei it is even several times higher than the latter. Such a behaviour of

the yields of the (d, p)- (d,n)-reactions contradicts the compound nucleus

mechanism because in the decay of the compound nucleus the emission of

protons is always more difficult than that of neutrons, and especially in the

case of heavy nuclei.

For the reactions induced by low energy deuterons at light nuclei it is as-

sumed that the main contribution to the cross section of the process is given

by the following three mechanisms: direct stripping when incident deuteron

leaves one of its nucleons in the target nucleus, resonant mechanism and in

some cases a compound nucleus statistical mechanism. It is accepted, that

the complete amplitude T of process is T=D + R, where D is the amplitude

of the direct process of stripping, which is calculated within the framework

of a method of deformed waves, and R-is the amplitude of resonant pro-

cess. The statistical compound mechanism contribution if any is incoherent

and it may be simply added. Complete and partial width of formation and

desintegration of resonances in the system, which are necessary in order to

calculate the amplitude of R, are defined by fitting the model predictions

to the available experimental cross sections of elastic deuteron scattering

and (d,p)-reaction. The satisfactory description of the experimental data

for 12C(d,po)13C reaction is feasible (see Fig.16). However, for a reliable


description of a whole set of (d,p)-reaction data a development of the model

in several directions is required.

0.8 1.0 1.2 1.4 1.6 1.8 2.00

20

40

60

80

100

120

lab=165o

12C(d,po)13C

d

/d

c.m

. (m

b/sr)

Energy (MeV)

Figure 16: Theoretical description of the 12C(d,p0)13C reaction cross section.

9 Conclusion

Nowadays low energy nuclear physics is a sufficiently studied field. Reac-

tion mechanisms are known and appropriate models have been developed.

However, satisfactory agreement between measured data and theoretical cal-

culations, which is sufficient as a rule in order to support a model, does not

provide a reliable base for cross section a priori prediction. In addition

nuclear reaction models use many adjustable parameters. Though some sys-

tematics and global sets of these parameters exist, fitting is always needed

in order to represent a particular cross section. Moreover, in some impor-

tant IBA cases reaction mechanisms are in general known but there is no

code which provides necessary calculations. Although nuclear physics theory

cannot provide sufficiently accurate cross section data when the calculations

are based simply on first principals, theory does provide a powerful tool for

data evaluation.

28 A.F. Gurbich

Acknowledgments

The author would like to thank Dr. N.N. Titarenko for useful comments.


References

[1] A.F. Gurbich, Evaluation and calculation of charged particle nuclear

data for ion beam materials analysis. In: Long term needs for nuclear

data development. INDC(NDS)-428 (2001) IAEA, Vienna p. 51.

[2] W. Hauser and H. Feshbach, Phys. Rev. 87 (1952) 366.

[3] P.G. Young, E.D. Arthur and M.B. Chadwick, Comprehensive nuclear

model calculations: Theory and use of the GNASH code, in Proc. of

the IAEA Workshop on Nuclear Reaction Data and Nuclear Reactors -

Physics, Design and Safety, Trieste, Italy, April 15 - May 17, 1996, p.

227, A. Gandini and G. Reffo, Ed. (World Scientific, Singapore, 1998).

[4] Handbook for Calculations of Nuclear Reaction Data. Reference input

parameter library. IAEA-TECDOC-1034 (IAEA, Vienna, 1998).

[5] T. Ericson and T. Mayer-Kuckuk, An. Rev. Nucl. Sci. 16, 183 (1966).

[6] P.E. Hodgson, Rep. Prog. Phys. 50, 1171 (1987).

[7] P.E. Hodgson, The Optical Model of Elastic Scattering (Clarendon

Press, Oxford, 1963).

[8] R.L. Varner, W.J. Thompson, T.L. McAbee, E.J. Ludwig and T.B.

Clegg, Phys. Rep. No.2, 57 (1991).

[9] F.D. Becchetti and G.W. Greenlees, Phys. Rev. 182, 1190 (1969).

[10] F.G. Perey, Phys. Rev. 131, 745 (1963).

[11] O. Bersillon, Centre dEtudes de Bruyeres-leChatel Note CEA-N-2227

(1981).

[12] S. Kailas, M.K. Mehta, S.K. Gupta, Y.P. Viyogi and N.K. Ganguly,

Phys. Rev. C 20, 1272 (1979).

[13] A.M. Lane and R.G. Thomas, Rev.Mod.Phys. 30, 257 (1958).

[14] R.O. Nelson, E.G. Bilpuch and G.E. Mitchell, Nucl. Instr. Meth. A 236,

128 (1985).

