Figure 1

(Color online) Single-photoionization cross section of Be as a function of the photon energy. The CCC calculations in the length and velocity gauges are shown by the black dotted and red solid lines, respectively. The RPAE calculation in two gauges (indistinguishable) is shown by the green dashed line. Experimental data of Wehlitz et al. [5] and Wehlitz and Whitfield [3] are displayed by filled circles and open squares, respectively. The arrow indicates the double-ionization threshold.Reuse & Permissions

Figure 2

(Color online) Double-to-single photoionization cross-section ratio in Be as a function of the photon energy. Present CCC calculation in the velocity gauge is plotted by red filled circles. Previous CCC results [4] are shown by blue open diamonds. Experimental data of Wehlitz et al. [5] and Wehlitz and Whitfield [3] are displayed by open circles and squares, respectively. The black solid line is the experimental double-to-single ratio in He [33] multiplied by 0.64 and plotted versus the excess energy in units of the ionization potential of ${\mathrm{He}}^{+}$.Reuse & Permissions

Figure 3

(Color online) Double-photoionization cross section of Be as a function of excess energy above the double-ionization threshold. The key is the same as in Fig. 2 except for the solid line, which shows the fit of experimental data with the fourth-order Wannier theory [34].Reuse & Permissions

Figure 4

(Color online) Single-photoionization cross sections of Mg (top) and Ca (bottom) as a function of the photon energy. RPAE calculations with one and three channels are shown by the black solid and blue dashed lines, respectively. The three-channel RPAE calculation for Ca is scaled down by a factor of 0.2. The CCC calculation is shown with the red circles. The corresponding double ionization thresholds are indicated by arrows.Reuse & Permissions

Figure 5

(Color online) Double-to-single photoionization cross-section ratios in Mg (top) and Ca (bottom) as functions of the photon energy. Present calculation (red filled squares) is shown in comparison with experimental data of Nagata [35]. The red solid line is a smooth interpolated curve to guide the eye through the calculated data.Reuse & Permissions

Figure 6

(Color online) Double-to-single photoionization cross-section ratios in He, Be, Mg, and Ca as functions of the reduced excess energy $\Delta E=E∕V_{\text{ion}}^{+}$ measured in units of the ionization potential of the corresponding singly charged ion. The photon energy in eV is indicated on the top horizontal scale of each plot. The data points (red filled squares) are interpolated by a smooth curve (red solid line) to guide the eye through the data. The experimental double-to-single ratio in He [33] (multiplied by 0.8, 0.4, and 1.5 to scale the Be, Mg, and Ca ratios, respectively) is shown on each plot as a thick solid line. Also shown are the electron-impact ionization cross sections of the corresponding ions (CCC calculation for He [37], black dotted line; experimental data for Be [40], Mg [41], and Ca [42], filled circles) and the theoretical dipole-only electron-impact ionization cross sections for each target ion (blue dashed lines).Reuse & Permissions

Figure 7

(Color online) Modulus of the symmetric amplitude $f^{+}$ of DPI on Be at equal energy sharings of $E_1=E_2=10$, 2, and $0.5\mathrm{eV}$ (from top to bottom) is shown by the red solid line. Each amplitude is fitted with a Gaussian displayed by a black dotted line. The arrow indicates the Gaussian width parameter $\Delta {\theta }_{12}$ in Eq. (11).Reuse & Permissions

Figure 8

(Color online) Gaussian width parameter $\Delta {\theta }_{12}$ as a function of the reduced excess energy $\Delta E=E∕V_{\text{ion}}^{+}$ for various alkaline-earth-metal atoms (Be, red circles; Mg, blue diamonds; Ca, green squares) and helium (black open circles). The correspondingly colored solid lines are smooth interpolated curves to guide the eye through the calculated data. The dotted line indicates the onset of the Wannier threshold law proportional to $E^{1∕4}$ for He.Reuse & Permissions

Figure 9

(Color online) Momentum profiles (squared momentum space wave functions) ${\mid R_{ns}\left(q\right)\mid }^2$ of the valence $ns$ state in ${\mathrm{He}}^{+}$ ($n=1$, black thick solid line), ${\mathrm{Be}}^{+}$ ($n=2$, red solid line), ${\mathrm{Mg}}^{+}$ ($n=3$, blue short-dashed line), and ${\mathrm{Ca}}^{+}$ ($n=4$, green long-dashed line). The momentum profiles are normalized to ${\mid R_{1s}\mid }^2$ of ${\mathrm{He}}^{+}$ at its maximum. The arrow indicates the extraction of the momentum width at half maximum $\Delta q$.Reuse & Permissions

Figure 10

(Color online) Gaussian angular width parameter $\Delta {\theta }_{12}$ taken at the reduced excess energy $\Delta E=E∕V_{\text{ion}}^{+}=0.3$ versus the momentum width at half maximum $\Delta q$ in He and various alkaline-earth-metal atoms. The dotted line is the fit with the power law $\Delta {\theta }_{12}\propto \Delta q^{3∕4}$.Reuse & Permissions

Figure 11

(Color online) Top panel: symmetric DPI amplitudes $f^{+}({\theta }_{12},E_1,E_2)$ at $E_1=E_2=12.5\mathrm{eV}$ in Ca (red solid line) and at $E_1=E_2=10\mathrm{eV}$ in He (blue dotted line) plotted versus the mutual electron angle ${\theta }_{12}$. Both amplitudes are fitted with the Gaussian ansatz Eq. (11) shown by the black solid (He) line and dotted (Ca) lines. Middle panel: DPI TDCS of Ca at $E_1=E_2=12.5\mathrm{eV}$ at fixed electron angle ${\theta }_1=0$ plotted versus ${\theta }_{12}$. Experimental data from Ref. 12 on relative scale are normalized to the calculation. Bottom panel: same for He at $E_1=E_2=10\mathrm{eV}$. Absolute experimental data are from Ref. 45. The black solid line on the middle panel and black dotted line on the bottom panel indicate the TDCS generated from the Gaussian fit to the corresponding symmetric amplitudes shown on the top panel.Reuse & Permissions