Exact relativistic analogues of the non-relativistic hyperfine structure operators

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Abstract
Exact relativistic analogues of the standard non-relativistic operators describing the nuclear spin-electron orbit (NSEO) and nuclear spin-electron spin contributions to the hyperfine hamiltonian for the interaction of an electron with a nuclear spin are derived. These relativistic operators, which are valid only for spatially extended descriptions of both the nuclear charge and magnetization, differ from their non-relativistic analogues solely by the additional multiplication by the Dirac β matrix. For the model in which the entire nuclear magnetization resides on the surface of a sphere of non-vanishing radius (rn ) the nuclear spin-electron spin interaction is the sum of a dipoledipole (SSD) contribution arising when the electron is outside the nuclear magnetization (r≥rn ) plus a contact type (SSC) term arising only when the electron is inside this magnetization (r⪯rn ). Both these SSD and SSC operators also differ from their non-relativistic counterparts by the appearance of the β matrix. It is shown that the relativistic hyperfine hamiltonian can be expressed exactly as the sum of the above relativistic analogues of the NSEO and spin-spin interactions augmented by a term containing the commutator with an arbitrary Dirac-Fock hamiltonian plus a purely relativistic term involving a commutator with the non-local part of the potential energy entering the arbitrary Dirac-Fock hamiltonian. When used to first order to calculate the hyperfine energy of a system containing only electrons but no positrons, the term contaiiaing the commutator with the Dirac-Fock hamiltonian is shown to yield contributions of higher order in (Z/c)2 than the dominant contributions which arise from the relativistic NSEO and spin-spin interactions. For Dirac orbitals having angular momentum (j) greater than one half, all these contributions to the hyperfine structure expectation value remain finite in the limit that rn tends to zero. For s and [pbar] orbitals the expectation values of the relativistic NSEO, SSC and SSD operators are shown in the limit of very small but non-vanishing rn to behave as r n/2γ−2 with γ=(1−Z 2/c2)1/2. Thus these three expectation values diverge with decreasing rn although their sum remains finite.

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