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The single backscattering of gamma rays
NUCLEAR I N S T R U M E N T S AND METHODS
144 ( 1 9 7 7 )
557-559
; ©
N O R T H - H O L L A N D PUBLISHING CO.
THE SINGLE BACKSCATTERING OF GAMMA RAYS JOSEF SEDA, JIR[ STRACHOTA, TOM,~S (2ECH,~K and JAROSLAV KLUSON
Department of Dosimeoy and Application ol Ionising Radiation, Faculty of Nuclear Science and Physical Engineering, Technical University o/ Prague, (zechoslovakia Received 9 December 1976 and in revised form 28 February 1977 The paper presents the results of the single backscattering investigations of gamma rays in various materials. A simplified mathematical model is given for determining the probability that a photon scattered back by a single Compton scattering is incident on the detector. The probability is calculated as a function of the primary photon energy and the atomic number Z of the scatterer.
The number of applications of the backscattering radiation of gamma rays requires theoretical and experimental investigations for determining the dependence of the scattered photon flux on primary photon energy, atomic number Z of the scatterer and the geometry used during the measurements. This problem has alreay been dealt with by several authorsla), who have pointed out that there exists only one maximum on the curve of the backscattered photon number indicated by the detector, N = f ( Z ) particularly for Z varying in the range from 25 to 30. However, the function N = f ( Z ) has been verified experimentally or theoretically only for a few values of Z. A more detailed analysis, performed for 89 elements, has shown a more complex dependence than that given previously. A single scattering proces has been considered as a basis for the theoretical model due to the fact rD
/
h
dc Fig. 1. Geometrical arrangement of sample source and detector.
that the analytical expression of a double or multiple scattering is rather complicated. The geometry shown in fig. 1 was chosen for deriving the probability that a photon having undergone a single Compton scattering in the material is incident back on the detector. This geometry (fig. 1) has been widely used both in the research and engineering applications of backscattering. Let us assume that a photon is emitted by an isotropic source located in the point O close to the surface of the scatterer. After passing the distance r in the material, the photon undergoes a Compton scattering in a volume element d V and the scattered photon is incident back on the detector in the point B. This process can be characterized by a definite probability. The absorption of the gamma radiation in the air can be neglected. The symbol r~ indicates the distance at which the intensity of the primary beam is attenuated to 0.3% of the original value. The probability PI that a photon being emitted by a point isotropic source will reach the volume element d V at point A is given by P1 -
1 exp(--/~ 1r)
4/Tr 2
r2
COSOdOd~o
(I)
where /.t~ is the linear attenuation coefficient of the material for the photon energy E and ~0 is the azimuthal angle around the axis of symmetry. The probability that the photon is scattered by a free electron in the volume dV through an angle 0-into the unit solid angle is Z N AP da(0) P2 =
A
d T dr,
(2)
558
et al.
J. S E D A
where Z, A, p are the proton number, mass number and the density of the scatterer respectively. N A is Avogadro's number and da(0)/d,Q is the Klein-Nishina formula. The probability that the photon, being scattered in the process described above, enters the surface element dS of the detector is P~ =
(3)
exp(--It2/),
where ~2 is the linear attenuation coefficient of the scatterer for the energy E ' of the scattered photon and t is the distance which the scattered photon has passed through the scatterer (fig. 1). The overall probability P is given by integrating the product o v e r the whole volume of the scatterer and the surface of the detector3). To simplify the calculation we have assumed the value ~2 to be equal to # corresponding to the energy of the photon backscattered at the angle O= re. Further, the integration over the detector surface is simplified by multiplying the integrand by the detector surface area S. When using the simplifying assumptions mentioned above the resuiting probability for a photon being emitted by a source at the point O and scattered back by the Compton effect into the detector at height h above the source (and surface of the scattering medium) is given by
PIP2P3
A
2
exp
-
#~+#2sinO
x
×[(h-krsinO)2+r2cos20]-1}1 /~-+;: sin 0
r
x
cos 3 (h + r sin O) da(O)drdO x [(t7+t" cosO) 2 + r 2 C0S20] ~ dO
(4)
P 3 . 1 0 -2 '
oo o o "
o
o
2.10-2.
