The Dirac operator on a finite domain and the R-matrix method

Abstract

Relativistic effects in electron–atom collisions and photo-excitation and -ionization processes increase in importance as the atomic number of the target atom grows and spin-dependent effects increase. A relativistic treatment in which electron motion is described using the Dirac Hamiltonian is then desirable. A version of the popular nonrelativistic R-matrix package incorporating terms from the Breit–Pauli Hamiltonian has been used for modelling such processes for some years. The fully relativistic Dirac R-matrix method has been less popular, but is becoming increasingly relevant for applications to heavy ion targets, where the need to use relativistic wavefunctions is more obvious. The Dirac R-matrix method has been controversial ever since it was first proposed by Goertzel (1948 Phys. Rev. 73 1463–6), and it is therefore important to confirm that recent elaborate and costly applications of the method, such as, Badnell et al (2004 J. Phys. B: At. Mol. Phys. 37 4589) and Ballance and Griffin (2007 J. Phys. B: At. Mol. Opt. Phys. 40 247–58), rest on secure foundations. The first part of this paper analyses the structure of the two-point boundary-value problem for the Dirac operator on a finite domain, from which we construct a unified derivation of the Schrödinger (nonrelativistic) and Dirac (relativistic) R-matrix methods. Suggestions that the usual relativistic theory is not well founded are shown to be without foundation.