ABSTRACT
A version of quantum-defect theory appropriate to the analysis of the spectra of highly ionised atoms is obtained from the single-electron Dirac equation. Quantum defects μ n are defined as the principal quantum number n increases along a Rydberg series using Sommerfeld’s relativistic one-electron energy level formula. Considering the analytic properties of solutions to the Dirac equation, we establish that (μ n can be extended smoothly away from the bound-state energies just as in the non-relativistic theory. Moreover, the relativistic μ(ϵ) can be analytically continued beyond the threshold ϵ = mc 2 and related to the non-Coulomb scattering phaseshift δ(ϵ). At the threshold we establish the result δ(mc 2) = πμ(mc 2) well known in non-relativistic quantum-defect theory. Several examples of the relativistic quantum-defect theory are given for highly stripped Na-like and Li-like ions.
