Magnetic field-induced electric quadrupole moment in the ground state of the relativistic hydrogen-like atom: 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 a relativistic hydrogen-like atom by a weak magnetic field, extending previous non-relativistic results using Sturmian expansion techniques.

Contribution

It provides the first relativistic derivation of the induced electric quadrupole moment using the Sturmian expansion of the Dirac-Coulomb Green function.

Findings

Derived a closed-form expression involving $_{3}F_{2}$ hypergeometric function

Confirmed the quadratic dependence of the quadrupole moment on magnetic field in the relativistic case

Extended non-relativistic results to relativistic hydrogen-like atoms

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 perturbing field, the only electric multipole moment induced by the field in the ground state of the atom is the quadrupole one. Using the Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825; erratum 30 (1997) 2747], we derive a closed-form expression for the induced electric quadrupole moment. The result contains the generalized hypergeometric function $_{3}F_{2}$ of the unit argument. Earlier calculations by other authors, based on the non-relativistic model of the atom, predicted in the low-field region the quadratic dependence of the induced electric quadrupole moment on $B$.