Transition energies of the $D$ lines in Na-like ions

The NIST electron beam ion trap (EBIT) was used to measure the $D_1$(3$s$-3$p_{1/2}$) and $D_2$(3$s$-3$p_{3/2}$) transitions in Na-like ions of xenon, barium, samarium, gadolinium, dysprosium, erbium, tungsten, platinum, and bismuth. The wavelengths are in the range 3–12 nm. Relativistic many-body perturbation theory calculations were carried out for the $D_1$ and $D_2$ lines for every element in the isoelectronic sequence from argon ($Z$ $=$ 18) to uranium ($Z$ $=$ 92), taking into account some higher-order terms in the quantum electrodynamics (QED) expansion. Uncertainties in the calculated values were carefully assessed by considering the uncertainties in the various contributions to the total calculated transition energies. We conclude that at the current level of accuracy, the calculated values can be taken to reliably represent the isoelectronic sequence from $Z$ $=$ 18 to 92. The agreement of theory and experiment for the $D_1$ line of bismuth ($Z$ $=$ 83) provides a test of QED at the level of 0.4$\%$. Our results are also sensitive to retardation effects due to the finite speed of light and to variations in the assumed nuclear size.

Figure 1

Bi spectrum near the $D_1$ line. Also shown are the neighboring 2$p_{1/2}$-3$d_{3/2}$ line in Al-like Bi and the 3$s^2$(1/2,1/2)${}_0$-3$s$3$p$(1/2,3/2)${}_1$ line in Mg-like Bi.

Figure 2

(a) Deviation of the $D_1$ line from present calculations [Δσ(th-expt.) from Table ], normalized to the second power of the screened nuclear charge. Red squares are the present results. Dotted lines bounding the shaded region are estimated uncertainties in the predictions (irregularities reflect variations in the uncertainties of the nuclear size). (b) As in (a), but for the $D_2$ line and normalized to the fourth power of the screened nuclear charge. For Gd ($Z$ $=$ 64) we show the weighted average of our result from Table  with that of Seely et al. [14].

Figure 3

Magnitudes of the QED shifts for the Bi $D_1$ line, given in Table . The distance of the points from the vertical line gives the magnitude of the discrepancy with experiment that would result if that point were not included in the sum of all calculated shifts. Dotted lines show the magnitude of the experimental uncertainty.

Figure 4

Fractional shift (solid red line) of the $D_1$ transition energy due to retardation as a function of $Z$. Black dots show the relative magnitudes of the experimental uncertainties.

Figure 5

Ratio of uncertainty in transition energy due to assumed nuclear size (Dirac Hartree Fock Unc. from Table ) to the total uncertainty in theory, for $D_1$ (this ratio is nearly the same for $D_2$). Ratios for the elements measured here are indicated by red squares.

Figure 6

First-order Breit interaction (dotted line) for a valence state $v$, involving a sum over the core states $c$.

Figure 7

A second-order Breit correction, B(rpa). The solid blob is the Breit-RPA vertex defined in Fig. 8; the crossed circle is the lowest-order one-body Breit vertex shown in Fig. 6.

Figure 8

A Dyson-type equation giving the matrix elements between states $i$ and $j$ of the Breit-RPA vertex. The dashed line is a Coulomb interaction, $a$ are core states, $r$ are excited states, and c.c. denotes the complex-conjugate diagrams of the last two terms on the right-hand side.

Figure 9

Second-order nonlinear Breit corrections in RMBPT. The solid blobs are the Breit-RPA vertices (Fig. 8), the dotted line is the Breit interaction, and c.c. denotes the complex conjugate of the second term. The states $a$ are core states, $r$ are excited states, and $i$ includes both core and excited states.

Figure 10

Brueckner-Goldstone diagrams for second-order two-Coulomb-photon corrections in RMBPT for a valence state $v$. Exchange variants of each diagram are also included.

Figure 11

(a) Box and (b) crossed-box Feynman diagrams. The state $v$ is the valence state, and the diagrams are summed over core states $c$. Exchange variants of the diagrams also exist, in which $v$ and $c$ at the top of the diagram are interchanged.

Figure 12

Feynman diagrams for (a) the valence SE and (b) the valence VP.

Figure 13

Feynman diagrams for the valence-exchange (“val-exch”) screening corrections to (a) the SE and (b) the VP. The state $v$ is the valence state, and the diagrams are summed over core states $c$. There are also complex-conjugate diagrams with the SE or VP unit above the exchanged photon.

Figure 14

Feynman diagrams for the core-relaxation (“core rlx”) corrections to (a) the SE and (b) the VP. The state $v$ is the valence state, and the diagrams are summed over core states $c$. There are also complex-conjugate diagrams with the SE or VP unit above the exchanged photon, and exchange variants in which the states $v$ and $c$ at the top of the diagram are interchanged.

Figure 15

Vertex-type diagrams omitted from the calculation of the valence-exchange corrections (Fig. 13). Analogous diagrams were also omitted in the calculation of core-relaxation effects (Fig. 14).