Breit and QED effects on the $3d{}^9{}^{2}D_{3/2}\to {}^2{D}_{5/2}$ transition energy in Co-like ions

The multiconfiguration Dirac–Hartree–Fock theory is used to calculate the $3d{}^9{}^{2}D_{3/2}\to {}^2{D}_{5/2}$ transition energy for Co-like ions with $Z=28–100$. We investigate how electron correlation, frequency-independent and -dependent Breit interactions, as well as QED corrections vary along the sequence. The well-understood frequency-independent Breit contribution has the largest contribution for all ions. Among the corrections to this, correlation decreases rapidly with $Z$, the frequency-dependent Breit contribution is important especially for high-$Z$ ions, and the self-energy contribution to the QED becomes the largest correction already for $Z>50$. We evaluate and compare results for the Self-energy in three different approximations, (i) the approach implemented in the grasp2k package, (ii) the method based on Welton's concept and (iii) a model operator approach recently developed by Shabaev and coworkers. Through comparison with experimental values, it seems that the third set of results have the best agreement with experiments, but the difference from experiments for high-$Z$ ions, is around 0.03%–0.04%, and therefore our results are outside the error bars of the experiments.

Figure 1

Convergence of fine-structure energy as a function of largest $n$ in the active set.

Figure 2

Differences [(a) in ${\mathrm{cm}}^{-1}$, (b) in %] in contributions of SE to the energy splittings for Co-like ions from calculations based on Welton's and Shabaev's concepts compared with the original grasp2k result. For an explanation of the jump in the curve for Welton's method, see Sec. 3b. The horizontal lines at 0 represents the grasp2k results.

Figure 3

Contributions from different effects to the fine-structure splitting of the ground term in Co-like ion, where Breit(0) represents frequency-independent Breit, Breit($\omega$) represents frequency-dependent Breit, SE represents self-energy, and VP represents vacuum polarization.

Figure 4

Comparison of fine-structure energy splittings from direct observations (see text) with the results of the present calculations. The horizontal lines at 0 represents the grasp2k results.