Differential Cross Sections for Elastic Scattering of

Protons and Helions from Light Nuclei

A.F. Gurbich

Institute of Physics and Power Engineering,

Obninsk, Russian Federation



Materials Analysis


LNS0822002

[email protected]

Abstract

The present status of the nuclear data for IBA is reviewed. Theconception of a so-called actual Coulomb barrier is shown to be un-availing. The principals of an evaluation procedure are described. Theresults obtained in the evaluation of the cross sections for IBA are dis-cussed. It is shown that the evaluation of cross sections by combininga large number of different data sets in the framework of the theoret-ical model enables excitation functions for analytical purposes to bereliably calculated for any scattering angle. A cross section calculatorSigmaCalc is presented.

Contents

1 Introduction 35

2 About the Actual Coulomb Barrier 37

3 Present Status of the Nuclear Data for IBA 38

4 Evaluation of the Cross Sections for IBA 40

4.1 The elastic scattering cross section for 1H+4He . . . . . . . . 41

4.2 Proton elastic scattering cross sections for carbon . . . . . . . 42

4.3 Proton elastic scattering cross section for oxygen . . . . . . . 44

4.4 Proton elastic scattering for aluminum . . . . . . . . . . . . . 45

4.5 Proton elastic scattering cross section for silicon . . . . . . . . 46

4.6 The cross section for elastic scattering of 4He from carbon . . 48

5 SigmaCalc - A Cross Section Calculator 50

6 Conclusion 51

References 53

Protons and Helions Elastic Scattering from Light Nuclei 35

1 Introduction

The utilization of proton and 4He beams with energies at which the elastic

scattering cross section for light elements, conditioned by nuclear rather

than electrostatic interaction, has become very common over the past years.

There are a number of benefits in the use of the elastic backscattering (EBS)

technique at higher-than-usual energies. First of all at higher energies

light ion elastic scattering cross section for light elements rapidly increases

whereas it still follows close to 1/E2 energy dependence for heavy nuclei.

Thus high sensitivity for determination of light contaminants in heavy matrix

is achieved (Fig.1). Besides, a depth of sample examination is enhanced.

However the cross section at these energies is no longer Rutherfordian and

consequently it cannot be calculated from an analytical formulae.

50 100 150 2000

500

1000

1500

2000

2500

3000

3500

O

FeE p=4.1 MeV

Measured Calculated for Rutherford

cross section

Coun

ts/C

hann

el

Channel Number

Figure 1: The EBS spectrum of protons scattered from an oxidized steel sample. Theenhancement of the oxygen signal due to non-Rutherford cross section is clearly seen.

At enhanced energies the excitation functions for elastic scattering of

protons and 4He from light nuclei have, as a rule, both relatively smooth

intervals convenient for elastic backscattering analysis and strong isolated

resonances suitable for resonance profiling. The linear dependence of the

registered signal on the atomic concentration and on the cross section results

in obvious constrains on the required accuracy of the employed data. It

is evident that the concentration cannot be determined with the accuracy

36 A.F. Gurbich

exceeded that of the cross section. Thus in order to take advantage of the

remarkable features of EBS the precise knowledge of the non-Rutherford

cross sections over a large energy region is required.

Since over the past few years non-Rutherford backscattering has been ac-

knowledged to be a very useful tool in material analysis the differential cross

sections for elastic backscattering of protons and helions from light nuclei

have become among the most important data for IBA. Cross section mea-

surements were reported for carbon, nitrogen, oxygen, sodium, magnesium,

aluminum, and many other nuclei. At the enhanced energy the cross-section

becomes non-Rutherford also for middleweight nuclei (see Fig.2). So not

only light element cross sections are needed for backscattering analysis but

also knowledge of energy at which heavy matrix scattering is no longer pure

RBS is important.

3700 3800 3900 4000 4100 4200 4300

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

56 Fe(p,po)56 Fe

lab =165o

d

/d

Ru

th

Energy, keV

Figure 2: The differential 56Fe(p,p0)56Fe cross section.

Although the officially accepted list of required nuclear data for IBA

does not exist it is a safe assumption that such a list should comprise first

of all (though not only) the differential cross sections for proton and 4He

non-Rutherford elastic scattering.


2 About the Actual Coulomb Barrier

In a series of papers by Bozoian and Bozoian et al. a classical model has

been developed to predict an energy threshold of cross section deviation from

Rutherford formulae. From the nuclear physics point of view it is evident

that this model treats the projectile-nucleus interaction in a quite irrelevant

way that cannot provide realistic results. It is occasionally consistent with

experimental data solely because of the fact that Coulomb barrier height is

involved in the model. On the other hand this model definitely disagrees

with experiment that was clearly shown in several papers. The detailed

discussion of the validity of Bozoians approach from the theoretical point of

view would lead far beyond the scope of the present lecture. It is sufficient

only to note that the classical approach is a priori inadequate in case of

resonance scattering whereas resonances often strongly influence the cross

section for light and middleweight nuclei. Hence, as far as an appropriate

physics is not involved one cannot rely upon the results obtained using this

model in any particular case.

