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The distribution of backscattered gamma ray photons in the scattering medium
NUCLEAR ENGINEERING AND DESIGN 24 (1973) 258-262. NORTH-HOLLANDPUBLISHINGCOMPANY
THE DISTRIBUTION OF BACKSCATTERED GAMMA RAY PHOTONS IN THE SCATTERING MEDIUM K. PREISS and R. LIVNAT Division o f Engineering Sciences, Negev Institute for Arid Zone Research, Beer Sheva, Israel Received 1 November 1972
Monte Carlo calculations were performed to simulate the behavlour of photons scattered back from a semiinfinite medium. The source considered was point isotropic on the surface of the medium, and the media simulated soils of various densities. The values of the number albedo obtained agreed well with those quoted in the literature. The radial distribution was found to fall off exponentially, with a dependence on distance from the source practically independent of source energy and material, provided the photo electric absorption cross section at the source energy is neghgtble. This conclusion is supported by calculation from results quoted in the literature. The distributions of the initial direction cosine relative to the radius joining source and point of exit, and of the number of scatters, were calculated and tabulated.
I. Introduction
2. Results
Backscattering of gamma radiation has been extensively dealt with in the past, and many experimental and theoretical results have been published. The information usually available gives some property of the scattered radiation (such as energy and angular distributions, or exposure rate) for some given material, geometry, and source. There is little or no information available giving details of how the radiation distribution is modified in the scattering medium. The details of how many times photons scatter, which photon paths are most probable, and so on, are of general interest in furthering understanding of gamma ray backscattering systems. The information is of particular interest in applications such as density measurements by gamma ray backscattering, where it can contribute to the understanding of the operation of such gauges. This paper gives such information, obtained by simple Monte Carlo calculations. The source considered was point isotroplc on the surface of a semi-infinite medium, and the media chosen simulated soils of various densitles.
A limited selection of results is presented here. These were chosen so as to emphasise certain aspects /CI"1
G
B2 AI
Bz
Z
lO'o
2 4 s= p (Eo)pr rnem free paths
Fig l. No. of photons emitted in an annulus of width 2.5 cm, as a function of the radius, S mean-free-paths, normalized to one source photon.
259
K. Preiss, R. Livnat, Backscattered 7-ray photons Table 1 Calculation data, cut off criteria, and output intervals. Calculation No.
Source energy Eo [MeV]
Density [g/cm3 ]
Z A"
Output tabulation intervals No. of Cut off criterta (a) histories Energy Weight Zmax Radius Meanfree Uinitial [MeVI [cm] (b) r[cml paths traversed
A1
1.5
1.8
0.5
l0 s
B1
1.25
2.75
0.4818
0.05
0.05
32.3
2.5
0.5
-
4 X 104 0.002
0.05
19.9
2.5
-
0.1
B2
1.25
1.00
0.4818
4 × 104 0.002
0.05
54.8
2.5
-
0.1
C1
0.663
2.75
0.4818
4 X 104 0.002
0.05
14.7
2.5
-
0.1
C2
0.663
1.00
0.4818
4 × 104 0.002
0.05
40.5
2.5
-
0.1
(a) Any one criterion suffices; (b) 3 mean free paths at the source energy.
of the behaviour of backscattered photons. Further results are available in ref. [ 1]. Table 1 gives a list of data, cut off criteria, and output parameters, for the calculations here reported. The following results are quoted in this paper: (a) Number albedo (table 2), for all cases A, B and C. (b) The number of photons scattered back in each annular ring on the surface of width 2.5 cm, normalized to one source photon (fig. 1), for all cases A, B and C. (c) A frequency distribution of the number of scatters a photon undergoes (fig. 2), for calculation A. (d) A frequency distribution of the direction cosine of the photons emitted from the source, relative to the line joining the source and exit points on the surface (fig. 3), for calculations B2 and C1 which are the extreme cases for B and C. The results referred to in (c) and (d) above are quoted for an infinite plane detector, and for an annular detector between radii of 20 cm and 25 cm. It is believed that the former interval is of general interest, and the latter of interest for backscatter density gauges. 3. Analysis of results
directions and intervals of radms r is the total number albedo, referred to aSAjN by Leimd6rfer [2]. Table 2 showsAjN obtained in the present work, compared with results for similar energies and materials by Berger and Raso [3], and Davisson and Beach quoted as a private communication by Leimd6rfer [2]. The agreement is reasonable.
