Dynamic scaling of photo-double-ionization to electron impact

We employ an ab initio theory to address several inconsistencies of the empirical scaling of photo-double-ionization (PDI) to electron impact ionization suggested by Samson [Phys. Rev. Lett. 65, 2861 (1990)]. We do so (i) by isolating the only relevant dipole component of the electron impact ionization and (ii) by replacing the static scaling of Samson, who used an effective cross-section area of the ion remainder, with dynamic scaling using the squared momentum of the photoelectron. The modified scaling is tested in the He $1s^2$ isoelectronic sequence and the combined inner shell $1s$ and valence shell $2p$ PDI of the negative ${\mathrm{F}}^{-}$ ion [Müller et al., Phys. Rev. Lett. 120, 133202 (2018)].

Figure 1

Graphical representation of PDI in RPAE. (a) The dipole matrix element $D_{1234}$ entering Eq. (4) (left half) and its complex conjugate $D_{1234}^{*}$ (right half). (b) The squared dipole matrix element $|D_{1234}{|}^2$ is incorporated into the imaginary self-energy part of the single-electron Green's function, Eq. (5). (c) Direct and exchange diagrams which represent the irreducible part of the Green's function. The single-electron Green's function consists of an infinite sequence of such diagrams (see Figs. 1 and 2 of [15]).

Figure 2

Electron impact ionization cross section ${\sigma }_{e,J}^{+}$ of the ${\mathrm{Li}}^{+}$ ion. RPAE calculations with various multiplicities $J=0,1,2,3$ and their sum ${\sum }_{J=0}^3$ are displayed along with the dipole $J=1$ and total ${\sum }_J$ CCC calculations from [21].

Figure 3

CCC calculations for the double-to-single ${\sigma }_{\gamma }^{2+}/{\sigma }_{\gamma }^{+}$ photoionization cross-section ratios in He, ${\mathrm{Li}}^{+}$ [12], and ${\mathrm{Mg}}^{10+}$ [5] (from top to bottom). The products of ${\sigma }_e^{+}$ with ${\left(\pi r^2\right)}^{-1}$ and $k^2$ are also displayed. The excess energy is shown in eV (bottom axis) and in units of the ionic binding energy $I_p^{+}$ (top axis). Experimental data for PDI of He are from [22]. Numerical values of $\langle r\rangle$ used for scaling theoretical PDI to $(e,2e)$ are 0.7 (He), 0.47 (${\mathrm{Li}}^{+}$), 0.36 (${\mathrm{Be}}^2+$, not shown), and 0.11 (${\mathrm{Mg}}^{10+}$) whereas the corresponding analytical values are 0.75, 0.5, 0.375, and 0.125.

Figure 4

RPAE calculations for ${\sigma }_{\gamma }^{2+}/{\sigma }_{\gamma }^{+}$ photoionization cross-section ratio (top) and PDI cross section ${\sigma }_{\gamma }^{2+}$ (bottom) in the ${\mathrm{F}}^{-}$ ion. Experimental data are displayed with blue absolute error bars, green statistical error bars, and black dots representing a normalized scan as in Fig. 4 of Ref. [9].