Another attempt to produce more realistic results has been published by

Bozoian in the Handbook of Modern Ion Beam Materials Analysis [1]. The

prediction of a so-called actual Coulomb barrier is grounded in the Handbook

on the optical model calculations. Unfortunately the utility of these data is

doubtful since a scattering angle for which the results have been obtained is

not known. Nor is quoted optical model parameters set that was used in the

calculations. It is known that the results of calculations strongly depend on

both of these input data. Besides it should be noted that the optical model

at low energy is not applicable in case of light nuclei (see [2]).

An example of the Handbook prediction of the proton energy at which

the scattering cross section deviates by 4% from its Rutherford value is

shown in Fig.3 by a dash vertical line. It is evident that the prediction is

unrealistic. The 4 percent deviation is expected at Ep=1.63 MeV, according

to the Handbook. In reality the cross section deviates by 4 percent from

pure Coulomb scattering at 1.3 MeV for the 170 scattering angle and is

about 40 percent lower than the Rutherford value at the 1.63 MeV point

indicated in the Handbook.

Summing up it should be concluded that introducing into practice the

conception of a so-called actual Coulomb barrier was irrelevant and the es-

timates based on this conception are misleading. The interaction of the

accelerated ions with combined electrostatic and nuclear fields is a process

38 A.F. Gurbich

800 1000 1200 1400 1600 18000

100

200

300

400 28 Si(p,p 0) lab =170o

A m 9 3 Ra85Sa93He00TheoryRutherford

d

/d

lab

(m

b/sr)

Energy (keV)

Figure 3: Comparison of the actual deviation of the cross section from Rutherford lawwith the prediction based on the concept of the actual Coulomb barrier (vertical dashline).

that is actually governed by quantum mechanics laws and it cannot be re-

duced to any classical model. On the other hand each nucleus has its unique

structure that influences projectile-nucleus interaction and so it is impossi-

ble to reliably predict a priori the energy at which the cross section starts

to deviate from Rutherford value. Unfortunately it becomes actually a rule

to refer to the actual Coulomb barrier in all the papers dealing with non-

Rutherford backscattering. It can hardly be imagined what merit has a

prediction that is only occasionally in agreement with reality. An argument

that it is the only available indication on the non-Rutherford threshold is

sometimes adduced. However, incorrect knowledge is worse than lack of

knowledge (A. Diesterweg).

3 Present Status of the Nuclear Data for IBA

To provide the charged particles cross sections for IBA is the task that re-

sembles the problem of nuclear data for other applications in all respects

save one. Differential cross sections rather than total ones are needed for

IBA. Whatever actual needs the requirements of analytical work favor the


use of only those reactions for which adequate information already exists.

Many differential nuclear reaction cross sections were measured in the fifties

and sixties. Most of those data are available from the literature but mainly

as graphs. Besides, the energy interval and angles at which measurements

were performed are often out of range normally used in IBA. Therefore,

although a large amount of cross section data seems to be available, most

of it is unsuitable for IBA. Because of lack of required data many research

groups doing IBA analytical work started to measure cross sections for their

own use every time when an appropriate cross section was not found. The

Internet site SigmaBase was developed for the exchange of measured data.

Previously published cross sections extracted from more than 100 references

were compiled in the PC-oriented database NRABASE. A great amount of

information published only in graphical form was digitized and presented

in NRABASE as tables. Accumulation of rough measured cross sections

in the database is only the first step towards establishing a reliable basis

for computer assisted IBA. The analysis of the compiled data revealed nu-

merous discrepancies in measured cross section values far beyond quoted

experimental errors. These discrepancies arise from inaccuracies in the ac-

celerator energy calibration, a cross section normalization procedure, etc. In

most cases the differential cross sections were measured at one selected scat-

tering angle and therefore they may be immediately used only in the same

geometry. Due to historical reasons charged particles detectors are fixed in

different laboratories at different angles in the interval approximately from

130 to 180. Meanwhile, the cross section may strongly depend on a scat-

tering angle. Fortunately in the field of IBA interests the mechanisms of

nuclear reactions are generally known and appropriate theoretical models

with adjustable parameters have been developed to reproduce experimental

results. Besides other advantages the extrapolation over all the range of

scattering angles can then be performed on the clear physical basis. Appli-

cability of such an approach for the evaluation of the proton non-Rutherford

elastic scattering cross sections has been clearly demonstrated in a number

of papers. Though in some cases measured data were parameterized using

empirical expressions it is essential that the parametrization should repre-

sent cross sections not only at measured energies and angles but also provide

a reliable extrapolation over all the range of interest. So a theoretical eval-

uation of the cross sections grounded on appropriate physics seems to be

the only way to resolve the problem of nuclear data for IBA. Generally, an

evaluation leans as far as possible on experimental data. But these data are

40 A.F. Gurbich

often insufficient, incoherent and sparse. This is the reason for which nuclear

reaction models are used to calculate cross sections taking advantage of the

internal coherence of the models.