3.2. Dependence o f backscattered intensity on distance from the source Monte Carlo calculations by Nakamura and Hyodo [4], and Davisson and Beach (quoted by Leirnd6rfer [2] ) showed that the number of photons emerging in an annulus of unit width at radius r, decreases exponentially with r. The results shown in fig. 1 also indicate an exponential dependence. I f N is the number backscattered in a ring of unit width at radius s (s = la(Eo)Pr), then one may apply the equation:
N = NO e-mS .
(1)
Table 3 shows the values of slope m obtained for the five curves of fig. 1 by least-squares fit, taking the standard errors of the backscattered intensity into account. The number Q backscattered in a circle of radius s is then
3.1. Total aibedo Q = (No/m) (1 - e -ms) The sum of the backscattered intensity over all
(2)
260
K. Preiss, R. Livnat, Backscattered "r-ray photons
Table 2 Total number albedo Th~s work Calculation
Albedo
Eo [MeVl
.
.
.
.
.
.
.
.
Ref [31 concrete
Davlsson and Beach (1964) + concrete
Eo
Albedo
Eo
Albedo
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Al
15
0 368
2.0
0.348
6.13
0.362
Bt or B 2
1.25
0.379
1.25
0 383
2,5
0.352
Cl or C2
0 663
0.419
1.0
0.413
1,0
0.413
0.5
0.447
0.662
0 437
0.2
0.438
0.2
0.449
0.1
0.333
t Ref. 12].
Table 3
and the number albedo is
This work
A j N = No/tTI .
Calculation
Slope m
Std error ofm
AI
0.83
0.03
Bt
0.79
0.03
B2
0.75
0.04
Ct
0.81
0.01
C2
0.78
0.02
Material
Assumed density [g/cm 3]
1.25 MeV
1.0 MeV
0.663 MeV
0.2 MeV
Davisson and Beach (1964) Water Concrete Iron Lead
1.0 2.35 7.87 11.35
-
-
0.82 0 82 0.86* 0.96*
0.70 0.81 1.05" 1.62"
0.72 0.88 1.28" 2.07*
-
0.63 0.77 0.99* 1 24*
Ref. [4] Polyethylene Aluminum Iron Lead
0.65 0.84 0.92* 1 14"
-
Where t~ph(Eo)/#s(Eo) > 0.005 an astenx is indicated.
These are similar to the equations shown by Leimd6rfer [2] and Nakamura and Hyodo [4]. Davisson and Beach as quoted by Leimd&fer [2], gave the coefficients a in the relation (r in cm) N o: e - a t "
Slope m
(3)
(4)
These values of a when divided by l a ( E o ) P to give slope in units of (mean free path)-1 are given in table 3. The values of density p assumed are shown in table 3, and the attenuation coefficients were assumed to be chosen of Grodstein [5]. Table 3 shows a similar calculation, using the resuits obtained by Nakamura and Hyodo [4] from Monte Carlo calculations, assuming that the densities used in their calculation were those used in their experiment, and assuming they used the attenuation coefficients of Grodstein [5]. In table 3, an asterisk is shown alongside those coefficients for which the photoelectric interaction cross section is not negligible at the source energy. It must also be noted that the calculated results of Nakamura and Hyodo [4] for polyethylene were found by them to be 10 to 20% below their experimental values. If one disregards those values and those results for which
K. Preiss, R. Livnat, Backscattered "r-rayphotons
261
/2ph (EO)/t,ts(Eo) > 0 . 0 0 5 ,
then it is seen that all the values of m cluster around the rounded-off value 0.8. Generalizing, it may be said as an empirical finding that if
1.0
O' '
!
ltph(Eo)/las(Eo) < 0.005, then m in eqs. (1) to (3) is approximately equal to 0.8. If/aph(E0) is not negligible then the backscattered intensity drops off more steeply than with the coefficient 0.8.
