Computational Methods for the Neutron Age

The purpose of this section is to display three computational methods for the neutron age that are especially useful for hand calculation. These methods are based on various approximate models which involve some rather gross idealizations of the physical system. It is convenient to discuss these at this point since they serve well to demonstrate the application of the Fermi age model and the transport-theory methods developed in the preceding sections. 

The particular utility of the first model, which was developed by Fermi and others, is in application to water-metal mixtures. The second model is a rationale for extending these results by similarity techniques, and the third is useful for materials of large nuclear mass. It should be pointed out that each of these methods was designed primarily as a fast hand tool. With the advent of the fast computing machine there is perhaps less interest in these methods nevertheless, the development of the analytical models is helpful for discussing some of the physical features of each system. 

We know from our earlier study of the Fermi age model that the age t u) is proportional to the quantity r (w), the mean radial distance squared at which fast neutrons from a specified source reach lethargy u the constant of proportionality in this relation is dependent upon the nature of the source [see, for example, Eq. (6.52)]. Our ultimate objective, then, is to compute r (w). For this purpose we introduce the following model  

The system defined by these assumptions is evidently one-dimensional and at steady state, and in this case the general Boltzmann relation reduces to a somewhat simpler form. The appropriate equation is obtained from (7.116) in terms of the lethargy variable this may be written 

Equation (7,336) gives a complete description of the neutron distribution in the system specified by the model (7.332). If the solution of this equation were obtained, the resulting expression for could be 

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