%*
o
$
*
°
o
%
o
°° o
o o
o
o
o
1.10-2 -
o
.
o
o oo
0
o*-°..~ . - ~
o o
°
%.
o
o
o
o~
o o
o
I
I
I
I
I
I
t
I
I
I
10
20
30
40
50
60
70
80
90
It~O
Fig. 3. Probability P as a function
Z
Z for E = 1 2 5 0 k e V ;
tl = 2 cm.
By using eq. (4), the dependence of the overall probability P on the atomic number Z and several values of the photon energy in the range 60-1250 keV has been calculated. The integration has been carried out numerically with the number of steps being chosen so that the final deviation is smaller than 4°/6. The probability function P--P(Z) is shown in figs. 2 and 3 for the photon energies 661 and 1250 keV respectively and for the detector located at h = 2 cm above the surface. The results clearly show that the shape of the calculated curve is primarily affected by the scatterer density p which depends on the atomic number Z. The curve P = P(Z) exhibits well defined minima in the regions corresponding to the positions of the gaseous elements in the periodic table of the elements. Another effect which plays an important role is a continuous decrease of the counting rate albedo with Z in the whole energy region having been investigated4). P
7.10"2 P 4 . I 0 "2
6 . 1 0 -2
e~ o
o 5 . 1 0 -2 •
3 . 1 0 -2
°
°
°~****
b
o
o
o
Fe
°o
4 . 1 0 -2 . o o
o
o
2 . 1 0 -2 •
2
o •
o
°
° 1 . 1 0 -2 •
o
3.10-2
~ o
o
°
o
°
o-~,~***.°__
o
o~ o
oo~o
o
oo
2.10"2.
.%
o 1,10 2 -
Fig.
I
I
4
I
I
i
[
I
I
10
20
30
40
50
60
70
80
90
2. Probability h = 2 cm.
P
as a function
Z
I
100 Z
for E = 6 6 1 k e V ;
0
0
200
400
600
800
1 000
Fig. 4. Probability P as a function E for Z = 6 , h=2cm.
1200
E [keV']
13, 26, 50, 82;
SINGLE B A C K S C A T T E R I N G
The values of the probability P have also been alculated as a function of the primary photon enrgy. The results are given in fig. 4 for Z = 6, 13, 6, 50 and 82 and for the detector location --2 cm above the scatterer surface. The shape of he curve P(E) is determined by the energy de,endence of the individual components of the toal attenuation coefficient p = f(E). The cross-secion of the photoeffect r decreases in the low enrgy region more steeply than that of the Compon effect o. Therefore, the fraction of the Compon scattering cross-section tr of the total value /~ ncreases with energy, resulting in an increase of he curve P(E) in the low energy region. For high:r energies where the photoeffect is negligible, the :haracter of the P(E) curve is determined mainly ly the Compton cross-section decreasing with en',rgy. The curve P(E) must therefore exhibit a naximum. The maximum for the carbon scatterer vould appear at the region of lower energies for vhich the calculation has not been performed (fig. 0. A simplified mathematical model has been used or the calculation of the probability that a photon roving undergone a single Compton scattering
OF GAMMA RAYS
559
hits the detector. This probability has been calculated as a function of the atomic number Z and energy E. The presented results lead to the conclusion that the function P(Z) is primarily affected by the density function p(Z). The curve P(Z) exhibits a maximum in each group of the periodic table of the elements and not only in the range of Z from 25 to 30 as has been previously published. The energy dependence of P is simpler. The shape of the curve is affected by the relative fraction of the Compton scattering cross-section which increases in the energy region up to 1 MeV. The absolute value of the Compton scattering crosssection, of course, decreases with increasing energy.
References l) E. Dahn, Isotopenpraxis 5 (1969) 252. 2) U. A. Ulmanis, and N. A. Dubinskaja, Atomnaja Energija 3 (1957) 59. 3) j. j. Fitzgerald, G. L. Brownell and F. J. Mahoney, Mathematical theory Of radiation dosimetry (J. Wiley, New York, 1967). 4) M. J. Berger and D. J. Raso, Rad. Res. 12 (1960) 20. 5) j. H. Hubbell, NSRDS-NBS 29 (1969).
