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Journal of Mathematical Physics

Two-Photon Ionization of Atomic Hydrogen. II

Wolfgang Zernik, R. W. Klopfenstein

Published Feb 1, 1965• 95 citations
Publisher

Abstract

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.

DOI
10.1063/1.1704277

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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.
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.
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{}.
Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.
The infinite summations over the complete set of unperturbed atomic states appearing in the NNNth-order perturbation theory for multiphoton ionization are performed using the Dalgarno-Lewis method. The relevant transition matrix elements are written in a closed integral form exhibiting all the analytic properties of the amplitude as a function of incident photon energy. The cross sections for two- and three-photon ionization for atomic hydrogen are calculated for both linearly and circularly polarized light with a wide range of photon energy spectrum including the near resonance, and numerical comparison is made with the values obtained by different methods.
Within the framework of nonrelativistic quantum mechanics, the analysis of the two-photon absorption by atomic hydrogen is generalized to the case of two incident beams with arbitrary direction and polarization (including circular polarization). In the dipole approximation, the second-order matrix elements responsible for two-photon absorption are transformed into a finite sum consisting of the product of a radial part and an angular part. Exact calculation of the angular part predicts that, for the s–s transition only, transparency is obtained regardless of the frequencies of the two beams with perpendicular polarizations. In addition, circularly polarized light is found to be more efficient (up to a factor of 1.5) for the s–d two-photon transition. The radial parts of the matrix elements are accurately evaluated by using the Coulomb Green’s function technique and an implicit technique of Dalgarno and Lewis. Their calculation indicates that “zeros” exist in the two-photon absorption spectrum, thereby predicting that absorption is not possible at certain frequencies. It should be noted that in the calculated spectrum, near or at resonance points, the linewidths of the intermediate levels must be included in order to obtain accurate results.
Two-photon ionization of atomic hydrogen with an excess photon is revisited. The non-relativistic dipole approximation and Coulomb Green function (CGF) formalism are applied. Using the CGF Sturmian expansion straightforwardly, one gets the radial transition amplitude in the form of an infinite sum over Gauss hypergeometric functions which are polynomials. It is convergent if all intermediate states are in the discrete spectrum. In the case of two-photon ionization with an excess photon, when photoionization is also possible, intermediate states are in the continuum. We performed the explicit summation over intermediate states and got a simple general expression for the radial transition amplitude in the form of a finite sum over Appell hypergeometric functions, which are not polynomials. An Appell function may be expressed as an infinite sum over Gauss functions. In the case of ionization by an excess photon, Gauss functions are transformed to give a convergent radial transition amplitude for the whole region. The generalized cross sections for two-photon above-threshold ionization of atomic hydrogen in the ground state and excited states calculated by us agree very well with results of previous calculations. Generalized cross sections for two-photon ionization of positronium in the ground state are obtained by scaling those for atomic hydrogen.