Relativistic calculations of double $K$-shell-photoionization cross sections for neutral medium-$Z$ atoms

Fully relativistic calculations are presented for the double $K$-shell photoionization cross section for several neutral medium-$Z$ atoms, from magnesium ($Z=10$) up to silver ($Z=47$). The calculations take into account all multipoles of the absorbed photon as well as the retardation of the electron-electron interaction. The approach is based on the partial-wave representation of the Dirac continuum states and uses the Green's-function technique to represent the full Dirac spectrum of intermediate states. The method is strictly gauge invariant, which is used as an independent cross-check of the computational procedure. The calculated ratios of the double-to-single $K$-shell ionization cross sections are compared with the experimental data and with previous computations.

Figure 1

Feynman diagrams that represent the double-photoionization process in the leading order of perturbation theory. $a$ and $b$ denote the bound electron states, ${\mathit{p}}_1$ and ${\mathit{p}}_2$ are the continuum electron states, and $\mathit{k}$ refers to the incoming photon. Double lines denote electrons that are propagating in a central potential (nuclear Coulomb field plus some screening potential). Proper antisymmetrization of the initial and the final two-electron wave functions is assumed.

Figure 2

Double $K$-shell photoionization cross section for neutral magnesium (Mg, $Z=12$). The double-to-single $K$-shell ionization ratio ($\mathcal{R}=Z^2{\sigma }^{++}/{\sigma }^{+}$) is plotted as function of the incoming photon energy $\omega$ in units of the double $K$-shell photoionization energy ${\omega }_{\mathrm{cr}}$. The results of the present calculation are plotted by a solid line (red) for the ${\mathrm{CH}}_1$ potential and by a dashed line (orange) for the ${\mathrm{CH}}_2$ potential, respectively. The experimental results of Refs. [12] and [13] are shown by black diamond points. On the left panel, the calculations by Kheifets et al. [14] in length, velocity, and acceleration gauge are shown by dotted lines (navy, purple, and blue, respectively). On the right panel, the present theory and experiment [12, 13] for neutral magnesium are compared with the computations for He-like magnesium ions. The relativistic results for the He-like ion are shown by the down-triangle points (dark green line); the nonrelativistic calculation is shown by the up-triangle points (light green line).

Figure 3

The same as Fig. 2 but for neutral calcium (Ca, $Z=20$). The crosses denote the experimental results of Oura et al. [28]. On the left panel, the violet dash-dotted line denotes the calculation by Mikhailov et al. [22].

Figure 4

The same as Fig. 2 but for neutral copper (Cu, $Z=29$). The experimental result is by Diamant et al. [29].

Figure 5

The same as in Fig. 4, but for neutral silver (Ag, $Z=47$). The experimental points are by Kanter et al. [11].