Application of One-velocity Model to Multiplying Media

Consider, therefore, planar distributions of isotropic neutron sources in an infinite homogeneous slab. The medium is of infinite extent in both y and z directions and of width 2a in the x direction (d is the extrapolated 

This model in which the neutron source is taken to be (which implies that neutrons from fission appear at the one velocity of diffusion) is not expected to apply directly to operating reactors however, the techniques to be used later are well illustrated by this formulation of the problem, and the results are useful, if properly adapted, to more general situations. We take then as our neutron-balance relation the time-dependent diffusion equation (5.21) along with the assumed source term 

The general one-velocity diffusion equation for the multiplying medium is then 

Equation (5.125) may be solved by the separation of variables therefore, define functions F(x) and G(t) such that 

The arbitrary constant c (here assumed positive) is determined by conditions on the time behavior of the flux. Note that a relation of this type must be satisfied by F and G inasmuch as x and t are independent variables, and (5.129) can hold only if c is a constant. Thus (5.129) may be separated into two ordinary differential equations 

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