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Effect of building-up factor on the optimum shape for a shielding barrier
ReactofScienceandjTechnology (Journalof NuclearEnergyPartsA/B) 1962,Vol. 16. pp. 375 to 378. PergamonPressLtd. Printedin NorthernIreland
LETTERS
TO THE EDITOR
Effect of building-up factor on the optimum shape for a shielding barrier
LESHCHINSKII (1960) has analysed the product e- prB(/~v,pr, z) and has suggested that this can be approximated by an equation of the form
(Received 8 December 1961)
f(r) = e-PrB(hv, ,ur, z) = ar2 + br + c (for large thicknesses) (2a) (2b) (fr) = e-fi7B(hv, ,ur, z) = br + 1 (for small thicknesses)
As SUGGESTEDby BLIZARD (1955) and later discussed for a particular case by KIMEL (1959), the most important problem of shield design is that of reducing the weight of the shield. KIMEL has considered the problem of?inding the optimum shape of the shieldine barrier for a linear isotrouic source. Theoretical work of FAN: et al. (1951, 1953) and the experimental work of WHITE (1950) indicate that one of the important factors, generally denoted as ‘build-up factor’, in the case of gamma-ray attenuation is due to the effects of multiple scattering in the shield. They have shown that the dose rate inside the shielding material, having p0 as its linear absorption coefficient, decreases e--W with distance less rapidly than a law of the form 7 . KIMEL has not considered this important factor in his calculations. A rod-shaped radioactive source or a long thin pipe containing radioactive material can be approximated as a line source. These types of sources, which prove particularly feasible in the design of movable gamma facilities, have been successfully used for several types -of industrial and experimental radiation facilities. Therefore. in this namer the effect of the build-tin factor in determining the o$cmum shape of the shielding barrier is discussed for a linear isotropic source. The gamma-ray dose rate, P, at a point A (Fig. 1) due to a linear isotropic source of length L, is given by the expression
s
LlZjY&
P=2
,,
2KM 40 Be-pusec 6 d# (1) B(hv, ,ur, z)e-pr dx = LH LRZ s0
where B is the build-up factor, ,u is the linear absorption coefficient for a narrow beam of gamma radiation in the shielding material, K is the y-constant of the isotope, M/L is the linear specific activity of the source, H is the distance of the source I from the point A and #,, = arc tan eH .
where a, b, and c are functions of the initial source energy (hv), atomic number (z) of the shielding material and the above defined linear absorption coefficient f$). In most practical situations where the problem of optimization of shielding material is applied, one deals with powerful sources and shields having thicknesses equivalent to several relaxation lengths. It rs, therefore, suggested that the most suitable approximated form of the function f(r) is given by (2a). Using this expression (2a) in (l), we get (3) Taking p as the density and t as the width of the shielding barrier, the expression for the weight of the shielding barrier is given by -W 90 ptydx=2 w=2 pt Hy se@ 4 d#. (4) s0 s0 In this variational problem, it is required to find y =f(#), such that for a specified value PO of the integral (3), the integral (4) has a minimum value. To solve this isoperimetric problem we form the auxiliary function
z=
2pt Hy
set*4 + 1 2g(ay2 se9 4
+ by set + + c) d+ I
where 3, is a Lagrange multiplier. The problem of minimizing Z leads to solving the Euler’s equation
where F is the integrand and y’ = du -.
d4
In this case, the integrand F is independent equation reduces to
of y’ so the above
aF 5=O
i.e.
2pt H Se@ 4 + 3,F(2ay
sec2 4 + b set #J)= 0.