The IBA groups often apply thick target measurements in order to deter-

mine absolute cross section against internal standard for which Rutherford

scattering is assumed. This method needs none of the quantities usually

defined with significant inaccuracy such as particle fluence or detection ge-

ometry but in this case errors are introduced by use of stopping power data.

Hence in both cases (thin and thick target measurements) a comparison of

the results obtained by different groups should be done in order to produce

reliable recommended cross section data.

Summing up the present status of the nuclear data for IBA is as follows.

Some raw measured data have been compiled in SigmaBase [4], NRABASE

(PC-oriented, [5]), handbooks (see e.g. Ref.[3] and Ref.[1] and note that

elastic scattering cross sections shown in the last handbook as graphs are

overestimated by about 10 percent in many cases), Nuclear Data Tables

(Ref.[6]), and internal reports (the most complete is Ref.[7]). Some cross

sections from SigmaBase (mainly measured in USA) were incorporated into

the EXFOR library maintained by IAEA. Strange enough, the information

compiled in different sources has never been compared.

4 Evaluation of the Cross Sections for IBA

The evaluation procedure consists of the following generally established steps.

First, a search of the literature and of nuclear data bases is made to compile

relevant experimental data. Data published only as graphs are digitized.

Then, data from different sources are compared and the reported exper-

imental conditions and errors assigned to the data are examined. Based

on this, the apparently reliable experimental points are critically selected.

Free parameters of the theoretical model, which involve appropriate physics

for the given scattering process, are then fitted in the limits of reasonable

physical constrains. The model calculations are finally used to produce the

optimal theoretical differential cross section, in a statistical sense. Thus, the

data measured under different experimental conditions at different scatter-

ing angles become incorporated into the framework of the unified theoretical

approach. The final stage is to compare the calculated curves to the experi-

mental points used for the model and to analyze the revealed discrepancies.

If no explanation for any disagreement can be found, then a new measure-


ment of the critical points should be made. The following scheme outlines

the procedure (Fig.4).

Critical Analysis

Data Compi lation

Theoretical Calculations

Analysis of Discrepancies

Cross SectionMeasurements

BenchmarkExperiments

Data Dissemination

Figure 4: The flowchart of the evaluation procedure.

The recommended differential cross sections are produced in result of

the evaluation. These data are based on all the available knowledge both

experimental and theoretical and so are reliable to the most possible extent.

4.1 The elastic scattering cross section for 1H+4He

This cross section is used in IBA for the analysis of helium by proton

backscattering and hydrogen by elastic recoil detection (ERD). It is evi-

dent that in the center of mass frame of reference the scattering process

is identical in both cases. Elastic scattering of protons by 4He was thor-

oughly studied in Ref.[8]. Based on different sets of experimental data the

R-matrix parametrization of the cross sections was produced. More recent

measurements reported in Ref.[9] and Ref.[10] are in reasonable agreement

with the theory. The analysis reported in [11] also supported the obtained

R-matrix parametrization. Thus for practical purposes the 1H+4He cross

42 A.F. Gurbich

section can be calculated using R-matrix theory with parameters listed in

table 8 of Ref.[8]. To calculate the cross sections for kinematically reversed

recoil process p(4He,p)4He, the identity of the direct and inverse processes

in the centre of mass frame of reference is utilized. The results of such

calculations along with available experimental data are shown in Fig.5.

1 2 3 4 5 6 7 8 9

200

400

600

800

1000

1200

1400

Recoil angle 40 o

1H( 4He, 1H)

Wa86 Bo01 Ya83 Na85 Evaluation

d

/d

, m

b/sr

Energy, MeV

Figure 5: The proton elastic recoil cross section at the laboratory angle of 40o as afunction of 4He laboratory energy.

The ratio between recoil cross section and scattering cross section in the

laboratory frame is given by the following relation (see Ref. [9] for details).

