Exact relativistic analogues of the non-relativistic hyperfine structure operators

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 (r_n ) the nuclear spin-electron spin interaction is the sum of a dipoledipole (SSD) contribution arising when the electron is outside the nuclear magnetization (r≥r_n ) plus a contact type (SSC) term arising only when the electron is inside this magnetization (r⪯r_n ). 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 r_n 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 r_n to behave as r n/2γ−2 with γ=(1−Z ²/c2)^1/2. Thus these three expectation values diverge with decreasing r_n although their sum remains finite.