3.3. The number of times photons scatter For all the photons, 39% scatter once, the rest up to ten times. For the interval 20 to 25 cm, there are about equal numbers of 1, 2 and 3 scatters (around 20%), the frequency decreasing to a negligible quantity at 10 scatters. No conclusions are drawn from these numbers, except to point out that the once-scattered model, although possibly useful in certain calculations, is not generally valid as sometimes assumed. The results quoted here permit some visualization of the behaviour of the backscattered photons in the scattered medium. 50 L L I
0
-- O
--~--4 2 6 K3 No. of scotters
Fig. 2. Frequency distribution of the number of scatters a photon undergoes, for calculation A (see table 1), for two detector dimensions.
3.4. Distribution of initial direction cosine In addition to the results shown in fig. 3, it may be noted that for the source energy 1.25 MeV, density 1.0 g/cm 3, and exit interval 0 to 5 cm, 36% of photons are from the interval 0.9 < Um.i~ < 1.0. For the source energy 0.663 MeV, density 2.75 g/cm 3, and exit interval 25 to oo cm, 59% of photons are in the same interval. The two results quoted are the extremes for the range of parameters used, and so it is seen that the distribution of UmitialIs not markedly dependent on source energy, material density, or
g
" ....
J
cos ~ Fig. 3. Frequency distribution o f the direction cosine o f the source p h o t o n relative to the line loimng source and exit point, for two extreme cases o f input parameters. The sohd hne is for the interval 0 < r < ~ , the broken line for the interval 20 cm < r < 25 cm.
distance from the source to the photon exit point. It may be noticed that the frequency drops rapidly as Uimt~1 decreases, and very few photons have a negative initial direction cosine. In general, over all intervals, about half the photons move in an initial direction which makes an angle of less than 30 ° with the radius on the material surface joining the source and exit points, and almost all make an angle of less than 90 ° .
4. Conclusions The values of number albedo obtained agree with those quoted in the literature. It has been reported in the literature that the radial distribution of radiation leaving the material follows a log-linear law. It was here shown from the present calculations and from the literature that in general, if /.tpb (Eo)/IJs(Eo) < 0.005,
then the e-folding length is 0.8 of a mean free path at the source energy. If/aph (E0) is appreciably greater than zero then the e-folding length is shorter, decreasing as ~tpb(E0) increases. Statistics were given on the number of scatters the photons undergo and on the distribution of initial directions. From the results given one may visualize the distribution of backscattered photons in the scattering medium.
262
K. Preiss, R. Livnat, Backscattered ~ray photons
Nomenclature r = radius from source, on surface [cm], s = radius from source, in mean free paths; s = U(Eo)Pr,
attenuation coefficient [cm2/g], p = density [g/cm3], E 0 = source energy, U i n i ~ = direction cosine of source photon relative to the line joining the source and point of exit, m = slope of the semi-log plot of intensity vs radius s, Q = number of photons backscattered in a circle of radms r or s, N = number of photons backscattered from an annulus of unit width at s, N O = N w h e n s = 0, AjN = number albedo, equal to Q when r = oo.
References [1 ] K. Prelss and R. Livnat, Note on the Behaviour of Backscattered Gamma Ray Photons in the Scattering Medium, Negev Institute for Arid Zone Research, Divn. of Engrg. Sci. Report 120 (1972). [2] M. Lelmdbrfer, The Backscattering of Photons, Chapter 4.4 of Engineering Compendium on Radiation Shielding, Springer-Verlag (1968). [3] M.J. Berger and D.J. Raso, Monte Carlo Calculations of Gamma Ray Backscattering, Radiation Research, 12,1 (1960) 20. [4] T. Nakamura and T Hyodo, Radial Distribution of Photons Backscattered from the Surface of a SemiInfimte Scatterer, J. Nucl. Sci. Tech., 6, 3 (1969) 143. [5] G.W. Grodstein, X-ray Attenuauon Coefficients from 10 keV to 100 MeV, USNBS Circular 583 (1957).
