Static electric multipole susceptibilities of the relativistic hydrogen-like atom in the ground state: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

Abstract

The ground state of the Dirac one-electron atom, placed in a weak, static electric field of definite $2^L$-polarity, is studied within the framework of the first-order perturbation theory. The Sturmian expansion of the generalized Dirac-Coulomb Green function [R. Szmytkowski, J. Phys. B 30 (1997) 825, erratum: 30 (1997) 2747] is used to derive closed-form analytical expressions for various far-field and near-nucleus static electric multipole susceptibilities of the atom. The far-field multipole susceptibilities --- the polarizabilities ${\alpha }_L$, electric-to-magnetic cross-susceptibilities ${\alpha }_{\mathrm{E}L\to \mathrm{M}(L\mp 1)}$ and electric-to-toroidal-magnetic cross-susceptibilities ${\alpha }_{\mathrm{E}L\to \mathrm{T}L}$ --- are found to be expressible in terms of one or two non-terminating generalized hypergeometric functions ${}_3{F}_2$ with the unit argument. Counterpart formulas for the near-nucleus multipole susceptibilities --- the electric nuclear shielding constants ${\sigma }_{\mathrm{E}L\to \mathrm{E}L}$, near-nucleus electric-to-magnetic cross-susceptibilities ${\sigma }_{\mathrm{E}L\to \mathrm{M}(L\mp 1)}$ and near-nucleus electric-to-toroidal-magnetic cross-susceptibilities ${\sigma }_{\mathrm{E}L\to \mathrm{T}L}$ --- involve terminating ${}_3{F}_2(1)$ series and for each $L$ may be rewritten in terms of elementary functions. Exact numerical values of the far-field dipole, quadrupole, octupole and hexadecapole susceptibilities are provided for selected hydrogenic ions. Analytical quasi-relativistic approximations, valid to the second order in $\alpha Z$, where $\alpha$ is the fine-structure constant and $Z$ is the nuclear charge number, are derived for all types of the far-field and near-nucleus susceptibilities considered in the paper.

66 pages, 19 tables

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Cited by in corpus (6)

• Relativistic calculations of quasi-one-electron atoms and ions using Laguerre and Slater spinors
• Magnetic-dipole-to-electric-quadrupole cross-susceptibilities for relativistic hydrogenlike atoms in some low-lying discrete energy eigenstates
• A scheme for quantum-logic based transfer of accuracy in polarizability measurement for trapped ions using a moving optical lattice
• Closed-form expression for the magnetic shielding constant of the relativistic hydrogenlike atom in an arbitrary discrete energy eigenstate: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function
• Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogen-like atom in the ground state
• Nuclear magnetic shielding constants of Dirac one-electron atoms in some low-lying discrete energy eigenstates