Stark-induced magnetic anapole moment in the ground state of the relativistic hydrogenlike atom: Application of the Sturmian expansion of the generalized Dirac-Coulomb Green function

Abstract

The Sturmian expansion of the generalized Dirac-Coulomb Green function R. Szmytkowski, J. Phys. B 30, 825 1997; 30, 2747E1997 is used to derive an analytical formula for the static magnetic anapole toroidal dipole moment induced in the ground state of the relativistic hydrogenlike atom by a weak, spatially uniform, static electric field. An expression for the anapole polarizability for the system in question is found. This expression contains a single generalized hypergeometric series 3 F 2 with the unit argument. In the non- relativistic limit our result agrees with that of Lewis and Blinder Phys. Rev. A 52, 4439 1995. Over the past three decades, in atomic and molecular physics an interest arose in effects associated with magnetic anapole moments. These effects may be divided into two principal categories. The first category comprises those atomic and molecular phenomena in which a nuclear mag- netic anapole moment, resulting from parity nonconservation in nuclear forces, manifests itself cf., e.g., Refs. 1-6. The second category includes those effects which evince the magnetic anapole moment of an atomic or a molecular elec- tronic cloud 7-22. In this paper, we shall be concerned with an effect falling into the second of the aforementioned categories. Specifi- cally, we shall be interested in the anapole moment induced in the ground state of the one-electron Dirac atom by a weak, spatially uniform, static electric field. This problem was al- ready considered a decade ago, both nonrelativistically and relativistically, by Lewis and Blinder 16; the reason for which we have decided to revisit it is that the relativistic calculations carried out in Ref. 16 were approximate in character. Exploiting the Sturmian expansion of the general- ized Dirac-Coulomb Green function 23-27, in the present paper we shall calculate the Stark-induced anapole moment exactly. The structure of the paper is as follows. In Sec. II we define the anapole moment. In Sec. III we consider the one- electron Dirac atom in the weak, spatially uniform, static electric field and apply the first-order Rayleigh-Schrodinger perturbation theory, together with the Green functions tech- nique, to derive an approximate expression for a perturbed electronic wave function. Next, in Sec. IV, this approximate wave function is used to determine an induced electric cur- rent density in the atom. Subsequently, the anapole moment of the resulting i.e., unperturbed plus induced current dis- tribution is considered. It is found that only the induced cur- rent contributes and the form of this contribution is calcu- lated analytically exploiting the Sturmian functions technique. The paper ends with conclusions, constituting Sec. V, and with several appendixes containing supplemen- tary material.