$S$-matrix calculations of energy levels of sodiumlike ions

A recent $S$-matrix-based QED calculation of energy levels of the lithium isoelectronic sequence is extended to the general case of a valence electron outside an arbitrary filled core. Emphasis is placed on modifications of the lithiumlike formulas required because more than one core state is present, and an unusual feature of the two-photon exchange contribution involving autoionizing states is discussed. The method is illustrated with a calculation of the energy levels of sodiumlike ions, with results for $3s_{1/2}, 3p_{1/2}$, and $3p_{3/2}$ energies tabulated for the range $Z=30–100$. Comparison with experiment and other calculations is given, and prospects for extension of the method to ions with more complex electronic structure discussed.

Figure 1

(a) One-photon exchange diagram between the valence and a core electron and (b) the action of a counterpotential, represented by a cross inside a circle, on a valence electron.

Figure 2

One-photon self-energy and vacuum polarization diagrams for the valence electron.

Figure 3

Two-photon exchange diagrams between the valence and up to two core electrons. A cross inside a circle represents a counterpotential. Where there is no photon exchanged between electrons (c), core electrons are treated as spectators only.

Figure 4

Screened self-energy (a)–(d) and vacuum polarization (e)–(g) diagrams between the valence and a core electron. A cross inside a circle represents a counterpotential. Where there is no photon exchanged between electrons (b),(d),(f), core electrons are treated as spectators only.

Figure 5

Experimental and theoretical energies relative to the present results for the $3s-3p_{1/2}$ transition. Solid circles, RMBPT [14]; crosses, NIST [42]; open triangles, Gillaspy et al. [21].

Figure 6

Experimental and theoretical energies relative to the present results for the $3s-3p_{3/2}$ transition. Solid circles, RMBPT [14]; open circles, RCI [20, 43]; crosses, NIST [42]; closed triangles, EBIT [20, 43].