Slab reactor. The diffusion equation reduces to a one-dimensional equation

where we have used a single absorption cross-section Xa, independent of x, and a plane neutron source at position x = 0. At the boundaries x = ±a/2, we require ‘(x = ±a/2, t) = 0. It is, therefore, convenient to use a Fourier development of ‘ and S,

with Bn = ш/a (n = 1,3,…). The coefficients An(t) are obtained by solving the equations

If S = 0, the solution is

A„0) =An(0) exp (^1 — 1 — B £avt.

For кж < 1 + B^(D/St) = 1 + (7r2D/a2Sa) all terms vanish exponentially. For кж > 1 + (7r2D/a2Sa), the first term, and possibly some other low order ones, increases exponentially. The reactor becomes critical for ki = 1 + (7r2D/a2Sa); in this case A1(t) becomes time independent, while higher order terms decrease exponentially. Therefore, the neutron flux distri­bution becomes time independent and is a solution of the time-independent diffusion equation

2

Ddx1 ‘(x, t) + ‘(x, t)^a(ki — 1)= 0

j2 2

d к

dX2 ‘(x, 0+^2 ‘(x, t) =0

which has the form

kx

‘(x) = A1 cos —.

Simple solutions are also obtained for spherical and cylindrical reactors.

Spherical reactor. For spherically symmetric systems the time-independent diffusion equation reads

IA (r2 d’ r2 dr dr

(3.36)

R being the radius of the reactor. The solution satisfying the boundary con­ditions is

(3.37)

(3.38)

Cylindrical reactor. Similarly, for infinite cylindrical systems

г d(r d)+B'(r)=0

whose solution is

‘(r) = AJ0(Br),

with J0 the ordinary Bessel function of order 0. J0(BR) = 0. This condition is fulfilled for a number of values, the smallest being BR = 2.405.