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Physical Review A

Two-photon above-threshold ionization of hydrogen over the photon energy range from 15 eV to 50 keV

Viorica Florescu, Olimpia Budrigă, H. Bachau

Published Sep 29, 2011• 26 citations
Publisher

Abstract

We investigate the absorption of two identical photons from the ground state of hydrogen-like atoms over an energy range that extends beyond that explored up to now. Our approach is based on a hybrid formula, valid in second-order perturbation theory, in which the ${\mathbf{A}}^{2}$ contribution from the nonrelativistic Hamiltonian is treated exactly, while the $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ contribution is calculated in dipole approximation. We find that, at least up to 50 keV, the nonrelativistic dipole approximation, based only on the $\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}$ contribution, determines the values of the total cross section. Our numerical results, covering photon energies from 90 nm (13.7 eV) to 0.0248 nm (50 keV) are in very good agreement with most previous theoretical works. Differences with recent results are discussed.

DOI
10.1103/physreva.84.033425

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We investigate the absorption of two identical photons from the ground state of hydrogen-like atoms over an energy range that extends beyond that explored up to now. Our approach is based on a hybrid formula, valid in second-order perturbation theory, in which the A2{\mathbf{A}}^{2}A2 contribution from the nonrelativistic Hamiltonian is treated exactly, while the A\ifmmode⋅\else\textperiodcentered\fiP\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}A\ifmmode⋅\else\textperiodcentered\fiP contribution is calculated in dipole approximation. We find that, at least up to 50 keV, the nonrelativistic dipole approximation, based only on the A\ifmmode⋅\else\textperiodcentered\fiP\mathbf{A}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{P}A\ifmmode⋅\else\textperiodcentered\fiP contribution, determines the values of the total cross section. Our numerical results, covering photon energies from 90 nm (13.7 eV) to 0.0248 nm (50 keV) are in very good agreement with most previous theoretical works. Differences with recent results are discussed.
This paper discusses the general theory of two-photon ionization of a hydrogenic state in the nonrelativistic dipole approximation. An application is made to the photoionization of the metastable 2S2S2S state of atomic hydrogen by optical radiation, for example, by the light from a ruby laser. The sum over intermediate states is carried out exactly by means of an implicit technique. The results are given for a range of wave-lengths from the threshold at 7290 down to 4556 \AA{}. At 6943 \AA{}, the total cross section per unit intensity is found to be 5.35\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}29}$ cm4{\mathrm{cm}}^{4}cm4 ${\mathrm{W}}^{\ensuremath{-}1}$. The lowest order damping corrections to the perturbation-theory results are worked out.
Two-photon above-threshold ionization processes in the x-ray regime are studied using atomic hydrogen as a model system. Within the minimal-coupling formalism of nonrelativistic quantum electrodynamics, two distinct interactions---$\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}\ensuremath{\cdot}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$ in second order and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}}^{2}$ in first order---contribute to the two-photon absorption amplitude. The relative importance of these two interactions is assessed. It is found that above a photon energy of 6.8 keV, the contribution from ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{A}}}^{2}$ to the total two-photon absorption cross section dominates. In this high-energy regime, above-threshold ionization is a nonsequential purely nondipole process. Rate equations are employed to calculate the probabilities of ionization by Compton scattering, one-photon absorption, and two-photon absorption.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.
We present the results of a numerical study of the invariant amplitudes of 1s-ns and 1s-nd two-photon transitions in hydrogenlike atoms in the nonrelativistic dipole approximation, based on compact analytic formulas. We report also results valid in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. The 1s-nd transitions with n>6 are studied for the first time. For other particular cases our numerical results are in agreement with previous calculations in the literature. The investigated energy region of the two-photon spectrum is extended. The position of the transparencies in the second half of the spectrum is found to be almost n independent.
This paper contains a generalization of results described in previous work. A general formula is given for the total cross section for two-photon ionization of a hydrogenic state. An implicit method for evaluating the second-order radial matrix elements that occur in this expression is described in detail. Numerical results were obtained for the states with principal quantum numbers n = 1 through 5. It is concluded that, when both one-photon and two-photon ionization are energetically possible, the effect of the latter may be expressed as an intensity-dependent correction to the Gaunt factor which may therefore be written as G = g1 + ζIλ3g2. Here g1 is the usual Gaunt factor, I is the intensity of the light in W cm−2, λ is the wavelength in cm, g2 is a dimensionless factor of order unity, and ζ is a constant given by ζ = 0.1504 W−1 cm−1. Graphs of g2 are given as a function of electron energy. These results do not include the effect of three-photon processes, which can also contribute to the first-order intensity-dependent correction to the Gaunt factor as a result of interference between the first- and third-order amplitudes.