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The variation of cross sections with energy for gamma rays in the atmosphere
Ann. Nucl. EnergJ>, Vol. 24. No. 1, pp. 65-69, 1997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0306-4549/97 $I 5.00 + 0.00
Pergamon 0306-4549(95)00130-l
TECHNICAL
NOTE
THE VARIATION OF CROSS SECTIONS WITH ENERGY FOR GAMMA RAYS IN THE ATMOSPHERE GOKAY KAYNAK and REMZIYE
ERGUL
Uludag University, Faculty of Sciences and Arts, 16059-Bursa, Turkey (Received 28 November 1995)
Abstract-It is known that for the study of low-energy gamma rays in the atmosphere, coherent scattering, incoherent scattering, photoelectric effect, and total coherent and incoherent cross-sections are needed. In this study, a number of interpolation functions are derived to relate energy to cross-section continuously. Copyright 0 1996 Elsevier Science Ltd
INTRODUCTION In theoretical and experimental studies involving gamma rays, one of the most important steps is the determination of the cross-section for a particular value of the energy. There are tables that enable us to do this in discrete steps (Hubbell, 1969; Storm and Israel, 1970), and although it is possible to make a simple interpolation for values between these steps, this would not be practical in many cases and the availabilty of an interpolation function that relates energy to cross-section in all intervals would be very convenient and time saving. This is especially true for the study of gamma rays in various atmospheric environments (Wecksung et al., 1971; Minato, 1973a,b, 1975; Doi and Chan, 1980; Sherbini et al., 1986). The normal approach for such a study is the use of the Monte Carlo Method on a computing platform. In such a case, the researcher would prefer to use functions rather than tables to reduce computing time and unwanted errors. In the case of nuclear experiments or nuclear accidents it is imperative to know beforehand how the gamma rays would disperse in the atmosphere (Simons and Comet, 1959; Swarup and Ganguly, 1975; Swarup 1979, 1980; Swarup and Minato, 1983; Momeni, 1985; Uehara et al., 1988.) This is why the atmosphere has been chosen as the environment for this study.
METHOD The energy of gamma rays of radioisotopes that disperse in the atmosphere after a nuclear experiment or accident varies between 30 keV and 1.6 MeV (Nuclear Data Project, 1976). The study described in this paper is based on these energy ranges. 65
66
Technical Note Table 1. Cross-sections of events investigated and corresponding
Gamma energy
Coherent scattering
Incoherent scattering
(MeV)
(cm’/g)
(cm*/g)
1.oo 1.50 2.00 3.00 4.00 5.00 6.00 8.00 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 1.oo 1.50
x x x x x x x x x x x x x x x x
10-2 10-2 10-2 10-2 10-2 10-2 1O-2 lo-’ 10-l 10-l lo-’ 10-l 10-l lo-’ 10-l 10-I
3.64 2.85 2.47 2.11 1.93 1.81 1.73 1.61 1.51 1.35 1.23 1.07 9.53 8.70 8.05
x x x x x x x x x x x x x x x
IO-’ lo-’ 10-l lo-’ 10-l lo-’ 10-I 10-l lo-’ 10-l 10-l 10-l 10-2 1O-2 10-2
1.93 1.89 1.86 1.80 1.74 1.69 1.64 1.56 1.48 1.33 1.22 1.06 9.52 8.70 8.04 7.07 6.36 5.17
x x x x x x x x x x x x x x x x x x
Photoelectric effect
Total coherent scattering (cm”/g)
(cm?g)
