78
2 Critical Mass, Efficiency, and Yield
For the tamper, the threshold solution of the diffusion equation is (2.39),
N tamp (r ) =
D
r
+ E.
(2.62)
(A, B, C, D, E) are constants of integration to be eliminated upon application of
boundary conditions, as described in what follows.
Five boundary conditions dictate the continuity of neutron density and flux
between the inner and outer cores, the outer core and the tamper, and at the outer
edge of the tamper. These are, from (2.43)–(2.45),
N 1 (R 1 ) = N 2 (R 1 ),
(2.63)
λ
tran
1
∂ N 1
∂r
R 1
= λ
tran
2
∂ N 2
∂r
R 1
,
(2.64)
N 2 (R 2 ) = N tamp (R 2 ),
(2.65)
λ
tran
2
∂ N 2
∂r
R 2
= λ
tran
tamp
∂ N tamp
∂r
R 2
,
(2.66)
and
N tamp
R tamp
= −
2
3
λ
tran
tamp
∂ N tamp
∂r
R tamp
.
(2.67)
The algebra to eliminate A, B, C, D, and E is lengthy; the result is an expression
which involves, aside from the usual nuclear parameters, only the three radii (R 1 ,
R 2 , R tamp ). A reader who wishes to test his or her algebra skills can verify that this
expression can be written as
U − QV = 0.
(2.68)
U, Q, and V are
U =
B
C
sin x 3
x 3
+
cos x 3
x 3
,
(2.69)
Q = ψ
B
C
[− sin x 3 + x 3 cos x 3 ] − [cos x 3 + x 3 sin x 3 ]
,
(2.70)
and
V =
2λ
tran
tamp
3R
2
tamp
−
1
R tamp
+
1
R 2
.
(2.71)