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Physical Review A

Photoionization cross section: Exact expansion over resonances and natural line shape

Gabriel Álvarez, Harris J. Silverstone

Published Oct 1, 1989• 10 citations
Publisher

Abstract

The photoionization cross section is analyzed as a sum of resonance contributions plus background. The contribution of each resonance is proportional to the square of a complex transition-dipole matrix element divided by a complex energy denominator and has a natural asymmetric line shape that is the superposition of a dispersive and an absorptive Lorentzian component.

DOI
10.1103/physreva.40.3690

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The photoionization cross section is analyzed as a sum of resonance contributions plus background. The contribution of each resonance is proportional to the square of a complex transition-dipole matrix element divided by a complex energy denominator and has a natural asymmetric line shape that is the superposition of a dispersive and an absorptive Lorentzian component.
Attenuation processes are examined from the viewpoint of collision theory, and the optical theorem is used to connect attenuation cross section $\ensuremath{\sigma}$ and refractive index nnn with diagonal elements of the TTT matrix. This approach provides resonance profiles (for "natural" lines shapes) of the form $\ensuremath{\sigma}(\ensuremath{\omega})=C+\frac{(\frac{\ensuremath{\Gamma}}{2})B+(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})A}{{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{2}+{(\frac{\ensuremath{\Gamma}}{2})}^{2}},$ $n(\ensuremath{\omega})\ensuremath{-}1=\frac{\mathrm{Nc}}{2\ensuremath{\omega}}\left[\frac{(\frac{\ensuremath{\Gamma}}{2})A\ensuremath{-}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})B}{{(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}^{2}+{(\frac{\ensuremath{\Gamma}}{2})}^{2}}\ensuremath{-}D\right],$ where the profile parameters AAA, BBB, CCC, DDD, $\ensuremath{\Gamma}$, ${\ensuremath{\omega}}_{0}$ are given in terms of atomic matrix elements.Part I reviews the notion of resonances. Part II summarizes the relevant results of collision theory, stressing physical interpretation, and gives a definition for excited (or resonance) states based on a simple partition of basis states into two classes. Part III applies perturbation theory to the calculation of resonance profiles. Part IV applies these results specifically to the attenuation and refraction of photons by tenuous gases, with particular attention paid to the profiles of autoionizing lines. The effects of degeneracy (the extension of the bound-state ${Z}^{\ensuremath{-}1}$ expansion theory) are noted.
This paper examines attenuation processes from t h e view- point of c o l l i s i o n theory, using the o p t i c a l theorem t o connect c a t t e n u a t i o n cross-section CJ and r e f r a c t i v i t y n with diagonal elements of the T matrix.This approach provides resonance prof i l e s (for '*natural*' l i n e s shapes) of the form where t h e p r o f i l e parameters A, B, C , D, r, w 0 are given i n t e r m s of atomic matrix elements.P a r t I reviews the notion of resonances.Part I1 summarizes the r e l e v a n t r e s u l t s of c o l l i s i o n theory, s t r e s s i n g physical ' 1
The shape of an isolated spectral transition is analyzed in terms of an approximation to the Waldmann–Snider kinetic equation. This equation is written in the form of a drift and collision operator acting on a density matrix. With the use of the spherical approximation, the collision operator is subdivided into an elastic Boltzmann-like collision term, an inelastic loss term, and a dephasing term. The Boltzmann-like term is responsible for Dicke narrowing of spectral lines, the inelastic loss term leads to line broadening and the dephasing term may contribute both to line broadening and shifting. Simple approximations to these terms are powerful enough to account for some of the details of experimental line shapes such as asymmetrical deviations from a Lorentzian line shape. Model numerical calculations are carried out assuming classical scattering potentials of the form 1/rn in the Boltzmann-like term and single complex frequencies for the other two. It was found that as long as the density was scaled to give the same diffusion constant, the exact form of the scattering cross section had little effect on the final line shape.
Asymmetric spectral line shapes are a hallmark of interference of a quasi-bound state with a continuum of states. Such line shapes are well known for multichannel systems, for example, in photoionization or Feshbach resonances in molecular scattering. On the other hand, in resonant single channel scattering, the signature of such interference may disappear due to the orthogonality of partial waves. Here, we show that probing the angular dependence of the cross section allows us to unveil asymmetric Fano profiles also in a single channel shape resonance. We observe a shift in the peak of the resonance profile in the elastic collisions between metastable helium and deuterium molecules with detection angle, in excellent agreement with theoretical predictions from full quantum scattering calculations. Using a model description for the partial wave interference, we can disentangle the resonant and background contributions and extract the relative phase responsible for the characteristic Fano-like profiles from our experimental measurements.