Applications of Spherical-harmonics Method

In this section we illustrate the application of the spherical-harmonics method by considering two problems of some practical interest. Both problems deal with steady-state one-dimensional systems, and the calculation is carried out on the basis of the one-velocity model developed in Sec. 7.2f. In the present applications the general time-dependent relations given in (7.84) reduce to the following set of differential equations  

The Pz approximation of the one-velocity transport model is obtained from Eqs. (7.271) by retaining all relations which involve the harmonics 4 n for n 3. These are 

In the hrst calculation which follows we compute in detail the angular distribution of the neutron flux in the vicinity of an interface between two dissimilar media. This example demonstrates the application of the interface-boundary condition (7.100). The second example deals with a slab of multiplying material, and in this calculation the emphasis is placed on the specification of the vacuum-interface condition and on the estimation of the critical width of the slab by means of the various models which have been developed. 

For this application the general steady-state solution to the transport equation in the P approximation (7.272) reduces to the somewhat simpler form. 

Equation (7.275) is an ordinary differential equation with constant coefficients. Its solution is easily shown to be 

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