THE DISTRIBUTION OF BACKSCATTERED GAMMA RAY PHOTONS IN THE SCATTERING MEDIUM K. PREISS and R. LIVNAT Division o f Engineering Sciences, Negev Institute for Arid Zone Research, Beer Sheva, Israel Received 1 November 1972
Monte Carlo calculations were performed to simulate the behavlour of photons scattered back from a semiinfinite medium. The source considered was point isotropic on the surface of the medium, and the media simulated soils of various densities. The values of the number albedo obtained agreed well with those quoted in the literature. The radial distribution was found to fall off exponentially, with a dependence on distance from the source practically independent of source energy and material, provided the photo electric absorption cross section at the source energy is neghgtble. This conclusion is supported by calculation from results quoted in the literature. The distributions of the initial direction cosine relative to the radius joining source and point of exit, and of the number of scatters, were calculated and tabulated.
I. Introduction
2. Results
Backscattering of gamma radiation has been extensively dealt with in the past, and many experimental and theoretical results have been published. The information usually available gives some property of the scattered radiation (such as energy and angular distributions, or exposure rate) for some given material, geometry, and source. There is little or no information available giving details of how the radiation distribution is modified in the scattering medium. The details of how many times photons scatter, which photon paths are most probable, and so on, are of general interest in furthering understanding of gamma ray backscattering systems. The information is of particular interest in applications such as density measurements by gamma ray backscattering, where it can contribute to the understanding of the operation of such gauges. This paper gives such information, obtained by simple Monte Carlo calculations. The source considered was point isotroplc on the surface of a semi-infinite medium, and the media chosen simulated soils of various densitles.
A limited selection of results is presented here. These were chosen so as to emphasise certain aspects /CI"1
G
B2 AI
Bz
Z
lO'o
2 4 s= p (Eo)pr rnem free paths
Fig l. No. of photons emitted in an annulus of width 2.5 cm, as a function of the radius, S mean-free-paths, normalized to one source photon.
259
K. Preiss, R. Livnat, Backscattered 7-ray photons Table 1 Calculation data, cut off criteria, and output intervals. Calculation No.
Source energy Eo [MeV]
Density [g/cm3 ]
Z A"
Output tabulation intervals No. of Cut off criterta (a) histories Energy Weight Zmax Radius Meanfree Uinitial [MeVI [cm] (b) r[cml paths traversed
A1
1.5
1.8
0.5
l0 s
B1
1.25
2.75
0.4818
0.05
0.05
32.3
2.5
0.5
-
4 X 104 0.002
0.05
19.9
2.5
-
0.1
B2
1.25
1.00
0.4818
4 × 104 0.002
0.05
54.8
2.5
-
0.1
C1
0.663
2.75
0.4818
4 X 104 0.002
0.05
14.7
2.5
-
0.1
C2
0.663
1.00
0.4818
4 × 104 0.002
0.05
40.5
2.5
-
0.1
(a) Any one criterion suffices; (b) 3 mean free paths at the source energy.
of the behaviour of backscattered photons. Further results are available in ref. [ 1]. Table 1 gives a list of data, cut off criteria, and output parameters, for the calculations here reported. The following results are quoted in this paper: (a) Number albedo (table 2), for all cases A, B and C. (b) The number of photons scattered back in each annular ring on the surface of width 2.5 cm, normalized to one source photon (fig. 1), for all cases A, B and C. (c) A frequency distribution of the number of scatters a photon undergoes (fig. 2), for calculation A. (d) A frequency distribution of the direction cosine of the photons emitted from the source, relative to the line joining the source and exit points on the surface (fig. 3), for calculations B2 and C1 which are the extreme cases for B and C. The results referred to in (c) and (d) above are quoted for an infinite plane detector, and for an annular detector between radii of 20 cm and 25 cm. It is believed that the former interval is of general interest, and the latter of interest for backscatter density gauges. 3. Analysis of results
directions and intervals of radms r is the total number albedo, referred to aSAjN by Leimd6rfer [2]. Table 2 showsAjN obtained in the present work, compared with results for similar energies and materials by Berger and Raso [3], and Davisson and Beach quoted as a private communication by Leimd6rfer [2]. The agreement is reasonable.
