Exact relativistic analogues of the non-relativistic hyperfine structure operators

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Abstract
For both one-electron and multi-electron atoms using a spatially extended nuclear charge with nuclear magnetization residing on the surface, the computed hyperfine structure has been decomposed exactly into contributions from the relativistic nuclear spin-electron orbit (NSEO), spin-spin dipole (SSD) and spin-spin contact (SSC) operators plus those arising from two further relativistic corrections. The latter two corrections, present only for multi-electron systems, are shown to be negligible when the hyperfine structure is calculated to first order as an expectation value from a relativistic wavefunction describing a system containing only electrons but no positrons. The nuclear size dependence of the dominant NSEO, SSD and SSC contributions to the hyperfine structure is investigated numerically. The results confirmed the predicted divergence of all three of these contributions for s and [pbar] orbitals as the nuclear size decreases, with the sum of the three contributions remaining finite. The hyperfine structure in many electron atoms is investigated for realistic values of the nuclear radii using Dirac-Fock wavefunctions. For unpaired s and [pbar] electrons, the relativistic correction to the sum of the NSEO, SSD and SSC contributions consists of two different effects. The first is the increase, originating from the difference between the relativistic large components and the non-relativistic wavefunction, of the expectation values of those operators yielding the hyperfine structure in the non-relativistic limit. The second, which introduces nonzero terms from operators which contribute zero in the non-relativistic case, is the entire contribution from the small components. For elements heavier than the third series of transition elements both these relativistic effects are significant, so that for such atoms with an unpaired s electron, the NSEO and SSD contributions originating from the second relativistic effect are comparable with the total hyperfine structure. The first effect, outweighing the second, can enhance the hyperfine structure by factors of more than three. Analogous results apply for atoms with an unpaired [pbar] electron.