Neue Lösungsformen der Boltzmann-Gleichung für monoenergetischen Neutronentransport in kugelförmiger Geometrie

Abstract

In Chapter I thesingular solution of the Boltzmann equation for neutron transport in spherical geometry will be derived. The calculation will be performed in two steps. First, a partial differential equation (7) with an assumed density (6) on its right hand side will be solved. But the partial solution found in this way will generally not yield the assumed density. Therefore on has to add a suitable solution of the homogeneous differential equation (10). This addition leads to an equation of compatibility which turns out to be a Sonine integral equation (12). The second step of the calculation is the solution of this integral equation. The total solution of the Boltzmann equation will be written down in two different representations, (15) and (31), but its uniqueness has been proved.

The main singularity at the center of the sphere is proportional to l/(ϱ√1 μ²). A term log ϱ does not appear, but a term proportional to log [(1+μ)/(1−μ)] does which, however, loses its importance at the center of the sphere ϱ=0 in comparison with the main singularity.

A characteristic equation needs not occur in this mathematical procedure; it may or may not be introduced. Therefore no hint at the spectrum of the Boltzmann operator in spherical geometry will be given.

In Chapter II it will be shown that there exists a remarkably short integral representation of theregular solution (38) which satisfies from the first all requirements, if the validity of the characteristic equation (3) is supposed. But there are also regular solutions, given by the difference of two singular solutions, which need not satisfy a characteristic equation.

In Chapter III both kinds of regular solutions in spherical geometry are given assuperpositions of solutions in plane geometry which belong to the discrete or to the continuous spectrum of the Boltzmann operator.

The regular solutions are identical with the corresponding well-known series of spherical harmonics, where the supposition of a characteristic equation needs also not necessarily be made for exact solutions in the infinite space. A preliminary discussion of this problem is given in the introduction.