lo-’ 10-l lo-’ 10-l 10-l 10-l 10-l 10-l lo-’ lo-’ 10-l 10-l lo-* 1om2 10-2 10-l lo-* 10-2
4.63 1.27 5.05 1.39 5.53 2.70 1.52 6.06 2.94 8.05 3.24 9.30 3.99 2.15 1.34 6.79 4.20 1.96
x x x x x x x x x x x x x x x x
total cross-sections
lo-’ lo-’ 1o-2 1O-2 lo-’ 1o-3 10m3 1oA 1OA 1o-5 10-5 lo-’ 1o-s 1O-6 1O-6 1O-6
4.99 1.55 7.52 3.49 2.48 2.08 1.88 1.67 1.54 1.36 1.23 1.07 9.54 8.70 8.05
x x x x x x x x x x x x x
10-l lo-’ 10-l lo-’ 10-l 10-l lo-’ 10-l 10-l 10-l 1o-2 1o-2 lo-*
Total incoherent scattering (cm*/g) 4.82 1.45 6.91 3.18 2.29 1.96 1.79 1.62 1.51 1.34 1.23 1.06 9.53 8.70 8.05 7.07 6.36 5.18
x x x x x x x x x x x x x x x x
10-l 10-l lo-’ 10-l lo-’ 10-l lo-’ lo-’ 10-l 10-l 10-2 10-2 10-2 10-2 1O-2 1o-2
The prevailing events in the interaction of matter with gamma rays are the photoelectric effect, pair production and the Compton effect. Since the minimum energy required for pair production is 1.02 MeV, this event is neglected in this study. The cross-sections of the events investigated and the corresponding total cross-sections given by Hubell (1969) are listed in Table 1. By studying the values given in this table, functions F(Ei,pi) are proposed. These are characterised by a parameter pi. A chi-square fit will be applied for these functions and cross-sections through systematic variation of pi: X2 =
i
[“i -
F(Ei9Pi)12*
i=l
The best fit is reached when x2=0 (Lyons, 1989). Below, different functions are proposed for each cross-section and the parameters of the proposed functions that minimize x are determined by varying them systematically. These functions and their parameters, together with their values, are listed below for coherent scattering, incoherent scattering, photoelectric effect and total coherent and incoherent scatterings.
Coherent scattering: (cm’/g)
o(E) =
1 P’ + pSp3
x2 = 0.6023 x 1O-3 p’ = -0.807
x 10-l
p2 = 20.751
p3 = 1.40 x 10-l.
Technical Note
61
Incoherent scattering: (cm’/g)
a(E) =
1 pl + pzE + p3E2 tp4E3
x1 = 1.32 x lo4 p, = 5.028
pz = 18.091
p1 = -11.948
p4 = 4.231.
Photoelectric effect: (cm’/g) g(E)
= expb,
+ P$
+
p3ev(p4EP5)l
x2 = 7.963 x 10m6 p2 = -1.018 x 10
p, = -15.108
p3 = 30.699
p4 = -2.362
ps = 2.921 x 10-I.
Total coherent scattering: (cml/g)
a(E) = expb, + p2dE + p3 exp(p.P)] + p,E + p,E’ x? = 1.99 x IO-2 pz = -1.018 X 10-l
pI = -1.931
p6 = -1.143
p3 = 198.966 x 10-j
p4 = -13.341
ps = 2.608 x 10 ’
p, = 3.859 x lo-‘.
Total incoherent scattering: (cm’lg)
dE)=ev
[P, + PS
+ p3
e~p(p4EfS)I +
P6
1 + p,E
+ psE2 + psE3
x2 = 6.469 X 10m3 p, = -15.062
p2 = -1.018
p6 = 1.968 X IO-’
x IO-’
p3 = 30.783
p7 = 3.235
p4 = -2.389
ps = -1.519
ps = 2.921 x IO-’
ps = 3.832 x 10-l.
CONCLUSIONS
In Figs 1-5, the functions proposed are plotted for coherent scattering, photoelectric effect, incoherent scattering, and total coherent and incoherent scatterings, respectively. On the same figures, the values given in Table 1 are also shown. It can be seen that the proposed functions do give accurate values for the cross-sections. They can therefore be used instead of the tables.
Technical Note
68
0.0
0.1
0.2
0.3
0.4
0.5
0.6
MeV
Fig. 1. The plot of the function that relates energy to cross-section for coherent scattering. The small triangles are the values given in Table 1.