ERD()

EBS()= 4 cos cos(cm )

sin2

sin2 cm

4.2 Proton elastic scattering cross sections for carbon

The evaluation of this cross section was described in Ref. [12]. The com-

parison of the obtained results with posterior measurements was made in

Refs.[13]-[15]. The reliability of the theoretical cross sections was confirmed

in all cases. The only significant difference reported in the work [15] was the

position of the strong narrow resonance which was placed in the calculations

at 1734 keV whereas in the last work it was found at 1726 keV. The position


of this resonance is actually well established due to numerous experimental

studies and the value used in the calculations is the adopted one taken from

the compilation of F.Ajzenberg-Selove. So very strong arguments are needed

in order to change its position. Thus the deviations from evaluated curves

observed in the posterior measurements do not necessarily mean that the

evaluation should be revised.

0

200

400

600

800

1000

1200

1400

o

o

o

o

o

o

o

o

Am93 170 Ra85 170

Li93 170 Sa93 170 Ya91 170

Ja53 168 Theory 170

0

200

400

600

800

o

o

o

o

o

o

o

Am93 150

Li93 155 Sa93 150 Me76 144

Theory 150 Theory 155 Theory 144

d

/d

lab(m

b/sr)

500 1000 1500 2000 2500 3000 35000

200

400

o

o

o

Am93 110

Me76 144 Theory 110 Theory 115

Energy (keV)

Figure 6: The evaluated differential cross section and the available experimental datafor proton elastic scattering from carbon.

The analysis of the proton elastic scattering cross sections for carbon

(Fig.6) revealed some discrepancies between available experimental data.

There is a set of data (Liu93) that significantly overestimates the cross

section in the vicinity of the peak observed in the excitation function at

44 A.F. Gurbich

Ep 1.735 MeV. The value of the cross section at the maximum of this

resonance exceeds values obtained in all other works by a factor of 1.5.

This is strange enough since both the energy resolution and energy steps

reported are comparable with those of other works. From the experimental

point of view, it would be easy to explain the result which is lower than a

true resonance maximum yield but it is hardly possible to imagine how to

obtain a greater value. This isolated strong peak provides favorable condi-

tions for resonance profiling. So the precise knowledge of the height of the

peak is of great importance. So far, as no confirmation for the singular set of

data was found, it is very probable that some unaccounted systematic error

influenced the results.

Theoretical calculations provide reliable evaluated cross sections for the

interval of angles from 110 to 170 for the proton energy range of 1.7 -

3.5 MeV and for the interval of angles from 150 to 170 in the whole en-

ergy range from Rutherford scattering up to 3.5 MeV. Extrapolation beyond

these intervals of the angles and the energy regions can be performed by the

calculations in the framework of the employed theoretical model.

4.3 Proton elastic scattering cross section for oxygen

There are several papers dealing with the proton elastic scattering cross

section for oxygen. The available experimental data are reviewed in Ref.[16]

where the evaluation of the cross section is reported. Except for two narrow

resonances at 2.66 and 3.47 MeV the cross section energy dependence is

rather smooth for the oxygen (p,p) elastic scattering up to approximately

4.0 MeV. Significant local variations due to resonances in p+16O system are

observed at higher energies. Hence the energy region Ep < 4 MeV is most

suitable for backscattering analysis and the evaluation was so made for this

region. It is worth noting that the oxygen (p,p) elastic cross section at 4

MeV exceeds its Rutherford value for a 170 scattering angle by a factor of

about 23.

As is seen from Fig.7, in the energy region greater than approximately 2

MeV the theoretical curves are in fair agreement with all the available data.

At lower energies theory is very close to all the experimental points except

for Braun83 and Amirikas93. The data from Braun83 at 110 scattering

angle disagree with theoretical predictions as well as other available data

in the region greater than 1.2 MeV. A discrepancy between theoretical

calculations and experimental results was obtained as well as published in


this paper for excitation functions at 135 and 160. A systematic deviation

of the Amerikas93 data at low energies from the other measurements and

theory is seen for all the three presented excitation functions. Since the

data from this paper were not included in the data set used for the model

parameters optimization an attempt has been made to reproduce these data

by adjusting the model parameters. The obtained results turned out to have

no physical meaning since the calculated single particle resonance parameters

as well as angular distributions disagreed with the experimentally observed

ones. Similar results were obtained in the case of Braun83 data. Because of

the obvious discrepancy with the other data and the inconsistency with the

theory there is reason to believe that the cross sections from the discussed

papers have some unaccounted experimental inaccuracy.

The evaluated differential cross sections are provided throughout the

energy region up to 4 MeV for any backward angle. The comparison with

posterior measurements (see [15]) shows an excellent agreement.

