A Study A Calculation of Neutron Cross Sections on Real Potential of
Optical Model
Wint Shwe War Hlaing*
Abstract
The optical model potential parameters which play an important role in the evaluation of
nuclear data for applied purposes are not fully satisfied at low energy neutron scattering.
They are fixed according to the target mass and incident energy. In this work the
calculated data are compared with those of IAEA data especially for Li-7 and Th-232.
Keywords: Neutron Cross-Sections, Optical Model, ABAREX, SCAT2
Introduction
Nuclear cross-section data are needed for scientific works as well as for reactor
construction. Nuclear cross-section data are provided by experimental work. Nuclear
experiments are very expensive, so modal calculations are used to predict cross-sections.
Various nuclear codes are developed to calculate nuclear cross-sections. Not all the codes
are perfect so we have to find correction factors. Generally, nuclear cross-sections depend
on target mass, incident energy and type of incident particle.
Nuclear optical model is found to be only successful applicable for energy about
6MeV, and intermediate and heavy nuclides. The improvement for low energy and low
mass nuclides is still of great interest for neutron cross sections calculation groups. The
theoretical estimation of optical model parameters is still to be modified in low energy,
low mass nuclides. The calculated neutron cross sections data were to be compared with
those of IAEA data especially for Li-7 and Th-232. In this work the effect of real potential
is
V r ( r )   V f ( r ),

 r  R1  

f ( r )   1  exp 
 
 a1


1
where R is the radius and a is the surface diffuseness parameter.
Statistical Model for Nuclear Reactions
In the study of nuclear reactions the concept of the compound nucleus (the
statistical model for nuclear reactions) is not always reliable. The statistical nature of the
compound nucleus theory implies that its predictions are at best averages and do not take
into account the differences between specific nuclei. Therefore a more detailed model is
needed for the description of nuclear reactions.
The statistical model assumes that the compound nucleus is formed immediately
when the incident neutron reaches the nuclear surface. The cross-section for reaching the
surface turns out to be a monotonically decreasing function of the energy, varying as E-1/2
for small energies and reaching the asymptotic value 2R2 for large energies. At the
neutron energies involved, between 0.1MeV and several MeV, and for intermediate or
1. Dr., Assistant Lecturer, Department of Geography, Yangon Institute of Education
2. Dr., Assistant Lecturer, Department of Geography, Yangon Institute of Education
heavy nuclei, individual resonances cannot be resolved, and the measured cross sections
are averages over many levels.
Optical Model for Nuclear Reactions
The optical model describes the effect of the nucleus on the incident particle by a
potential well –V0 (r), but allows for the possibility of compound nucleus formation by
adding to the potential a negative imaginary part, -iV1(r). This part produces absorption of
the incident particle within the nucleus, and this absorption is supposed to represent the
formation of the compound nucleus.
On the optical model, compound nucleus formation does not occur immediately or
with complete certainty. Even if the incident particle has entered the nucleus, it is removed
from its free particle state only with some delay and with a certain probability. The model
has also been extended and applied with more complicated potential functions to other
nuclear reactions and cross sections.
The feature of nuclear scattering cross sections can be explained by a very simple
model in which we represent the interaction between the incident nucleons and the nucleus
by a one-body potential that depends only on the nuclear radius. This is called the optical
model.
Methods and Materials
Calculation Procedure
As the ABAREX runs under DOS, we have to load the Dos. After loading DOS,
we have to load the ABAREX program. Under ABAREX program we open INPUT.DAT
and make some changes to input data file.
C:\> ABAREX > EDIT INPUT 
The file is saved and closed. The calculation is done when we type this command.
C:\> ABAREX >DEL OUTPUT 
C:\> ABAREX >DEL PUNCH 
C:\> ABAREX > ABAREX 
To see and edit output results, we open ABAREX and rename it.
C:\> ABAREX >EDIT OUTPUT 
Results and Discussions
The calculated data and respective graphs are shown in the following tables and figures.