0.05
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
MeV
Fig. 2. The plot of the function that relates energy to cross-section for incoherent scattering. The small triangles are the values given in Table 1. 5.0 3.0
$ E v
1.0 -1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
MeV
Fig. 3. The plot of the function that relates energy to cross-section for photoelectric effect. The small triangles are the values given in Table 1. 6.0 r WJ 4.0
2
3
2.0 t 0.0
(
0.0
0.1
0.2
0.3
0.4
0.5
0.6
MeV
Fig. 4. The plot of the function that relates energy to cross-section for total coherent scattering. The small triangles are the values given in Table 1.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
MeV
Fig. 5. The plot of the function that relates energy to cross-section for total incoherent scattering. The small triangles are the values given in Table 1.
Technical Note
69
Acknowledgements-The
authors would like to acknowledge the input and encouragement of Professor Dr t)mtir Akytiz (Bogazici University, Istanbul).
REFERENCES Doi, K. and Chan, H. P. (1980) Radiology 135 199. Hubbell, J. H. (1969) National Bureau of Standards, NSRDS-NBS, 29. Lyons, L. (1989) Statistics for Nuclear and Particle Physicists, p. 74. Cambridge University Press, Cambridge. Minato, S. (1973a) NucZ. Sci. Engng 51, 32. Minato, S. (1973b) Rad. Res. 56, 1. Minato, S. (1975) Nucl. Instrum. Meth. 131, 157. Momeni, M. H. (1985) Health Phys. 49, 3 10. Nuclear Data Project (1976), Report ORNL-5114. Simons, H. A. B. and Comet, S. (1959) Phys. in Med. and Biol. 3, 233. Sherbini, S., Tamasanis, D., Sykes, J. and Porter, S. W. (1986) Health Phys. 51, 699. Storm, E. and Israel, H. I. (1970) Nucl. Data Tables A7, 565. Swarup, J. (1979) Indian J. Pure Appl. Phys. 17, 381. Swarup, J. (1980) Nucl. Instrum. Meth. 172, 559. Swarup, J. and Ganguly, A. K. (1975) Indian J. Pure Appl. Phys. 13, 595. Swarup, J. and Minato, S. (1983) Indian J. Pure Appl. Phys. 21, 702. Uehara, S. Hoshi, M., Sawada, S., Nagatomo, T. and Ichikawa, Y. (1988) Health Phys. 54, 249.
Wecksung, G. W., Walker, J. J. and Brown, R. T. (1971) Nucl. Instrum. Meth. 95, 605.
Pergamon 0306-4549(95)00130-l
TECHNICAL
NOTE
THE VARIATION OF CROSS SECTIONS WITH ENERGY FOR GAMMA RAYS IN THE ATMOSPHERE GOKAY KAYNAK and REMZIYE
ERGUL
Uludag University, Faculty of Sciences and Arts, 16059-Bursa, Turkey (Received 28 November 1995)
Abstract-It is known that for the study of low-energy gamma rays in the atmosphere, coherent scattering, incoherent scattering, photoelectric effect, and total coherent and incoherent cross-sections are needed. In this study, a number of interpolation functions are derived to relate energy to cross-section continuously. Copyright 0 1996 Elsevier Science Ltd
INTRODUCTION In theoretical and experimental studies involving gamma rays, one of the most important steps is the determination of the cross-section for a particular value of the energy. There are tables that enable us to do this in discrete steps (Hubbell, 1969; Storm and Israel, 1970), and although it is possible to make a simple interpolation for values between these steps, this would not be practical in many cases and the availabilty of an interpolation function that relates energy to cross-section in all intervals would be very convenient and time saving. This is especially true for the study of gamma rays in various atmospheric environments (Wecksung et al., 1971; Minato, 1973a,b, 1975; Doi and Chan, 1980; Sherbini et al., 1986). The normal approach for such a study is the use of the Monte Carlo Method on a computing platform. In such a case, the researcher would prefer to use functions rather than tables to reduce computing time and unwanted errors. In the case of nuclear experiments or nuclear accidents it is imperative to know beforehand how the gamma rays would disperse in the atmosphere (Simons and Comet, 1959; Swarup and Ganguly, 1975; Swarup 1979, 1980; Swarup and Minato, 1983; Momeni, 1985; Uehara et al., 1988.) This is why the atmosphere has been chosen as the environment for this study.