4.4 Proton elastic scattering for aluminum

The 27Al(p,p0)27Al cross section has a lot of narrow resonances in the whole

energy range used in EBS. The detailed 27Al(p,p0)27Al excitation function

was obtained in the high resolution proton resonance measurements [17].

The R-matrix fit to the data was shown to be in excellent agreement with the

measured points. The measurements of this cross section was also reported

in Refs.[18] and [19]. The results of the cross section from [17] retrieved

by the R-matrix calculations along with measured points of Ref.[18] and

Ref.[19] are shown in Fig.8.

As is seen from Fig.8 the measured points are in a reasonable mutual

agreement as well as in a fair agreement with the retrieved high resolution

data, however the fine structure of the excitation function is completely

missed both in the sparse points measurements of [19] and in the cross sec-

tions derived from a thick target yield [18].

It follows from the results presented in Ref.[20] that EBS spectrum can be

adequately simulated in the case when the excitation function has a strong

fine structure. However, detailed knowledge of the cross section is needed in

this case. It means that in the thin target measurements the cross section

should be measured with an energy step not exceeding the target thickness

whereas extraction of the cross section fine structure from the thick target

yield is hardly possible.

46 A.F. Gurbich

Figure 7: The evaluated differential cross section and the available experimental datafor proton elastic scattering from oxygen.

4.5 Proton elastic scattering cross section for silicon

The evaluation is described in Ref.[21]. At energy lower than 1.5 MeV the

theory predicts higher cross sections for the 150 and 170 scattering angles

as compared with the data from Am93 (see Fig.3). The most prominent dis-

crepancy (up to factor 1.5) is observed for 110 scattering angle at energies

lower than 1.2 MeV. The discrepancy has been thoroughly studied but no

reasons for such a deviation of the cross section from Rutherford one was

found in the present analysis. Because of the lack of another experimental

information an additional measurement was made to clear up the problem

([22]). New results appeared to be in good agreement with theoretical cal-


1000 1200 1400 1600 1800 2000

50

100

150

200

250

300

Rauhala89 Chiari01 Theory

=170o 27Al(p,po)27Al

d

/d

c.m

. (m

b/sr)

Energy (keV)

Figure 8: The 27Al(p,p0)27Al differential elastic scattering cross section.

culations (see Fig.9).

The cross section for natural silicon is a sum of the cross sections for

its three stable isotopes weighted by the relative abundance. The detailed

evaluation of the cross section for proton elastic scattering from the minor

isotopes of the silicon was not made. A complicated resonance structure

is observed for proton scattering from 29Si and 30Si in the energy range

under investigation. The resonances are too weak and too close to be used

in resonance profiling of isotopically enriched targets. On the other hand

it has been generally realized that such a resonance behaviour of the cross

section is inconvenient for the conventional backscattering technique. If one

undertakes say tracing experiments with 29Si or 30Si, other methods rather

than elastic proton backscattering should be employed. It is worth noting

that the contribution of the minor silicon isotopes to the total cross section

is significant when 28Si cross section is far from the Rutherford value. For

instance, the 29Si and 30Si isotopes give in sum about a half of the observed

cross section for 170 excitation function at the center of the broad dip near

2.8 MeV.

The evaluated differential cross sections are provided in the energy range

up to 3.0 MeV. The comparison with posterior measurements was reported

in Ref.[15].

48 A.F. Gurbich

900 1050 1200 1350 1500 1650 18000

100

200

300

400

500

600

700

800

lab =110o

A m 9 3 He00TheoryRutherford

d

/d

lab

(m

b/sr)

Energy (keV)

Figure 9: The 28Si(p,p0)28Si differential elastic scattering cross section.

4.6 The cross section for elastic scattering of 4He from car-

bon

The differential cross sections for elastic backscattering of 4He ions from

light nuclei are among the most important data for IBA. The evaluated

curves d(E)/d and the available experimental data at scattering angles

lab 165 are shown in Figs.10 and 11 for the energy ranges of 2.5 - 4.0

MeV and of 4.0 - 8.0 MeV, respectively. Reproducing the narrow resonances

at 3.577 MeV (c.m.=0.625 keV), at 5.245 MeV (c.m.=0.28 keV), and at

6.518 MeV (c.m.=1.5 keV) in the measurements strongly depends on the

energy spread of the beam. For this reason and since these resonances are

hardly of interest for IBA because of their relative weakness they are not

shown in Fig.10. The resonance at 3.577 MeV is only shown in Fig.10, for

example. As is seen from Fig.10 fair agreement is observed between available

experimental data and theoretical excitation function in energy range of 2.5

- 4.0 MeV except the height of the narrow resonance.

