Convergence method for calculating solutions to the 3D invariant embedding integro-differential equations describing electron transport processes

TL;DR

This paper introduces a convergence method for solving 3D invariant embedding integro-differential equations, enabling analytical solutions for electron transport processes relevant in spectroscopy, with applications in calculating backscattered and absorbed electron fractions.

Contribution

A novel convergence method that provides analytical solutions to complex 3D electron transport equations within invariant embedding theory.

Findings

Successfully calculated electron backscattered fraction dependence on atomic number and energy.

Determined absorbed electron fractions as a function of incident angles.

Provided a tool for testing physical parameters like cross sections in transport theory.

Abstract

The electron and photon transport processes in spectroscopy techniques described by the invariant embedding theory is here revisited. We report a convergence method to obtain closed analytical solutions to the 3D integro-differential equations. This method was successfully used in calculating the dependence of the electron backscattered fraction on the atomic number and on the energy. Also the fraction of absorbed electron as a function of incident angles was calculated. Using a states ladder model for the electron energies, this method provides a tool for testing physical parameters involved in the transport theory, such as the elastic and inelastic cross sections. The outstanding feature of the invariant embedding differential equations of considering observable quantities (such as the emergent flux of particles) as independent variables makes them a suitable tool to describe experimental situations.