Magnetic-field-induced electric quadrupole moments for relativistic hydrogenlike atoms: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

TL;DR

This paper derives a closed-form expression for the electric quadrupole moment induced in relativistic hydrogenlike atoms by a weak magnetic field using Sturmian expansion, confirming previous results for the ground state.

Contribution

It introduces a new analytical method employing Sturmian expansion to calculate induced electric quadrupole moments in relativistic atoms.

Findings

Derived a closed-form expression involving hypergeometric functions.

Confirmed previous results for the ground state.

Extended the calculation to arbitrary discrete energy states.

Abstract

We consider a Dirac one-electron atom placed in a weak, static, uniform magnetic field. We show that, to the first order in the strength B of the external field, the only electric multipole moments, which are induced by the perturbation in the atom, are those of an even order. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30, 825 (1997); 30, 2747(E) (1997)], we derive a closed-form expression for the electric quadrupole moment induced in the atom in an arbitrary discrete energy eigenstate. The result, which has the form of a double finite sum involving the generalized hypergeometric functions 3F2 of the unit argument, agrees with the earlier relativistic formula for that quantity, obtained by us for the ground state of the atom.