Eur. Phys. J. D (2017) 71: 8
https://doi.org/10.1140/epjd/e2016-60706-8

Regular Article

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

Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, 48 Vasileos Constantinou Avenue, 11635 Athens, Greece

^a e-mail: ykomn@eie.gr

Received: 14 December 2015
Received in final form: 5 April 2016
Published online: 17 January 2017

Abstract

The present study examines the mathematical properties of the free-free (f − f) matrix elements of the full electric field operator, O_E(κ, r̅), of the multipolar Hamiltonian. κ is the photon wavenumber. 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 O_E(κ, r̅) have singularities of first order, i.e., as ε′ → ε, they behave as (ε − ε′)^-1. The numerical applications are for f − f transitions in hydrogen and neon, obeying electric dipole and quadrupole selection rules. In the limit κ = 0, O_E(κ, r̅) reduces to the length form of the electric dipole approximation (EDA). It is found that the results for the EDA agree with those of O_E(κ, r̅), with the exception of a wave-number region k′ = k ± κ about the point k′ = k.