METHOD The energy of gamma rays of radioisotopes that disperse in the atmosphere after a nuclear experiment or accident varies between 30 keV and 1.6 MeV (Nuclear Data Project, 1976). The study described in this paper is based on these energy ranges. 65
66
Technical Note Table 1. Cross-sections of events investigated and corresponding
Gamma energy
Coherent scattering
Incoherent scattering
(MeV)
(cm’/g)
(cm*/g)
1.oo 1.50 2.00 3.00 4.00 5.00 6.00 8.00 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 1.oo 1.50
x x x x x x x x x x x x x x x x
10-2 10-2 10-2 10-2 10-2 10-2 1O-2 lo-’ 10-l 10-l lo-’ 10-l 10-l lo-’ 10-l 10-I
3.64 2.85 2.47 2.11 1.93 1.81 1.73 1.61 1.51 1.35 1.23 1.07 9.53 8.70 8.05
x x x x x x x x x x x x x x x
IO-’ lo-’ 10-l lo-’ 10-l lo-’ 10-I 10-l lo-’ 10-l 10-l 10-l 10-2 1O-2 10-2
1.93 1.89 1.86 1.80 1.74 1.69 1.64 1.56 1.48 1.33 1.22 1.06 9.52 8.70 8.04 7.07 6.36 5.17
x x x x x x x x x x x x x x x x x x
Photoelectric effect
Total coherent scattering (cm”/g)
(cm?g)
lo-’ 10-l lo-’ 10-l 10-l 10-l 10-l 10-l lo-’ lo-’ 10-l 10-l lo-* 1om2 10-2 10-l lo-* 10-2
4.63 1.27 5.05 1.39 5.53 2.70 1.52 6.06 2.94 8.05 3.24 9.30 3.99 2.15 1.34 6.79 4.20 1.96
x x x x x x x x x x x x x x x x
total cross-sections
lo-’ lo-’ 1o-2 1O-2 lo-’ 1o-3 10m3 1oA 1OA 1o-5 10-5 lo-’ 1o-s 1O-6 1O-6 1O-6
4.99 1.55 7.52 3.49 2.48 2.08 1.88 1.67 1.54 1.36 1.23 1.07 9.54 8.70 8.05
x x x x x x x x x x x x x
10-l lo-’ 10-l lo-’ 10-l 10-l lo-’ 10-l 10-l 10-l 1o-2 1o-2 lo-*
Total incoherent scattering (cm*/g) 4.82 1.45 6.91 3.18 2.29 1.96 1.79 1.62 1.51 1.34 1.23 1.06 9.53 8.70 8.05 7.07 6.36 5.18
x x x x x x x x x x x x x x x x
10-l 10-l lo-’ 10-l lo-’ 10-l lo-’ lo-’ 10-l 10-l 10-2 10-2 10-2 10-2 1O-2 1o-2
The prevailing events in the interaction of matter with gamma rays are the photoelectric effect, pair production and the Compton effect. Since the minimum energy required for pair production is 1.02 MeV, this event is neglected in this study. The cross-sections of the events investigated and the corresponding total cross-sections given by Hubell (1969) are listed in Table 1. By studying the values given in this table, functions F(Ei,pi) are proposed. These are characterised by a parameter pi. A chi-square fit will be applied for these functions and cross-sections through systematic variation of pi: X2 =
i
[“i -
F(Ei9Pi)12*
i=l
The best fit is reached when x2=0 (Lyons, 1989). Below, different functions are proposed for each cross-section and the parameters of the proposed functions that minimize x are determined by varying them systematically. These functions and their parameters, together with their values, are listed below for coherent scattering, incoherent scattering, photoelectric effect and total coherent and incoherent scatterings.
Coherent scattering: (cm’/g)
o(E) =
1 P’ + pSp3
x2 = 0.6023 x 1O-3 p’ = -0.807
x 10-l
p2 = 20.751
p3 = 1.40 x 10-l.
Technical Note
61
Incoherent scattering: (cm’/g)
a(E) =
1 pl + pzE + p3E2 tp4E3
x1 = 1.32 x lo4 p, = 5.028
pz = 18.091
p1 = -11.948
p4 = 4.231.