Table 1. Comparison of Calculation data and IAEA Data for Li-7
Total Cross Sections (b)
E(MeV)
SCAT2
Vr = 46 MeV
JENDL
EXFOR
1
1.5571
1.2094
1.3200
1.5779
2
1.3728
1.1483
1.3938
1.7424
3
1.3875
1.1764
1.8628
2.0074
Total Cross Sections (b)
E(MeV)
SCAT2
Vr = 46 MeV
JENDL
EXFOR
4
1.4546
1.2434
2.1298
2.3836
5
1.5084
1.3148
2.0989
2.4544
6
1.5304
1.3683
2.0292
2.1725
7
1.5276
1.3980
1.9356
1.9662
8
1.5114
1.4081
1.8514
1.8078
9
1.4896
1.4056
1.7752
1.7786
10
1.4666
1.3965
1.6987
1.6628
11
1.4444
1.3846
1.6277
1.5808
12
1.4233
1.3723
1.5608
1.5813
13
1.4033
1.3608
1.4937
1.4968
14
1.3837
1.3504
1.4313
1.4391
15
1.3639
1.3410
1.3100
1.4533
16
1.3437
1.3324
1.3328
1.3562
17
1.3226
1.3240
1.2854
1.2684
18
1.3008
1.3154
1.2381
1.1828
19
1.2782
1.3064
1.1977
1.1718
20
1.2550
1.2976
1.1706
1.1858
Comparison of Cross-sections Data for Li-7
Total Cross-section (b)
3.00
2.50
SCAT2
2.00
Vr = 46 MeV
1.50
JENDL
EXFOR
1.00
0.50
0.00
0
5
10
15
20
25
Neutron Incident Energy (MeV)
Figure 1. Comparison of Calculation data and IAEA Data for Li-7
The comparison of total cross-sections for Li-7 using real potential 46 MeV with
those of SCAT2 data, JENDL data and EXFOR data is given in Table 1 and the respective
graph is shown in Figure 1. It can be seen that ABAREX data agree with SCAT2 data, and
EXFOR data agree with JENDL data.
Table 2. Comparison of Calculated Data and IAEA Data for Th-232
Total Cross Sections (b)
E(MeV)
SCAT2
Vr = 46 MeV
JENDL
EXFOR
1
7.8896
8.6491
6.9350
7.3790
2
6.8074
6.8441
7.0400
7.2210
3
6.9426
6.7317
7.7100
7.9620
4
7.2415
7.1399
7.8800
8.0210
5
7.3122
7.4845
7.5100
7.6910
6
7.1488
7.6346
6.9800
7.1490
7
6.7833
7.6056
6.4900
6.5900
8
6.3204
7.4540
6.0800
6.3500
9
5.8968
7.2212
5.7600
6.1550
10
5.5879
6.9347
5.6200
5.9480
11
5.4103
6.6367
5.6150
5.9360
12
5.3363
6.3787
5.6300
5.8680
13
5.3300
6.1880
5.6800
5.8150
14
5.3727
6.0227
5.7400
5.9330
15
5.4584
5.8236
5.8250
6.0120
16
5.5817
5.6618
5.9900
6.0380
17
5.7325
5.5833
6.1500
6.1090
18
5.8936
5.5833
6.2700
6.2600
19
6.0437
5.6497
6.3500
6.2640
20
6.1645
5.7617
6.4100
6.4370
Comparison of Cross-sections Data for Th-232
Total Cross-section (b)
10.00
9.00
8.00
7.00
6.00
SCAT2
5.00
4.00
JENDL
Vr = 46 MeV
EXFOR
3.00
2.00
1.00
0.00
0
5
10
15
20
25
Neutron Incident Energy (MeV)
Figure 2. Comparison of Calculated Data and IAEA Data for Th-232
The comparison of total cross-sections for Th-232 using real potential 46 MeV
with those of SCAT2 data, JENDL data and EXFOR data is given in Table 2 and the
respective graph is shown in Figure 2. It can be seen that calculated cross sections show
some difference from other below 14 MeV.
Table 3. Best Fixed Real Potential for Total Cross-sections of Li-7
Total Cross Sections (b)
E(MeV)
Vr = 54MeV
Vr = 46 MeV
JENDL
EXFOR
1
1.5213
1.2094
1.3200
1.5779
2
1.1354
1.1483
1.3938
1.7424
3
1.0994
1.1764
1.8628
2.0074
4
1.1507
1.2434
2.1298
2.3836
5
1.2049
1.3148
2.0989
2.4544
6
1.2387
1.3683
2.0292
2.1725
7
1.2549
1.3980
1.9356
1.9662
8
1.2606
1.4081
1.8514
1.8078
9
1.2616
1.4056
1.7752
1.7786
10
1.2614
1.3965
1.6987
1.6628
11
1.2621
1.3846
1.6277
1.5808
12
1.2645
1.3723
1.5608
1.5813
13
1.2689
1.3608
1.4937
1.4968
14
1.2747
1.3504
1.4313
1.4391
15
1.2809
1.3410
1.3100
1.4533
16
1.2866
1.3324
1.3328
1.3562
17
1.2910
1.3240
1.2854
1.2684
18
1.2936
1.3154
1.2381
1.1828
19
1.2942
1.3064
1.1977
1.1718
20
1.2927
1.2976
1.1706
1.1858
Best Fixed Real Potential for Total Cross-sections of Li-7
Total Cross-section (b)
3.00
2.50
Vr = 54MeV
2.00
Vr = 46 MeV
1.50
JENDL
EXFOR
1.00
0.50
0.00
0
5
10
15
20
25
Neutron Incident Energy (MeV)
Figure 3. Best Fixed Real Potential for Total Cross-sections of Li-7
The comparison of total cross-sections for Li-7 using real potential value of 46
MeV and 54 MeV, JENDL data and EXFOR data is given in Table 3 and the respective
graph is shown in Figure 3. It can be seen that increasing the real potential makes the data
more agree with EXFOR and JENDL data.
