The Variational Method Applied to the Monoenergetic Boltzmann Equation. Part I

Abstract

The monoenergetic integro-differential Boltzmann equation with an arbitrary scattering kernel is transformed to a self-adjoint form and the corresponding Lagrangian written. It is shown that this transformation results in a loss of the continuity (neutron conservation) information contained by the Boltzmann equation. This information is recovered by writing the directionai flux as the sum of an even and odd function (in angie) and considering a self- adjoint Lagrangian for only one portion (even or odd) of the directional flux. This procedure is shown to be equivalent to separating the nonself-adjointness from the Boltzmann operator. Further, it is shown that this self-adjoint principle is an extremum principle if the mean number of secondaries per collision is less than one. This self-adjoint formalism is appiied to the angular expansion of the directionai flux which results in an improved diffusion theory. Numerical results for the linear extrapolation distance and diffusion coefficient are compared with the ciassical (P -- 1) diffusion theory. (auth)