Spin-dependent polarizabilities of hydrogenic atoms in magnetic fields of arbitrary strength

Abstract

Utilizing the magnetic field-dependent spin-orbit interaction, the relativistic correction to the Zeeman energy, and the usual diamagnetic interaction, we have calculated spin-dependent electrical polarizabilities of hydrogenic atoms using the Hass\'e variational approach. The polarizabilities $\ensuremath{\alpha}(\ensuremath{\uparrow})$ and $\ensuremath{\alpha}(\ensuremath{\downarrow})$ for the two spin directions have been obtained for the electric field both parallel and perpendicular to the magnetic field ${H}_{z}$ in the weak-field ($\ensuremath{\gamma}\ensuremath{\ll}1$), intermediate-field ($\ensuremath{\gamma}\ensuremath{\sim}1$), and strong-field ($\ensuremath{\gamma}\ensuremath{\gg}1$) limits, where $\ensuremath{\gamma}=(\frac{{\ensuremath{\epsilon}}^{2}{\ensuremath{\hbar}}^{3}{H}_{z}}{m^{*2}e^{3}c})$, with $\ensuremath{\epsilon}$ a static dielectric constant and ${m}^{*}$ an isotropic effective mass. The results for hydrogen atoms ($\ensuremath{\epsilon}=1 \mathrm{and} {m}^{*}=m$) in the weak-field limit yield $\frac{[\ensuremath{\alpha}(\ensuremath{\downarrow})\ensuremath{-}\ensuremath{\alpha}(\ensuremath{\uparrow})]}{\ensuremath{\alpha}(0)\ensuremath{\approx}2.31{\ensuremath{\alpha}}_{\mathrm{f}}^{2}_{\mathrm{s}}\ensuremath{\gamma}} ({\ensuremath{\alpha}}_{\mathrm{fs}}=\frac{1}{137})$ with a negligible anisotropy. In the strong-field limit [${\ensuremath{\alpha}}_{\ensuremath{\perp}}(\ensuremath{\downarrow})\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\perp}}(\ensuremath{\uparrow})$] falls precipitously while $[{\ensuremath{\alpha}}_{\ensuremath{\parallel}}(\ensuremath{\downarrow})\ensuremath{-}{\ensuremath{\alpha}}_{\ensuremath{\parallel}}(\ensuremath{\uparrow})]$ continues to increase up to at least $\ensuremath{\gamma}={10}^{4}$, but more slowly than linearly with $\ensuremath{\gamma}$. The spin-independent quantities [${\ensuremath{\alpha}}_{\ensuremath{\parallel}}(\ensuremath{\downarrow})+{\ensuremath{\alpha}}_{\ensuremath{\parallel}}(\ensuremath{\uparrow})$] and [${\ensuremath{\alpha}}_{\ensuremath{\perp}}(\ensuremath{\downarrow})+{\ensuremath{\alpha}}_{\ensuremath{\perp}}(\ensuremath{\uparrow})$] are discussed in the intermediate- and high-field limits and represent an extension of the earlier low-field results obtained by Dexter. The implications of these results for shallow-donor impurity atoms in semiconductors and for hydrogen-atom atmospheres of magnetic white dwarfs and neutron stars are briefly considered. The effects of the dramatic shrinkage of the electron's wave function on the spin Zeeman energy and the electron-proton hyperfine interaction are also discussed.