Integral transport in a three region sphere with associated benchmark problems

We derive, by the method of characteristics, an integral form of the transport equation with isotropic scattering for spherical symmetry in which the spatial dependence of the cross-sections is arbitrary. The sources of particles (photons or neutrons) can be either incident on the surface with an arbitrary angular distribution or distributed within the body. A specific example is given for the case of a three region sphere in the form of a central spherical core surrounded by a concentric shell followed by an outer shell. By allowing the total cross-section to go to zero in any one of the regions we may evaluate the effect of voids. Some numerical results are given which enable the limitations of void representation in a standard computer code to be highlighted. We also consider the case of a multiplying system and calculate the effect of an annular void on the multiplication factor.