Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy state

TL;DR

This paper derives analytical formulas for the diamagnetic and paramagnetic contributions to the magnetizability of a Dirac one-electron atom in any discrete energy state, using the Sturmian expansion of the Dirac--Coulomb Green function.

Contribution

It provides a new closed-form, analytical decomposition of magnetizability into diamagnetic and paramagnetic parts for Dirac one-electron atoms in arbitrary states, extending previous results.

Findings

Derived closed-form formulas for $oldsymbol{ ext{chi}_d}$ and $oldsymbol{ ext{chi}_p}$

$oldsymbol{ ext{chi}_p}$ involves hypergeometric functions ${}_3F_2$

Results reduce to known formulas for the ground state

Abstract

We present a Gordon decomposition of the magnetizability of a Dirac one-electron atom in an arbitrary discrete energy eigenstate, with a pointlike, spinless, and motionless nucleus of charge $Ze$. The external magnetic field, by which the atomic state is perturbed, is assumed to be weak, static, and uniform. Using the Sturmian expansion of the generalized Dirac--Coulomb Green function proposed by Szmytkowski in 1997, we derive a closed-form expressions for the diamagnetic ($\chi_{d}$) and paramagnetic ($\chi_{p}$) contributions to $\chi$. Our calculations are purely analytical; the received formula for $\chi_{p}$ contains the generalized hypergeometric functions ${}_3F_2$ of the unit argument, while $\chi_{d}$ is of an elementary form. For the atomic ground state, both results reduce to the formulas obtained earlier by other author. This work is a prequel to our recent article, where the numerical values of $\chi_{d}$ and $\chi_{p}$ for some excited states of selected hydrogenlike ions with $1 \leqslant Z \leqslant 137$ were obtained with the use of the general formulas derived here.