Analytical solution for the Doppler Broadening Function using the Tsallis distribution

Abstract

The development of models that generalise the Maxwell–Boltzmann distribution has been the object of much study and research. The statistics of Tsallis and Kaniadakis that use the q and κ-deformed functions attempt to encompass physical phenomena that lie outside the thermal equilibrium. One application of these deformed functions is in the derivation of the modified Doppler Broadening Function that may be used to calculate broadened neutron multigroup cross-sections. The numerical solution of the broadened cross sections demands considerable computing time and because of that several methods were developed to obtain analytical approximations. Taking this perspective into account, this paper obtains an analytical solution for the Doppler Broadening Function using Tsallis statistics. First of all, the integral equation of this function is converted into an approximated differential equation by using a Taylor series expansion of one of the functions in the integrand. In the mathematical formulation towards obtaining the differential equation, the integrals with the q-deformed function were solved analytically. Finally, the overall solution of the approximated differential equation for the Doppler Broadening Function using Tsallis deformed statistics was obtained, with such results shown to be consistent. The deviations obtained in comparison to the numerical reference values were satisfactory, except for the |x⋅ξ|≥6 domain.