(5)
Determining y from the equation (5), we get
y
1
Al T
+
4 ~0s #
(6)
where and Substituting PO,we get
FIG. 4
l.-Diagram
showing the notations calculations.
used in the 315
,4*=--b 2a’ this value of y in (3) and using the value of P as
376
Letters to the editor
Integrating this and solving for 11,we get
Substituting
(Received 2 February 1962)
this value of rl in (6), we get
J
$$jPO-
y=f
Measurements of the temperature dependence of thermal neutron diffusion parameters in water and ice
(a& + bA2 -I- c) Co + Aa cos $. (7)
a tan &
The choice of the sign of the first term in (7) can only be determined by analysing as to which of the two values of y will minimiie the integral (4). In order to discuss, in general, the effect of build-up factor in determining the optimum shape of the shielding barrier, we have calculated the values of y for a particular case of a linear isotropic Co60 source using the values of a, b and c for water as given by LESHCHINSKII (1960). In this case the minus sign of the first term in (7) will minimize the integral (4). Figure 2 shows the general shapes of the shielding barrier with or without taking
Y
II
A PULSEDneutron source was used in the papers by VON DARDEL (1954) and ANTONOVet al. (1955) for measurements of the neutron diffusion coefficient D,, neutron diffusion length L, mean lifetime of neutrons T,, and dilfusion cooling constants C in the moderator. It is well known, that after a certain time lapse from the beginning of the neutron pulse, the neutron density in the moderator decreases as e-at, where
The geometrical pressions :
buckling
in rectangular-form dimensions a, b, c.
moderator a=
x-x
Without
-
With
FIG. 2.-Shape
build-up
build-up
systems
with
ex-
extrapolated
2405 2 + (R 1 +G
factor
factor
of the optimum shielding barrier.
in to consideration the effect of multiple scattering in the shielding material. It should, however, be noted here that in these above calculations the effect of the scattering in air is neglected. Defence Science Laboratory Delhi, India
n is given by the following
J. P. JAIN B. L. SHARMA
REFERENCES BLIZARD E. P. (1955) Reference Material on Atomic Energy (USAEC) 1,709. FA~O U. and SPENCERL. V. (1951) J. Res. Nat. bur. Stand. 46, 446. FANO U. (1953) Nucleonics 11,(8)8; 11,(9)55. KIMEL D. P. (1959) Atomnuya Energiya 7,265. LESHCHINSKII N. I. (1960) Atomnaya Energiya 8, 62. WHITE G. R. (1950) Phys. Rev. 80, 154.
in cylindrical-form moderator system having the extrapolated radius R and extrapolated height h. The neutron transport length It, for water at 20°C and ice was taken as 0.345 cm. For water at the temperature of 0°C this value was 0.315 cm. The method of measuring the decay constant tc = ~((a) was similar to that described earlier by ANTONOVet al. The moderator was placed both in a cylindrical and rectangular vessel. The vessels were made of aluminium the thickness of which was 1.5 mm. The aluminium was covered from without by a 2-mm thick layer, around which a cooling tube was soldered. This system was surrounded by 5-cm thick insulating material (Styrofoam) and a 6-cm thick boric acid layer. The diffusion parameters were measured at the water temperatures 0°C and 2O”C, for ice at the temperature 0°C and also at several between 0°C and -8°C. No variations of the decay constant cc were observed when the insulator and cooler were removed. The moderator temperature was measured with a Cuconstantan-Cu thermocouple, one end of which was placed in the moderator and the other in melting ice. The thermocouple in the moderator was placed at the distance of 2/3 of the radius from the centre of the vessel. Such a method of measuring temperature made it possible to determine the moderator temperature with an accuracy of f0.5”C. The presence of the thermocouple did not have any affect on the values of the decay constant c(. The values .G for water were determined from the vessel dimensions and weight before and after the measurements. Those for ice were determined from the known weight of the water, vessel dimensions and from the known difference of the water and ice densities at 0°C which was taken as 8.963 per cent. Besides this, the values of G for ice were