Solution of Boltzmann Equation for Hydrogenous Systems

The present analysis is a more general development of the ideas prev i-ously considered in Sec. 7.6a. For this purpose we require a statement of the Boltzmann equation for a system containing a mixture of N nuclear species. The direct extension of the steady-state form of (7.110) is 

The conditional frequency is defined according to (7.114), and the superscript (i) means that the scattering collision occurred with a nucleus of type i. In the present application we consider the case of one-dimensional symmetry so that spatial variation occurs only in the X direction, and the only angular variable of interest is the cosine of the angle between the x axis and the direction of motion of the neutron. The appropriate one-dimensional form of (12.11) is [cf. Eq. (7.336)] 

We obtain the solution to Eq. (12.12) by expanding the flux and the frequency functions in infinite series of Legendre functions. For the frequency function we use the general form (7,129), and for the flux [cf. Eq. (7.74)], 

A similar series is defined also for the source function S(a ,u,M)- If these expansions are used in Eq. (12.12), along with the addition theorem 

To proceed further we require a specification of the coefficients For this purpose we introduce the assumption that the scattering collisions are isotropic in the center-of-mass system of coordinates. In that case we obtain 

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