Theory and computation of electromagnetic transition matrix elements in the continuous spectrum of atoms

TL;DR

This paper investigates the mathematical properties of free-free electromagnetic transition matrix elements in atomic spectra, develops computational methods, and compares full operator results with electric dipole approximation for hydrogen and neon.

Contribution

It introduces specialized methods for calculating free-free matrix elements with energy-normalized wavefunctions in atomic potentials, analyzing singularities and comparing with electric dipole approximation.

Findings

Full operator matrix elements have first-order singularities at equal energies.

Results for electric dipole approximation agree with full operator except near singularities.

Numerical applications performed for hydrogen and neon transitions.

Abstract

The present study examines the mathematical properties of the free-free ( f-f) matrix elements of the full electric field operator, of the multipolar Hamiltonian. Special methods are developed and applied for their computation, for the general case where the scattering wavefunctions are calculated numerically in the potential of the term-dependent (N-1) electron core, and are energy-normalized. It is found that, on the energy axis, the f-f matrix elements of the full operator have singularities of first order in the case of equal photoelectron energies. The numerical applications are for f-f transitions in Hydrogen and Neon, obeying electric dipole and quadrupole selection rules. In the limit of zero photon wave-number, the full operator reduces to the length form of the electric dipole approximation (EDA). It is found that the results for the EDA agree with those of the full operator, with the exception of a photon wave-number region about the singularity.