Table 4. Best Fixed Real Potential for Total Cross-sections of Th-232
Total Cross Sections (b)
E(MeV)
Vr = 45 MeV
Vr = 46 MeV
JENDL
EXFOR
1
8.3845
8.6491
6.9350
7.3790
2
6.8296
6.8441
7.0400
7.2210
3
6.8877
6.7317
7.7100
7.9620
4
7.3272
7.1399
7.8800
8.0210
5
7.6222
7.4845
7.5100
7.6910
6
7.6772
7.6346
6.9800
7.1490
7
7.5439
7.6056
6.4900
6.5900
8
7.3021
7.4540
6.0800
6.3500
9
7.0211
7.2212
5.7600
6.1550
10
6.7253
6.9347
5.6200
5.9480
11
6.4405
6.6367
5.6150
5.9360
12
6.2044
6.3787
5.6300
5.8680
13
6.0456
6.1880
5.6800
5.8150
14
5.9522
6.0227
5.7400
5.9330
Total Cross Sections (b)
E(MeV)
Vr = 45 MeV
Vr = 46 MeV
JENDL
EXFOR
15
5.8497
5.8236
5.8250
6.0120
16
5.7167
5.6618
5.9900
6.0380
17
5.6350
5.5833
6.1500
6.1090
18
5.6342
5.5833
6.2700
6.2600
19
5.7042
5.6497
6.3500
6.2640
20
5.8257
5.7617
6.4100
6.4370
Best Fixed Real Potential for Total Cross-sections of Th-232
10.00
9.00
Total Cross-section (b)
8.00
7.00
6.00
Vr = 45 MeV
5.00
Vr = 46 MeV
4.00
JENDL
EXFOR
3.00
2.00
1.00
0.00
0
5
10
15
20
25
Neutron Incident Energy (MeV)
Figure 4. Best Fixed Real Potential for Total Cross-sections of Th-232
The comparison of total cross-sections for Th-232 using real potential value of 45
MeV and 46 MeV, JENDL data and EXFOR data is given in Table 4 and the respective
graph is shown in Figure 4. It shows that Vr = 45 MeV much agree with EXFOR and
JENDL than default value Vr = 46 MeV for Th-232.
Conclusion
Generally, from the results, the optical model fitting procedure for light nuclides
and for heavy nuclides need more factors other than real potential, and for intermediate
nuclides need less factors. And then, when it is compare with IAEA data, the calculated
results are a bit difference for light nuclides and intermediate nuclides. So, they also need
more effective parameters in the low energy range (below 5MeV). But, for heavy nuclides,
the results are agreed with IAEA data. It is found that the calculated real potential is
applicable for neutron energy range (1MeV to 20MeV)
So in conclusion it can be said that calculation of cross-sections on the real
potential of optical model well agree with those obtained given in JENDL and EXFOR
especially for above 5MeV of neutron energy by using IAEA nuclear codes. The results
are valid for various targets. The codes are also applicable for light nuclides, intermediate
nuclides and heavy nuclides. Thus, the codes are the efficient tools for the determination
of nuclear cross- section and other useful nuclear information.
Acknowledgements
I would like to express my profound thanks to Dr Lwin Lwin Soe and Dr Pyay Thein, Pro-Rectors
of Yangon Institute of Education, for their constructive suggestions for the manuscript.
I would also like to extend my gratitude and deep appreciation to Professor Dr Khin Mu Soe, Head
of the Department of Physics, Yangon Institute of Education, and Dr Khin Tint, Associated Professor,
Department of Physics, Yangon Institute of Education, for their kind permission and encouragement to carry
out this research work.
Special thanks should also go to Dr Win Sin, Lecturer, Department of Physics, Yangon University,
for his kind help in writing my research paper.
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