An Improved Single-Parameter Escape Probability Function

It is known that, based on Wigner's rational approximation, the escape probability function can be improved by the insertion of geometry dependent constants. Bonalumi improved this method by replacing these constants by a function. The formulas derived here, based on general considerations, justify the form given by Bonalumi and generalize it to spheres as well. The results for a cylinder and a sphere are compared to the exact tabulated values, and show an error of <0.3% through the whole spectral region. Only one parameter is needed here. The method is shown to be insensitive to this parameter to a certain extent. Comparison is also made with the results achieved by the P₀ + AP₂ method. The treatment here is limited to an isolated lump, for cylinders and spheres only.