determined from the measured ice height in the vessel. The values of Q for ice, obtained by bothmethods, were in good agreement. The values of the parameters D,, T., C were calculated by the least squares method using equation (1) from the measured time dependence of the density of neutrons leaving the moderator. The values of the diffusion length L were calculated using the formula Lz = Da . T,. The results are given in Table 1. For ice only, the results for the temperature T = 0°C are listed, since in the small temperature interval used (between 0°C and - 8°C) the quantity CLwas practically constant. The diffusion coefficient D, and the mean lifetime of neutrons T, for water at the temperature 20°C are in good agreement with
LETTERS
TO THE EDITOR
Effect of building-up factor on the optimum shape for a shielding barrier
LESHCHINSKII (1960) has analysed the product e- prB(/~v,pr, z) and has suggested that this can be approximated by an equation of the form
(Received 8 December 1961)
f(r) = e-PrB(hv, ,ur, z) = ar2 + br + c (for large thicknesses) (2a) (2b) (fr) = e-fi7B(hv, ,ur, z) = br + 1 (for small thicknesses)
As SUGGESTEDby BLIZARD (1955) and later discussed for a particular case by KIMEL (1959), the most important problem of shield design is that of reducing the weight of the shield. KIMEL has considered the problem of?inding the optimum shape of the shieldine barrier for a linear isotrouic source. Theoretical work of FAN: et al. (1951, 1953) and the experimental work of WHITE (1950) indicate that one of the important factors, generally denoted as ‘build-up factor’, in the case of gamma-ray attenuation is due to the effects of multiple scattering in the shield. They have shown that the dose rate inside the shielding material, having p0 as its linear absorption coefficient, decreases e--W with distance less rapidly than a law of the form 7 . KIMEL has not considered this important factor in his calculations. A rod-shaped radioactive source or a long thin pipe containing radioactive material can be approximated as a line source. These types of sources, which prove particularly feasible in the design of movable gamma facilities, have been successfully used for several types -of industrial and experimental radiation facilities. Therefore. in this namer the effect of the build-tin factor in determining the o$cmum shape of the shielding barrier is discussed for a linear isotropic source. The gamma-ray dose rate, P, at a point A (Fig. 1) due to a linear isotropic source of length L, is given by the expression
s
LlZjY&
P=2
,,
2KM 40 Be-pusec 6 d# (1) B(hv, ,ur, z)e-pr dx = LH LRZ s0
where B is the build-up factor, ,u is the linear absorption coefficient for a narrow beam of gamma radiation in the shielding material, K is the y-constant of the isotope, M/L is the linear specific activity of the source, H is the distance of the source I from the point A and #,, = arc tan eH .
where a, b, and c are functions of the initial source energy (hv), atomic number (z) of the shielding material and the above defined linear absorption coefficient f$). In most practical situations where the problem of optimization of shielding material is applied, one deals with powerful sources and shields having thicknesses equivalent to several relaxation lengths. It rs, therefore, suggested that the most suitable approximated form of the function f(r) is given by (2a). Using this expression (2a) in (l), we get (3) Taking p as the density and t as the width of the shielding barrier, the expression for the weight of the shielding barrier is given by -W 90 ptydx=2 w=2 pt Hy se@ 4 d#. (4) s0 s0 In this variational problem, it is required to find y =f(#), such that for a specified value PO of the integral (3), the integral (4) has a minimum value. To solve this isoperimetric problem we form the auxiliary function
z=
2pt Hy
set*4 + 1 2g(ay2 se9 4
+ by set + + c) d+ I
where 3, is a Lagrange multiplier. The problem of minimizing Z leads to solving the Euler’s equation
where F is the integrand and y’ = du -.
d4
In this case, the integrand F is independent equation reduces to
of y’ so the above
aF 5=O
i.e.
2pt H Se@ 4 + 3,F(2ay
sec2 4 + b set #J)= 0.
