ANALYTICAL CROSS SECTIONS FOR MOMENTUM TRANSFER TO INNER SHELL ELECTRONS

Abstract

The Plane Wave Born Approximation (PWBA), the most prevalent First Born Approximation (FBA), serves as input into theories well beyond the scope of the PWBA. Using Mathematica's advanced symbolic programming capabilities and integrated C++ code, algorithms are designed and developed that generate analytical formulas for electron excitation and ionization of atomic constituents of biological materials ranging from hydrogen to argon. Singly-differential cross sections with respect to momentum transfer, obtained here analytically, can be integrated over energy transfer with its zeroth and higher powers to yield cross sections, stopping power, and straggling; they provide input for studies of energy deposition in biomedical media by Monte Carlo or other techniques. Mathematica's existing subroutines and functions do not analytically integrate products of non-trivial functions that are the core of the plane wave framework of the FBA toward calculations for the ionizing and/or excitation interactions. Computational procedures and programs developed here in Mathematica overcome this obstacle, and result in cross sections, differential with respect to the momentum transfer, for excitation and ionization of 1s, 2s, 2p, 3s, 3p, and 3d screened hydrogenic states of said biological constituents.