Green's Functions for the One-Speed Transport Equation in Spherical Geometry

Abstract

Several problems in one-speed neutron transport theory for spherically symmetrical systems are discussed. The singular eigenfunction expansion technique is used to construct a solution for a specific finite-slab Green's function. This slab solution is then used to construct the finite-medium spherical Green's function by extending the point-to-plane transformation concept. For the general case, the expansion coefficients are shown to obey a Fredholm equation, and first-order solutions are obtained; however, in the infinite-medium limit the solution is represented in closed form. In addition, the solution for the angular density in an infinite-medium due to an isotropic point source is developed directly from the set of normal modes of the transport equation. A proof that the result so obtained obeys the proper source condition at the origin is given.

Singular Eigenfunction Expansions in Neutron Transport Theory

“…(63). A derivation with the aid of spherical eigenmodes is also possible [ (12,59,60); see Section V]. The angular flux may then be found by using the integral form of the transport equation.…”

Technique for Finding the Moment Equations of a Nonlinear Stochastic System

“…Equation (9) can be written as (11) A(~t) = (p) LAt LAtR,(t l -t 2 ) dt l dt 2 , provided that f(t) satisfies the three conditions given earlier.…”

TwoGroup Neutron Transport Theory in Spherical Geometry

“…If we multiply Eq. (10) by P m(P), integrate over p from -I to I, and make the identification (11) we note that G m ('/'}) will be a solution to Eq. (8).…”