The criticality of the transport system in a slab geometry with partially reflecting boundary conditions

In this paper we investigate the critical solution of transport System in a slab geometry with partially reflecting boundary conditions. We shed the properties of the principal eigenvalue (i.e. the critical parameter) of the corresponding Boltzmann integral operator, by using the functional analysis, especially the operator theory in LP spaces. WE) prove that the corresponding transport system is in the subcritical state if k<1, and the critical stab exists if k<1, where c(x). Sigma (x) is the total cross section, scattering-fission cross section, respectively and k = varisup {c (x) [Sigma (x)](-1)}.