(5)
Determining y from the equation (5), we get
y
1
Al T
+
4 ~0s #
(6)
where and Substituting PO,we get
FIG. 4
l.-Diagram
showing the notations calculations.
used in the 315
,4*=--b 2a’ this value of y in (3) and using the value of P as
376
Letters to the editor
Integrating this and solving for 11,we get
Substituting
(Received 2 February 1962)
this value of rl in (6), we get
J
$$jPO-
y=f
Measurements of the temperature dependence of thermal neutron diffusion parameters in water and ice
(a& + bA2 -I- c) Co + Aa cos $. (7)
a tan &
The choice of the sign of the first term in (7) can only be determined by analysing as to which of the two values of y will minimiie the integral (4). In order to discuss, in general, the effect of build-up factor in determining the optimum shape of the shielding barrier, we have calculated the values of y for a particular case of a linear isotropic Co60 source using the values of a, b and c for water as given by LESHCHINSKII (1960). In this case the minus sign of the first term in (7) will minimize the integral (4). Figure 2 shows the general shapes of the shielding barrier with or without taking
Y
II
A PULSEDneutron source was used in the papers by VON DARDEL (1954) and ANTONOVet al. (1955) for measurements of the neutron diffusion coefficient D,, neutron diffusion length L, mean lifetime of neutrons T,, and dilfusion cooling constants C in the moderator. It is well known, that after a certain time lapse from the beginning of the neutron pulse, the neutron density in the moderator decreases as e-at, where
The geometrical pressions :
buckling
in rectangular-form dimensions a, b, c.
moderator a=
x-x
Without
-
With
FIG. 2.-Shape
build-up
build-up
systems
with
ex-
extrapolated
2405 2 + (R 1 +G
factor
factor
of the optimum shielding barrier.
in to consideration the effect of multiple scattering in the shielding material. It should, however, be noted here that in these above calculations the effect of the scattering in air is neglected. Defence Science Laboratory Delhi, India
n is given by the following
J. P. JAIN B. L. SHARMA
REFERENCES BLIZARD E. P. (1955) Reference Material on Atomic Energy (USAEC) 1,709. FA~O U. and SPENCERL. V. (1951) J. Res. Nat. bur. Stand. 46, 446. FANO U. (1953) Nucleonics 11,(8)8; 11,(9)55. KIMEL D. P. (1959) Atomnuya Energiya 7,265. LESHCHINSKII N. I. (1960) Atomnaya Energiya 8, 62. WHITE G. R. (1950) Phys. Rev. 80, 154.
in cylindrical-form moderator system having the extrapolated radius R and extrapolated height h. The neutron transport length It, for water at 20°C and ice was taken as 0.345 cm. For water at the temperature of 0°C this value was 0.315 cm. The method of measuring the decay constant tc = ~((a) was similar to that described earlier by ANTONOVet al. The moderator was placed both in a cylindrical and rectangular vessel. The vessels were made of aluminium the thickness of which was 1.5 mm. The aluminium was covered from without by a 2-mm thick layer, around which a cooling tube was soldered. This system was surrounded by 5-cm thick insulating material (Styrofoam) and a 6-cm thick boric acid layer. The diffusion parameters were measured at the water temperatures 0°C and 2O”C, for ice at the temperature 0°C and also at several between 0°C and -8°C. No variations of the decay constant cc were observed when the insulator and cooler were removed. The moderator temperature was measured with a Cuconstantan-Cu thermocouple, one end of which was placed in the moderator and the other in melting ice. The thermocouple in the moderator was placed at the distance of 2/3 of the radius from the centre of the vessel. Such a method of measuring temperature made it possible to determine the moderator temperature with an accuracy of f0.5”C. The presence of the thermocouple did not have any affect on the values of the decay constant c(. The values .G for water were determined from the vessel dimensions and weight before and after the measurements. Those for ice were determined from the known weight of the water, vessel dimensions and from the known difference of the water and ice densities at 0°C which was taken as 8.963 per cent. Besides this, the values of G for ice were determined from the measured ice height in the vessel. The values of Q for ice, obtained by bothmethods, were in good agreement. The values of the parameters D,, T., C were calculated by the least squares method using equation (1) from the measured time dependence of the density of neutrons leaving the moderator. The values of the diffusion length L were calculated using the formula Lz = Da . T,. The results are given in Table 1. For ice only, the results for the temperature T = 0°C are listed, since in the small temperature interval used (between 0°C and - 8°C) the quantity CLwas practically constant. The diffusion coefficient D, and the mean lifetime of neutrons T, for water at the temperature 20°C are in good agreement with