144 ( 1 9 7 7 )
557-559
; ©
N O R T H - H O L L A N D PUBLISHING CO.
THE SINGLE BACKSCATTERING OF GAMMA RAYS JOSEF SEDA, JIR[ STRACHOTA, TOM,~S (2ECH,~K and JAROSLAV KLUSON
Department of Dosimeoy and Application ol Ionising Radiation, Faculty of Nuclear Science and Physical Engineering, Technical University o/ Prague, (zechoslovakia Received 9 December 1976 and in revised form 28 February 1977 The paper presents the results of the single backscattering investigations of gamma rays in various materials. A simplified mathematical model is given for determining the probability that a photon scattered back by a single Compton scattering is incident on the detector. The probability is calculated as a function of the primary photon energy and the atomic number Z of the scatterer.
The number of applications of the backscattering radiation of gamma rays requires theoretical and experimental investigations for determining the dependence of the scattered photon flux on primary photon energy, atomic number Z of the scatterer and the geometry used during the measurements. This problem has alreay been dealt with by several authorsla), who have pointed out that there exists only one maximum on the curve of the backscattered photon number indicated by the detector, N = f ( Z ) particularly for Z varying in the range from 25 to 30. However, the function N = f ( Z ) has been verified experimentally or theoretically only for a few values of Z. A more detailed analysis, performed for 89 elements, has shown a more complex dependence than that given previously. A single scattering proces has been considered as a basis for the theoretical model due to the fact rD
/
h
dc Fig. 1. Geometrical arrangement of sample source and detector.
that the analytical expression of a double or multiple scattering is rather complicated. The geometry shown in fig. 1 was chosen for deriving the probability that a photon having undergone a single Compton scattering in the material is incident back on the detector. This geometry (fig. 1) has been widely used both in the research and engineering applications of backscattering. Let us assume that a photon is emitted by an isotropic source located in the point O close to the surface of the scatterer. After passing the distance r in the material, the photon undergoes a Compton scattering in a volume element d V and the scattered photon is incident back on the detector in the point B. This process can be characterized by a definite probability. The absorption of the gamma radiation in the air can be neglected. The symbol r~ indicates the distance at which the intensity of the primary beam is attenuated to 0.3% of the original value. The probability PI that a photon being emitted by a point isotropic source will reach the volume element d V at point A is given by P1 -
1 exp(--/~ 1r)
4/Tr 2
r2
COSOdOd~o
(I)
where /.t~ is the linear attenuation coefficient of the material for the photon energy E and ~0 is the azimuthal angle around the axis of symmetry. The probability that the photon is scattered by a free electron in the volume dV through an angle 0-into the unit solid angle is Z N AP da(0) P2 =
A
d T dr,
(2)
558
et al.
J. S E D A
where Z, A, p are the proton number, mass number and the density of the scatterer respectively. N A is Avogadro's number and da(0)/d,Q is the Klein-Nishina formula. The probability that the photon, being scattered in the process described above, enters the surface element dS of the detector is P~ =
(3)
exp(--It2/),
where ~2 is the linear attenuation coefficient of the scatterer for the energy E ' of the scattered photon and t is the distance which the scattered photon has passed through the scatterer (fig. 1). The overall probability P is given by integrating the product o v e r the whole volume of the scatterer and the surface of the detector3). To simplify the calculation we have assumed the value ~2 to be equal to # corresponding to the energy of the photon backscattered at the angle O= re. Further, the integration over the detector surface is simplified by multiplying the integrand by the detector surface area S. When using the simplifying assumptions mentioned above the resuiting probability for a photon being emitted by a source at the point O and scattered back by the Compton effect into the detector at height h above the source (and surface of the scattering medium) is given by
PIP2P3
A
2
exp
-
#~+#2sinO
x
×[(h-krsinO)2+r2cos20]-1}1 /~-+;: sin 0
r
x
cos 3 (h + r sin O) da(O)drdO x [(t7+t" cosO) 2 + r 2 C0S20] ~ dO
(4)
P 3 . 1 0 -2 '
oo o o "
o
o
2.10-2.
