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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
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.
Protons and Helions Elastic Scattering from Light Nuclei 37
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
Protons and Helions Elastic Scattering from Light Nuclei 39
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-
Protons and Helions Elastic Scattering from Light Nuclei 41
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
Protons and Helions Elastic Scattering from Light Nuclei 43
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
Protons and Helions Elastic Scattering from Light Nuclei 45
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-
Protons and Helions Elastic Scattering from Light Nuclei 47
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
Protons and Helions Elastic Scattering from Light Nuclei 49
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-
Protons and Helions Elastic Scattering from Light Nuclei 51
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.
Protons and Helions Elastic Scattering from Light Nuclei 53
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).
[16] A.F. Gurbich, Nucl. Instr. and Meth. B 129, 311 (1997).
[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).
[21] A.F. Gurbich, Nucl. Instr. and Meth. B 145, 578 (1998).
[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).