Static-field ionization model of He-like ions for diagnostics of light-field intensity

Erik Lötstedt, Marcelo F. Ciappina, and Kaoru Yamanouchi

Phys. Rev. A 102, 013112 – Published 27 July 2020

Abstract

We study static-field ionization of He-like ions with nuclear charge number in the range of $2 \leq Z \leq 36$ . Both the tunneling and over-the-barrier regimes are considered. We calculate the ionization rates by three approximate methods: a fitting formula based on the Perelemov-Popov-Terent'ev (PPT) formula, a perturbative expansion in powers of $1 / Z$ , and a single-active electron approximation, and compare them with reference ionization rates computed by the multiconfiguration time-dependent Hartree-Fock (MCTDHF) approach. The relative deviation of the rates computed by the PPT-based fitting formula and the third-order perturbation theory from the rates computed by the MCTDHF approach is found to be around 10%. We discuss quantitatively the contribution from the exchange interaction during the ionization by comparing the single-active electron rates and the reference rates in which multielectron effects are included. We find that a single-active electron approximation, where the exchange interaction is neglected, results in the overestimation of the ionization rates by 30% for He and 2% for He-like Kr, showing that the magnitude of the effect of the exchange interaction scales approximately as $1 / Z$ .

Received 2 May 2020
Revised 2 July 2020
Accepted 7 July 2020

DOI:https://doi.org/10.1103/PhysRevA.102.013112

Physics Subject Headings (PhySH)

Atomic & molecular processes in external fields Multiphoton or tunneling ionization & excitation

Hartree-Fock methods

Atomic, Molecular & Optical

Authors & Affiliations

Erik Lötstedt ^1,*, Marcelo F. Ciappina ^2,3, and Kaoru Yamanouchi ¹

¹Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
²Institute of Physics of the ASCR, ELI-Beamlines Project, Na Slovance 2, 182 21 Prague, Czech Republic
³ICFO–Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, E-08860 Castelldefels (Barcelona), Spain

^*lotstedt@chem.s.u-tokyo.ac.jp

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Vol. 102, Iss. 1 — July 2020

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Images

Figure 1
Ionization rates $Γ = Z^{2} Γ^{'} / 4$ calculated with the MCTDHF method for (a) He, (b) He-like Be ( $Z = 4$ ), and (c) He-like Ar ( $Z = 18$ ) are shown with filled circles. The ionization rates are shown as a function of the scaled field $F^{'} = 8 F / Z^{3}$ . Comparisons are made with the ionization rates calculated using the PPT formula [solid line; see Eq. (C1)] and the PPT-based fitting formula suggested in [38] [dotted line; see Eq. (22)]. In panel (a), the dash-dotted line shows the results obtained by Scrinzi, Geissler, and Brabec in [31]. The scaled barrier suppression field $F_{BS}^{'}$ is indicated with a vertical solid line. The blue shaded area indicates the interval where the ionization probability during one laser cycle $P_{ion}$ [defined in Eq. (25)] is in the range between 10% and 90%. In terms of peak laser field intensities, $F^{'} = 0.4$ a.u. corresponds to $5.6 \times 10^{15} W / {cm}^{2}$ for $Z = 2, 3.6 \times 10^{17} W / {cm}^{2}$ for $Z = 4$ , and $3.0 \times 10^{21} W / {cm}^{2}$ for $Z = 18$ .
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Figure 2
(a) Ionization rate $Γ = Z^{2} Γ^{'} / 4$ at $F = F_{BS}$ calculated with the MCTDHF method (blue solid circles), the rate $Γ_{fit}$ calculated with the PPT-based fitting formula (red triangles), and a fit of $Γ$ to a quadratic function $a Z^{2}$ with $a = 1.4 \times 10^{- 3}$ (thick solid gray line) are shown as a function of $Z$ . (b) Ionization rates evaluated at $F = {(2 I_{p})}^{3 / 2} \times 0.05$ , such that the reduced field $\tilde{F} = 0.05$ a.u. The coefficient $a$ in the quadratic fit is $a = 1.3 \times 10^{- 4}$ .
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Figure 3
Ionization rates for (a) $Z = 10$ and (b) $Z = 18$ obtained by the MCTDHF method (filled circles) and those obtained by perturbation theory as a function of the scaled field $F^{'} = 8 F / Z^{3}$ . The perturbative rate $Γ_{pert}^{N}$ contains corrections up to the order $1 / Z^{N}$ . In panel (c), we show the rates as a function of $Z$ at a fixed value $F^{'} = 0.4$ a.u. of the scaled field.
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Figure 4
Ionization rates $Γ = Z^{2} Γ^{'} / 4$ as a function of $F^{'}$ for (a) He and (b) He-like Ne ( $Z = 10$ ), calculated with the MCTDHF method and the SAE model, using potential $V_{1}$ [see Eq. (16)]. In (a) we additionally show the rates obtained with the SAE model including the polarization potential [see Eq. (18)], and the FC model including a correlated He ground-state wave function [see Eq. (21)] In (c), we show the relative deviation $Δ Γ / Γ = (Γ_{SAE} - Γ) / Γ$ of the rate $Γ_{SAE}$ obtained with the SAE approximation from the MCTDHF rate $Γ$ as a function of $Z$ , evaluated at $F^{'} = 0.4$ a.u. and $F = F_{BS}$ . Also shown are fits to the function $b / Z$ , with $b (F^{'} = 0.4) = 0.68$ , and $b (F = F_{BS}) = 0.84$ . In the case of $F = F_{BS}$ , only $Δ Γ / Γ$ values for $Z \geq 10$ were included in the fit.
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Figure 5
(a) Zeroth-order expansion coefficent $λ_{0}^{'}$ and eight times the ionization rate $Γ_{H}$ of hydrogen, taken from Trinh et al. [29]. (b) Expansion coefficients $λ_{n}^{'}$ for the orders $n = 1$ , 2, and 3.
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Figure 6
Static-field ionization rate $Γ_{sym}$ of He calculated by the Hartree-Fock model defined in Eq. (F1) where exchange symmetry is fulfilled and the ionization rate $Γ_{nonsym}$ calculated by the Hartree-Fock model defined in Eq. (F2) where no exchange symmetry is considered.
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