3.2. Dependence o f backscattered intensity on distance from the source Monte Carlo calculations by Nakamura and Hyodo [4], and Davisson and Beach (quoted by Leirnd6rfer [2] ) showed that the number of photons emerging in an annulus of unit width at radius r, decreases exponentially with r. The results shown in fig. 1 also indicate an exponential dependence. I f N is the number backscattered in a ring of unit width at radius s (s = la(Eo)Pr), then one may apply the equation:
N = NO e-mS .
(1)
Table 3 shows the values of slope m obtained for the five curves of fig. 1 by least-squares fit, taking the standard errors of the backscattered intensity into account. The number Q backscattered in a circle of radius s is then
3.1. Total aibedo Q = (No/m) (1 - e -ms) The sum of the backscattered intensity over all
(2)
260
K. Preiss, R. Livnat, Backscattered "r-ray photons
Table 2 Total number albedo Th~s work Calculation
Albedo
Eo [MeVl
.
.
.
.
.
.
.
.
Ref [31 concrete
Davlsson and Beach (1964) + concrete
Eo
Albedo
Eo
Albedo
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Al
15
0 368
2.0
0.348
6.13
0.362
Bt or B 2
1.25
0.379
1.25
0 383
2,5
0.352
Cl or C2
0 663
0.419
1.0
0.413
1,0
0.413
0.5
0.447
0.662
0 437
0.2
0.438
0.2
0.449
0.1
0.333
t Ref. 12].
Table 3
and the number albedo is
This work
A j N = No/tTI .
Calculation
Slope m
Std error ofm
AI
0.83
0.03
Bt
0.79
0.03
B2
0.75
0.04
Ct
0.81
0.01
C2
0.78
0.02
Material
Assumed density [g/cm 3]
1.25 MeV
1.0 MeV
0.663 MeV
0.2 MeV
Davisson and Beach (1964) Water Concrete Iron Lead
1.0 2.35 7.87 11.35
-
-
0.82 0 82 0.86* 0.96*
0.70 0.81 1.05" 1.62"
0.72 0.88 1.28" 2.07*
-
0.63 0.77 0.99* 1 24*
Ref. [4] Polyethylene Aluminum Iron Lead
0.65 0.84 0.92* 1 14"
-
Where t~ph(Eo)/#s(Eo) > 0.005 an astenx is indicated.
These are similar to the equations shown by Leimd6rfer [2] and Nakamura and Hyodo [4]. Davisson and Beach as quoted by Leimd&fer [2], gave the coefficients a in the relation (r in cm) N o: e - a t "
Slope m
(3)
(4)
These values of a when divided by l a ( E o ) P to give slope in units of (mean free path)-1 are given in table 3. The values of density p assumed are shown in table 3, and the attenuation coefficients were assumed to be chosen of Grodstein [5]. Table 3 shows a similar calculation, using the resuits obtained by Nakamura and Hyodo [4] from Monte Carlo calculations, assuming that the densities used in their calculation were those used in their experiment, and assuming they used the attenuation coefficients of Grodstein [5]. In table 3, an asterisk is shown alongside those coefficients for which the photoelectric interaction cross section is not negligible at the source energy. It must also be noted that the calculated results of Nakamura and Hyodo [4] for polyethylene were found by them to be 10 to 20% below their experimental values. If one disregards those values and those results for which
K. Preiss, R. Livnat, Backscattered "r-rayphotons
261
/2ph (EO)/t,ts(Eo) > 0 . 0 0 5 ,
then it is seen that all the values of m cluster around the rounded-off value 0.8. Generalizing, it may be said as an empirical finding that if
1.0
O' '
!
ltph(Eo)/las(Eo) < 0.005, then m in eqs. (1) to (3) is approximately equal to 0.8. If/aph(E0) is not negligible then the backscattered intensity drops off more steeply than with the coefficient 0.8.
