First flight escape probability for a spherical shell with an empty central region

First flight transport kernels are derived for a spherical shell region where the central zone is empty. Assuming a uniformly distributed, isotropic source throughout the annulus the uncollided angular flux distribution is obtained enabling the formulation of the current at any point in the medium. It is shown that the current at the outer radius of the shell, and therefore also the escape probability P_e of the shell, can be put in a simple analytical form. Numerical values of P_e are given for different inner radii and different shell thicknesses and comparison is made with the escape probabilities for spherical shells with a black central region. Finally, an upper limit is given for the relative error introduced in a shell collision probability by the common assumption of a perfectly absorbing inner zone.