Ab initio calculations of the $2p_{3/2}-2p_{1/2}$ fine-structure splitting in boronlike ions

We have performed ab initio QED calculations of the ${\left(1s\right)}^2{\left(2s\right)}^{2}2p_{3/2}-{\left(1s\right)}^2{\left(2s\right)}^{2}2p_{1/2}$ fine-structure splitting along the boron isoelectronic sequence for all ions with $17\le Z\le 100$. This level splitting was evaluated within the extended Furry picture and by making use of four different screening potentials in order to estimate the effects of interelectronic correlations. The accuracy of the predicted transition energies has been improved significantly when compared with previous computations.

Figure 1

The interelectronic interaction diagrams. The double lines denote the bound electrons moving in the effective potential (2), whereas, the dashed lines with triangles denote the total screening potential but taken with the opposite sign.

Figure 2

First- and second-order QED diagrams (without the one-electron two-loop diagrams) for systems with a single valence electron. The notation is the same as in Fig. 1.

Figure 3

Decomposition of the one-electron self-energy diagram into various contributions. Whereas, the upper line represents the traditional decomposition by means of a zero-, first-, and higher-order term $\Delta E_0^{\mathrm{SE}}+\Delta E_1^{\mathrm{SE}}+\Delta E_{\ge 2}^{\mathrm{SE}}$, the lower line is used in the present paper. It gives rise to a second-order term $\Delta E_2^{\mathrm{SE}}$, which is calculated by using the analytical representation of the free-electron Green's function and a higher-order term $\Delta E_{\ge 3}^{\mathrm{SE}}$. This decomposition ensures an improved convergence and, thus, makes the second-order SE computation feasible for many-electron ions; see Sec. 3b for further details. The dashed lines with crosses at the ends denote the sum of the nuclear and screening potentials.