On the Stochastic Theory of Neutron Transport

We consider the probability, pn(R,tf; ,,t), that in a multiplying system, a neutron with position , velocity , at time t leads to exactly n neutrons in region R of , space at time tf. By formulating pn in terms of first collision probabilities we derive a non-linear (Boltzmann-like) integro-differential equation for the probability generating function, G. The linearized equation for = 1 - G is shown to be adjoint to the usual Boltzmann equation for the average neutron flux. The behavior of for subcritical and supercritical systems is analyzed. For large tf-t, it is shown that for subcritical systems approaches zero exponentially, while for supercritical systems → which is a solution of the time-independent non-linear equation for and equals the probability of getting a divergent chain reaction from the initial neutron.In section B, one-velocity theory with isotropic scattering is described in some detail while in section C are outlined the extensions to 1) energy-dependent problems with anisotropic scattering 2) multiple final states, 3) random sources, 4) counting problems, and 5) delayed neutron precursors. In section D methods for solution of equations for G are briefly discussed, and it is shown that the asymptotic behavior may be found from solutions of linear time-independent ‘adjoint α’ and ‘adjoint k’ calculations. Derivation of a point model independent of space and velocity is carried out by an expansion in adjoint α eigenfunctions and the model parameters are shown to differ from those usually assumed in point models.