Applications of Fermi Age Theory

Consider, then, the problem defined by the following set of assumptions  

The general solution to this equation was developed in Sec. 5.2d by use of the kernel method [see Eq. (5.97)]. In the present case the source S(z) from (6.125) may be substituted directly into (5.97), and the integration will yield 0ih(2). We note, however, that the system in question involves only one space variable z. Thus although the general form (5.97) is applicable here, it is by no means necessary, and, in fact, some labor can be eliminated by using the one-dimensional kernel relation [see Eq. (5.78)], namely, 

This result may be written in terms of the error function erf(x), defined by 

Equation (6.129) gives the thermal-flux distribution, in an infinite medium, that results from a plane source of neutrons at the origin 2 = 0. The resonance-escape probability and the age which appear in this expression are measured from w = 0, which corresponds to the energy Eo. 

As previously indicated, the only interest in this result is in its form for large values of the variable z. We expect that the asymptotic solution will give the correct form for the thermal flux in the case of the actual geometry for regions far from the source. We obtain the asymptotic form for (6.129) by introducing the expansion for the error function in the case of large x, namely, 

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