Above 4.0 MeV the theoretical curve is very close to the data from the

classical work of Bittner et al.[23] (see Fig.11). The experimental points

marked as Cheng94 and Davies94 are systematically higher by 20% being

in good agreement with each other. If renormalized these points appear to be


in close agreement with Ref.[23] and with the calculated curve. Therefore all

the difference originates from the normalization of the original experiments.

The experimental points marked as Feng94 are close to the data Cheng94

and Davies94 up to approximately 6.0 MeV and consequently they disagree

with the theory. At higher energies the data Feng94 are close to the data

from [23] and to the evaluated curve. Such a behaviour of the excitation

function is rather strange and the suspicion consequently arises that some

unaccounted error influenced the experimental results. As compared with

evaluated cross sections the points derived from the thick target yield at 5.4

and 6.16 MeV (marked as Gosset89) are underestimated by 14% and 17%,

respectively.

2.5 3.0 3.5 4.00

2

4

6

8

10

12

14

[5] 170.5 o

[8] 165.0 o

[11] 166.9 o

Theory 166.9 o

d

/d

R

Energy, MeV

Figure 10: The available experimental data and the evaluated excitation function for4He elastic scattering from carbon in the energy range from 2.5 to 4.0 MeV.

Summing up it can be concluded that except for normalization fair agree-

ment is in general observed between the available sets of experimental data

(excluding the data Feng94) in a wide energy range. An additional calibra-

tion experiment is needed to resolve the discrepancy of the normalization.

Now that the differential cross sections for 12C(4He,4He)12C scattering has

been evaluated the required excitation functions for analytical applications

may be calculated in the energy range from Coulomb scattering up to 8

MeV at any scattering angle. Calculations show that the cross section at

backward angles has a strong angular dependence that should be taken into

50 A.F. Gurbich

4 5 6 7 80

20

40

60

80

100

120

140

160

180

Theory 170.0o

Theory 166.9 o Feng 94 165.0o

Gosset 89 165.0 o

Bittner 54 166.9o

Cheng 94 170.0 o

Davies 94 170.0o

Leavitt 91 170.5o

Somatri 96 172.0 o

d

/d

R

Energy, MeV

Figure 11: The available experimental data and the evaluated excitation function for4He elastic scattering from carbon in the energy range from 4.0 to 8.0 MeV.

account while designing an experiment. The results of the posterior mea-

surements [24] appeared to be in satisfactory agreement with the evaluated

cross sections in a wide angular interval forward scattering angles included.

5 SigmaCalc - A Cross Section Calculator

When the evaluation of the cross section is completed and recommended

data are produced they are ready for dissemination among users. In practice

this is usually made through establishing a database of the evaluated cross

sections for one or another particular field of application. As was already

mentioned IBA differs from practically all other nuclear physics applications

by using differential rather than total cross sections. As one can see from the

above figures an angular dependence of the cross section can be very strong.

This is especially often the case for regions in the vicinity of resonances and

for large scattering angles. As far as a detector in IBA can be fixed at any

scattering angle the problem arises how to arrange access to users to the

data. Databases of experimental cross sections established for IBA contain

measured data for selected scattering angles. Distinct of experimental cross

sections the evaluated data being generated in result of theoretical calcula-


tions can be produced for any scattering angle. It is evident that to fill a

database with cross sections for all the possible scattering angles is imprac-

tical. In principle it is possible to create a database of the model parameters

fitted in course of the evaluation and a collection of the programs used for

the calculations. However, being rather complicated such calculations are

hardly expected to be carried out without problems by everyone who needs

the data.

In order to provide the IBA scientist with a tool for computing the dif-

ferential cross sections required for an analytical work, a software SigmaCalc

has been developed. The SigmaCalc calculator is based on the already pub-

lished and some new results of the data evaluation. The cross sections are

calculated using nuclear reaction models fitted to the available experimen-

tal data. A user friendly environment enables the IBA scientist having no

expertise in nuclear physics to perform the calculations of the required dif-

ferential cross sections for any scattering angle and for energy range and

elements of interest to Ion Beam Analysis. Taken into account the diversity

of the spectra processing programs used in IBA different formats for output

data are provided. Tools to show the results of the calculations in tabular

and graphical forms are included.

It is normal practice that recommended cross sections are changed from

time to time. This usually happens when new experiments undertaken at a

higher level of experimental accuracy give rise to the revision of the present

results. In order to facilitate updating of the SigmaCalc parameter sets it

would be desirable to make this software accessible via Internet. In this case

a user could perform remote calculations using every time the last version

of the evaluation.

