NUCLEONICS DATA SHEET No. 19 
Induced Radiation 
Radiation from Neutron-Activated 
Slabs and Spheres 
By W. B.-LEWIS 
Atomic Energy Division 
Phillips Petroleum Co. 
Idaho Falls, Idaho 
THE RADIATION from an object acti- 
vated by irradiation in a neutron flux 
that is uniform and isotropic depends 
on the manner in which the induced 
activity is distributed through the 
sample volume and on the absorption 
coefficient of the object for the radia- 
tion under consideration. The distri- 
bution of induced activity depends on 
the shape of the sample and on the 
coefficient of absorption for neutrons. 
The effective activity, defined as the 
ratio of the amount of radiation leaving 
the surface to the total amount induced 
in the object, can be determined by 
solving two problems sequentially. 
First, the distribution of the radioactive 
nuclides must be determined. Sec- 
ond, the fraction of radiation reaching 
the surface from an element of volyme 
must be calculated, and an integration 
made over the entire volume. 
These problems have been solved for 
two simple cases: (1) a sphere of radius 
a and (2) an infinite slab of half- 
thickness a. Three quantities are dis- 
played: the density distribution Z, the 
effective activity s, and contours of 
constant s. Figure 1 shows these 
for spheres, and Fig. 2 for slabs. 
Analytical expressions can be ob- 
tained for four quantities. These are 
tabulated below. The general functions 
s(B,,B,) are obtained by interpolation. 
The contours of s(B,,B,) offer a 
quick means to evaluate the effective 
activity. It is necessary only to 
evaluate B, and B, from the physical 
constants of the sample and locate the 
corresponding point on the map. The 
accuracy of the value obtained is about 
+10%. The contours cannot be used 
when both B’s are close to unity. In 
such cases recourse to numerical inte- 
gration is necessary, or, better yet, just 
irradiate and see what happens. 
Example: Suppose we irradiate a 
typical iridium sphere having the fol- 
lowing characteristics: mass = 800 mg, 
Hn = 30.8cm™, up, = 8.0cm™, a = 0.204 
cm, bn = 6.08, b, = 1.63, B, = 0.9977, 
B, = 0.805. 
Since b, is large, most of the activity 
is produced in a thin outer shell, and 
since 0, is fairly large, the efficiency of 
recovery should be about 50%. The 
problem is complicated by the large 
range of gamma energies. There are 
11 gammas listed for Ir with energies 
ranging from 0.136 to 0.613 Mev. The 
value chosen for yp, is a good estimate 
for four gammas that account for 80% 
of the total energy: 0.296, 0.308, 0.316, 
0.468 Mev. 
Values of s are 0.65 from Fig. 1 and 
0.59 from numerical integration. 
Quantities Important in Evaluating Neutron-Induced Radiation 
Quantity Value for sphere Value for slab 
Z(buz) = > 5 [cosh Banal bao aad o bel 2) p e* cosh bax + 5 {ba(l +2) Ei [—ba(1 + 2)] 
+ ba(1 — z) Ei [—b,(1 — z)]} 
Zava(bs) aes [2,2 — 1 + (2bn + 1) e728] — 2s) + by Ei (—2bn) 
(0,By) i = — 1 + (2b, + 1) e-™] a “ + a (I — e-) +b, Bi (—2,) 
s(1,Br) 5 4 4 (1 — e>*) - (1 + e-*%) + b, Ei (—2b,) 
8(Bn,0) 1 1 
3(Bp,1) 0 0 
a e 
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