The application of the free-gas scattering kernel to the problem of calculating rethermalization cross sections has been extended by the inclusion of velocity dependence in the cross section within the free-gas scattering kernel. The cross section within the scattering kernel is a function of the relative velocity between neutrons and moderator and is hereafter referred to as the relative cross section. The scattering cross section which is calculated from the free-gas scattering kernel is shown to obey a differential equation of the same form as the one-dimensional heat-flow equation with the relative cross section occupying the position of the initial condition.

The relative cross section is numerically determined and its use is shown to give improved agreement with Parks' kernel for both the elements and the moments of the scattering kernel. The moments of the improved free-gas kernel are used to calculate thermal-flux spectra as well as rethermalization cross sections which are both in better agreement with experimental values than is obtainable with the free-gas kernel. The rethermalization cross sections, which are based upon the improved free-gas kernel, are used to predict power profiles at temperature and absorption discontinuities and the results are in better agreement with experimental values than was obtainable using the free-gas scattering kernel.