%*
o
$
*
°
o
%
o
°° o
o o
o
o
o
1.10-2 -
o
.
o
o oo
0
o*-°..~ . - ~
o o
°
%.
o
o
o
o~
o o
o
I
I
I
I
I
I
t
I
I
I
10
20
30
40
50
60
70
80
90
It~O
Fig. 3. Probability P as a function
Z
Z for E = 1 2 5 0 k e V ;
tl = 2 cm.
By using eq. (4), the dependence of the overall probability P on the atomic number Z and several values of the photon energy in the range 60-1250 keV has been calculated. The integration has been carried out numerically with the number of steps being chosen so that the final deviation is smaller than 4°/6. The probability function P--P(Z) is shown in figs. 2 and 3 for the photon energies 661 and 1250 keV respectively and for the detector located at h = 2 cm above the surface. The results clearly show that the shape of the calculated curve is primarily affected by the scatterer density p which depends on the atomic number Z. The curve P = P(Z) exhibits well defined minima in the regions corresponding to the positions of the gaseous elements in the periodic table of the elements. Another effect which plays an important role is a continuous decrease of the counting rate albedo with Z in the whole energy region having been investigated4). P
7.10"2 P 4 . I 0 "2
6 . 1 0 -2
e~ o
o 5 . 1 0 -2 •
3 . 1 0 -2
°
°
°~****
b
o
o
o
Fe
°o
4 . 1 0 -2 . o o
o
o
2 . 1 0 -2 •
2
o •
o
°
° 1 . 1 0 -2 •
o
3.10-2
~ o
o
°
o
°
o-~,~***.°__
o
o~ o
oo~o
o
oo
2.10"2.
.%
o 1,10 2 -
Fig.
I
I
4
I
I
i
[
I
I
10
20
30
40
50
60
70
80
90
2. Probability h = 2 cm.
P
as a function
Z
I
100 Z
for E = 6 6 1 k e V ;
0
0
200
400
600
800
1 000
Fig. 4. Probability P as a function E for Z = 6 , h=2cm.
1200
E [keV']
13, 26, 50, 82;
SINGLE B A C K S C A T T E R I N G
The values of the probability P have also been alculated as a function of the primary photon enrgy. The results are given in fig. 4 for Z = 6, 13, 6, 50 and 82 and for the detector location --2 cm above the scatterer surface. The shape of he curve P(E) is determined by the energy de,endence of the individual components of the toal attenuation coefficient p = f(E). The cross-secion of the photoeffect r decreases in the low enrgy region more steeply than that of the Compon effect o. Therefore, the fraction of the Compon scattering cross-section tr of the total value /~ ncreases with energy, resulting in an increase of he curve P(E) in the low energy region. For high:r energies where the photoeffect is negligible, the :haracter of the P(E) curve is determined mainly ly the Compton cross-section decreasing with en',rgy. The curve P(E) must therefore exhibit a naximum. The maximum for the carbon scatterer vould appear at the region of lower energies for vhich the calculation has not been performed (fig. 0. A simplified mathematical model has been used or the calculation of the probability that a photon roving undergone a single Compton scattering
OF GAMMA RAYS
559
hits the detector. This probability has been calculated as a function of the atomic number Z and energy E. The presented results lead to the conclusion that the function P(Z) is primarily affected by the density function p(Z). The curve P(Z) exhibits a maximum in each group of the periodic table of the elements and not only in the range of Z from 25 to 30 as has been previously published. The energy dependence of P is simpler. The shape of the curve is affected by the relative fraction of the Compton scattering cross-section which increases in the energy region up to 1 MeV. The absolute value of the Compton scattering crosssection, of course, decreases with increasing energy.
References l) E. Dahn, Isotopenpraxis 5 (1969) 252. 2) U. A. Ulmanis, and N. A. Dubinskaja, Atomnaja Energija 3 (1957) 59. 3) j. j. Fitzgerald, G. L. Brownell and F. J. Mahoney, Mathematical theory Of radiation dosimetry (J. Wiley, New York, 1967). 4) M. J. Berger and D. J. Raso, Rad. Res. 12 (1960) 20. 5) j. H. Hubbell, NSRDS-NBS 29 (1969).