diﬀerential cross sections for elastic scattering of protons and...

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

Lectures given at the

Workshop on Nuclear Data for Science and Technology:

Materials Analysis

Trieste, 19-30 May 2003

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.

50 100 150 200 0

500

1000

1500

2000

2500

3000

3500

O

Fe E p =4.1 MeV

Measured

Calculated for Rutherford

cross section

Cou

nts/

Cha

nnel

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,p

o )

56 Fe

θ lab

=165 o

d σ /

d σ R

uth

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 Bozoian’s 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 1800 0

100

200

300

400 28 Si(p,p

0 ) θ

lab =170

o

A m 9 3

Ra85

Sa93

He00

Theory

Rutherford

d σ /

d Ω

lab (

mb/

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 Section Measurements

Benchmark Experiments

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

1 H(

4 He,

1 H)

Wa86

Bo01

Ya83

Na85

Evaluation

d σ /

d Ω

, mb/

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).

σERD(ϕ)

σEBS(θ)= 4 cos ϕ cos(θcm − θ)

sin2 θ

sin2 θcm

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 σ /

d Ω

la

b (m

b/sr

)

500 1000 1500 2000 2500 3000 3500 0

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

θ =170 o 27

Al(p,p o ) 27

Al

d σ /

d Ω

c.m

. (

mb/

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 1800 0

100

200

300

400

500

600

700

800

θ lab

=110 o

A m 9 3

He00

Theory

Rutherford

d σ /

d Ω

lab (

mb/

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 dσ(E)/dΩ and the available experimental data at scattering angles

θlab ≥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 (Γc.m.=0.625 keV), at 5.245 MeV (Γc.m.=0.28 keV), and at

6.518 MeV (Γc.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 4.0 0

2

4

6

8

10

12

14

[5] 170.5 o

[8] 165.0 o

[11] 166.9 o

Theory 166.9 o

d σ /

d σ 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 8 0

20

40

60

80

100

120

140

160

180

Theory 170.0 o

Theory 166.9 o Feng 94 165.0

o

Gosset 89 165.0 o

Bittner 54 166.9 o

Cheng 94 170.0 o

Davies 94 170.0 o

Leavitt 91 170.5 o

Somatri 96 172.0 o

d σ /

d σ 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.


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