Second-order Stark effect and polarizability of a relativistic two-dimensional hydrogen-like atom in the ground state

TL;DR

This paper derives an analytical expression for the second-order Stark effect and polarizability of a relativistic two-dimensional hydrogen-like atom in the ground state, providing numerical results and connecting to nonrelativistic limits.

Contribution

It presents a new closed-form formula for the polarizability of a relativistic 2D hydrogen-like atom using Sturmian series expansion and Green functions.

Findings

Derived a closed-form expression involving hypergeometric functions.

Provided numerical polarizability values for Z=1 to 68.

Reproduced the nonrelativistic polarizability formula in the appropriate limit.

Abstract

The second-order Stark effect for a planar Dirac one-electron atom in the ground state is analyzed within the framework of the Rayleigh-Schr\"odinger perturbation theory, with the use of the Sturmian series expansion of the generalized Dirac-Coulomb Green function. A closed-form analytical expression for the static dipole polarizability of that system is found. The formula involves a generalized hypergeometric function ${}_{3}F_{2}$ with the unit argument. Numerical values of the polarizabilities for relativistic planar hydrogenic atoms with atomic numbers $1\leqslant Z\leqslant68$ are provided in a tabular form. A simple formula for the polarizability of a nonrelativistic two-dimensional hydrogenic atom, reported previously by several other authors, is recovered from our result in the nonrelativistic limit.