6 Conclusion

It should be stressed that exact knowledge of the cross-section cannot be

extracted from any experiment or calculation. Given by nature, these data

could only be estimated with some degree of confidence. It is sometimes said

that all the IBA community needs from nuclear physics is reliable measured

excitation functions. However, it remains unclear what criteria for reliability

are implied and if this is the case, perhaps the excitation functions should be

measured at all possible scattering angles for IBA applications. Meanwhile,

it has already been clearly shown in numerous papers that evaluating cross

sections by combining a large number of different data sets in the framework

52 A.F. Gurbich

of the theoretical model enables excitation functions for analytical purposes

to be calculated for any scattering angle, with reliability exceeding that of

any individual measurement. It is when experiment and theory lock together

into a coherent whole that one knows that a reliable result has been obtained.


References

[1] J.R. Tesmer and M. Nastasi, eds., Handbook of Modern Ion Beam Ma-

terials Analysis, MRS, Pittsburg, PA, 1995.

[2] A.F. Gurbich, Physics of the Interaction of Charged Particles with Nu-

clei, Lecture given at the Workshop on Nuclear Data for Science and

Technology: Material Analysis, Trieste, 19-30 May 2003, ICTP LNS,

2003.

[3] J.W. Mayer and E. Rimini, eds., Ion Beam Handbook for Material

Analysis, Academic Press, New York, 1977.

[4] (Internet address http://ibaserver.physics.isu.edu/sigmabase

[5] A.F. Gurbich and A.V. Ignatyuk, in: G. Reffo, A. Ventura, C. Grandy

(Eds.), Nuclear Data for Science and Technology, Conf. Proc., 59, SIF,

Bologna, 1740 (1997).

[6] H.J. Kim, W.T. Milner and F.K. McGowan, Nuclear Data Tables A 2,

353 (1966); 3, 123 (1967);

[7] R.A. Jarjis, Nuclear Cross Section Data for Surface Analysis, Depart-

ment of Physics, University of Manchester Press, Manchester, 1979.

[8] D.C. Dodder, G.M. Hale, N. Jarmie, J.H. Jett, P.W. Keaton, Jr., R.A.

Nisley and K. Witte, Phys. Rev. C 15, 518 (1977).

[9] A. Nurmela, J. Raisanen and E. Rauhala, Nucl. Instr. and Meth. B

136-138, 77 (1998).

[10] I. Bogdanovic Radovic and O. Benka, Nucl. Instr. and Meth. B 174, 25

(2001).

[11] S.K. Kim and H.D. Choi, Nucl. Instr. and Meth. B 174, 33 (2001).

[12] A.F. Gurbich, Nucl. Instr. and Meth. B 136-138, 60 (1998).

[13] A.F. Gurbich, Nucl. Instr. and Meth. B 152, 403 (1999).

[14] S. Mazzoni, M. Chiari, L. Giuntini, P.A. Mando and N. Taccetti, Nucl.

Instr. and Meth. B 136-138, 86 (1998); 159, 191 (1999).

54 A.F. Gurbich

[15] A.R.L. Ramos, A. Paul, L. Rijniers, M.F.da Silva and J.C. Soares, Nucl.

Instr. and Meth. B 190, 95 (2002).


[17] R.O. Nelson, E.G. Bilpuch and C.R. Westerfeldt, Phys. Rev. C 29, 1656

(1984).

[18] E. Rauhala, Nucl. Instr. and Meth. B 40/41, 790 (1989).

[19] M. Chiari, L. Giuntini, P.A. Mando and N. Taccetti, Nucl. Instr. and

Meth. B 174, 259 (2001).

[20] A.F. Gurbich, N.P. Barradas, C. Jeynes and E. Wendler, Nucl. Instr.

and Meth. B 190, 237 (2002).


[22] M.J.F. Healy and A.F. Gurbich, Nucl. Instr. and Meth. B 161-163, 136

(2000).

[23] J.W. Bittner and R.D. Moffat, Phys. Rev. 96, 374 (1954).

[24] I. Bogdanovic Radovic, M. Jaksic, O. Benka and A.F. Gurbich, Nucl.

Instr. and Meth. B 190, 100 (2002).

Rutherford Backscattering Spectrometry (RBS)

M. Mayer

Max-Planck-Institut fur Plasmaphysik, EURATOM Association,

Garching, Germany



Materials Analysis


LNS0822003

[email protected]

Abstract

Rutherford Backscattering Spectrometry (RBS) is a widely usedmethod for the surface layer analysis of solids. This lecture gives abrief introduction into the method, and describes scattering kinematics,scattering cross-section data, stopping power data, detector resolutionissues, and electronic energy loss straggling in some detail. Computercodes for the simulation of RBS spectra and RBS data analysis arepresented, and supply sources of codes and cross-section data are given.Practical applications of RBS and RBS data analysis are demonstratedin some

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