Photoelectric effect: (cm’/g) g(E)
= expb,
+ P$
+
p3ev(p4EP5)l
x2 = 7.963 x 10m6 p2 = -1.018 x 10
p, = -15.108
p3 = 30.699
p4 = -2.362
ps = 2.921 x 10-I.
Total coherent scattering: (cml/g)
a(E) = expb, + p2dE + p3 exp(p.P)] + p,E + p,E’ x? = 1.99 x IO-2 pz = -1.018 X 10-l
pI = -1.931
p6 = -1.143
p3 = 198.966 x 10-j
p4 = -13.341
ps = 2.608 x 10 ’
p, = 3.859 x lo-‘.
Total incoherent scattering: (cm’lg)
dE)=ev
[P, + PS
+ p3
e~p(p4EfS)I +
P6
1 + p,E
+ psE2 + psE3
x2 = 6.469 X 10m3 p, = -15.062
p2 = -1.018
p6 = 1.968 X IO-’
x IO-’
p3 = 30.783
p7 = 3.235
p4 = -2.389
ps = -1.519
ps = 2.921 x IO-’
ps = 3.832 x 10-l.
CONCLUSIONS
In Figs 1-5, the functions proposed are plotted for coherent scattering, photoelectric effect, incoherent scattering, and total coherent and incoherent scatterings, respectively. On the same figures, the values given in Table 1 are also shown. It can be seen that the proposed functions do give accurate values for the cross-sections. They can therefore be used instead of the tables.
Technical Note
68
0.0
0.1
0.2
0.3
0.4
0.5
0.6
MeV
Fig. 1. The plot of the function that relates energy to cross-section for coherent scattering. The small triangles are the values given in Table 1.
0.05
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
MeV
Fig. 2. The plot of the function that relates energy to cross-section for incoherent scattering. The small triangles are the values given in Table 1. 5.0 3.0
$ E v
1.0 -1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
MeV
Fig. 3. The plot of the function that relates energy to cross-section for photoelectric effect. The small triangles are the values given in Table 1. 6.0 r WJ 4.0
2
3
2.0 t 0.0
(
0.0
0.1
0.2
0.3
0.4
0.5
0.6
MeV
Fig. 4. The plot of the function that relates energy to cross-section for total coherent scattering. The small triangles are the values given in Table 1.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
MeV
Fig. 5. The plot of the function that relates energy to cross-section for total incoherent scattering. The small triangles are the values given in Table 1.
Technical Note
69
Acknowledgements-The
authors would like to acknowledge the input and encouragement of Professor Dr t)mtir Akytiz (Bogazici University, Istanbul).
REFERENCES Doi, K. and Chan, H. P. (1980) Radiology 135 199. Hubbell, J. H. (1969) National Bureau of Standards, NSRDS-NBS, 29. Lyons, L. (1989) Statistics for Nuclear and Particle Physicists, p. 74. Cambridge University Press, Cambridge. Minato, S. (1973a) NucZ. Sci. Engng 51, 32. Minato, S. (1973b) Rad. Res. 56, 1. Minato, S. (1975) Nucl. Instrum. Meth. 131, 157. Momeni, M. H. (1985) Health Phys. 49, 3 10. Nuclear Data Project (1976), Report ORNL-5114. Simons, H. A. B. and Comet, S. (1959) Phys. in Med. and Biol. 3, 233. Sherbini, S., Tamasanis, D., Sykes, J. and Porter, S. W. (1986) Health Phys. 51, 699. Storm, E. and Israel, H. I. (1970) Nucl. Data Tables A7, 565. Swarup, J. (1979) Indian J. Pure Appl. Phys. 17, 381. Swarup, J. (1980) Nucl. Instrum. Meth. 172, 559. Swarup, J. and Ganguly, A. K. (1975) Indian J. Pure Appl. Phys. 13, 595. Swarup, J. and Minato, S. (1983) Indian J. Pure Appl. Phys. 21, 702. Uehara, S. Hoshi, M., Sawada, S., Nagatomo, T. and Ichikawa, Y. (1988) Health Phys. 54, 249.
Wecksung, G. W., Walker, J. J. and Brown, R. T. (1971) Nucl. Instrum. Meth. 95, 605.