3.3. The number of times photons scatter For all the photons, 39% scatter once, the rest up to ten times. For the interval 20 to 25 cm, there are about equal numbers of 1, 2 and 3 scatters (around 20%), the frequency decreasing to a negligible quantity at 10 scatters. No conclusions are drawn from these numbers, except to point out that the once-scattered model, although possibly useful in certain calculations, is not generally valid as sometimes assumed. The results quoted here permit some visualization of the behaviour of the backscattered photons in the scattered medium. 50 L L I
0
-- O
--~--4 2 6 K3 No. of scotters
Fig. 2. Frequency distribution of the number of scatters a photon undergoes, for calculation A (see table 1), for two detector dimensions.
3.4. Distribution of initial direction cosine In addition to the results shown in fig. 3, it may be noted that for the source energy 1.25 MeV, density 1.0 g/cm 3, and exit interval 0 to 5 cm, 36% of photons are from the interval 0.9 < Um.i~ < 1.0. For the source energy 0.663 MeV, density 2.75 g/cm 3, and exit interval 25 to oo cm, 59% of photons are in the same interval. The two results quoted are the extremes for the range of parameters used, and so it is seen that the distribution of UmitialIs not markedly dependent on source energy, material density, or
g
" ....
J
cos ~ Fig. 3. Frequency distribution o f the direction cosine o f the source p h o t o n relative to the line loimng source and exit point, for two extreme cases o f input parameters. The sohd hne is for the interval 0 < r < ~ , the broken line for the interval 20 cm < r < 25 cm.
distance from the source to the photon exit point. It may be noticed that the frequency drops rapidly as Uimt~1 decreases, and very few photons have a negative initial direction cosine. In general, over all intervals, about half the photons move in an initial direction which makes an angle of less than 30 ° with the radius on the material surface joining the source and exit points, and almost all make an angle of less than 90 ° .
4. Conclusions The values of number albedo obtained agree with those quoted in the literature. It has been reported in the literature that the radial distribution of radiation leaving the material follows a log-linear law. It was here shown from the present calculations and from the literature that in general, if /.tpb (Eo)/IJs(Eo) < 0.005,
then the e-folding length is 0.8 of a mean free path at the source energy. If/aph (E0) is appreciably greater than zero then the e-folding length is shorter, decreasing as ~tpb(E0) increases. Statistics were given on the number of scatters the photons undergo and on the distribution of initial directions. From the results given one may visualize the distribution of backscattered photons in the scattering medium.
262
K. Preiss, R. Livnat, Backscattered ~ray photons
Nomenclature r = radius from source, on surface [cm], s = radius from source, in mean free paths; s = U(Eo)Pr,
attenuation coefficient [cm2/g], p = density [g/cm3], E 0 = source energy, U i n i ~ = direction cosine of source photon relative to the line joining the source and point of exit, m = slope of the semi-log plot of intensity vs radius s, Q = number of photons backscattered in a circle of radms r or s, N = number of photons backscattered from an annulus of unit width at s, N O = N w h e n s = 0, AjN = number albedo, equal to Q when r = oo.
References [1 ] K. Prelss and R. Livnat, Note on the Behaviour of Backscattered Gamma Ray Photons in the Scattering Medium, Negev Institute for Arid Zone Research, Divn. of Engrg. Sci. Report 120 (1972). [2] M. Lelmdbrfer, The Backscattering of Photons, Chapter 4.4 of Engineering Compendium on Radiation Shielding, Springer-Verlag (1968). [3] M.J. Berger and D.J. Raso, Monte Carlo Calculations of Gamma Ray Backscattering, Radiation Research, 12,1 (1960) 20. [4] T. Nakamura and T Hyodo, Radial Distribution of Photons Backscattered from the Surface of a SemiInfimte Scatterer, J. Nucl. Sci. Tech., 6, 3 (1969) 143. [5] G.W. Grodstein, X-ray Attenuauon Coefficients from 10 keV to 100 MeV, USNBS Circular 583 (1957).