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Physical Review A

Relativistic effects on low-frequency Rayleigh scattering including spin flip for hydrogenic atoms

Viorica Florescu, M. Marinescu, Richard H. Pratt

Published Oct 1, 1990• 16 citations
Publisher

Abstract

We present an approximate analytic expression for the low-frequency ground-state relativistic Rayleigh scattering matrix elements for hydrogenic atoms, including the possibility of electron spin flip as a result of the elastic photon atom scattering. Our expression is obtained by taking the long-wavelength approximation in the relativistic Coulomb matrix elements, without neglecting the photon frequency dependence in the Coulomb Green operator. By comparison with existing exact numerical calculations, we demonstrate that our approximation is appropriate for photon frequencies below the photoelectric threshold, at least for the non-spin-flip amplitude. Using our expressions, we investigate two issues: (i) the comparison of static and dynamical polarizabilities, and (ii) the behavior of the electron spin-flip amplitude, related to magnetic scattering.

DOI
10.1103/physreva.42.3844

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We present an approximate analytic expression for the low-frequency ground-state relativistic Rayleigh scattering matrix elements for hydrogenic atoms, including the possibility of electron spin flip as a result of the elastic photon atom scattering. Our expression is obtained by taking the long-wavelength approximation in the relativistic Coulomb matrix elements, without neglecting the photon frequency dependence in the Coulomb Green operator. By comparison with existing exact numerical calculations, we demonstrate that our approximation is appropriate for photon frequencies below the photoelectric threshold, at least for the non-spin-flip amplitude. Using our expressions, we investigate two issues: (i) the comparison of static and dynamical polarizabilities, and (ii) the behavior of the electron spin-flip amplitude, related to magnetic scattering.
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.
We have calculated the cross section for Raman scattering of light from a hydrogenic atom, for a wide range of photon energies. The calculation assumes the photon encounters the atom in its ground state and excites it to the 2s2s2s state. Since various contributions to the matrix element may be isolated in the computation, the calculation allows one to obtain a notion of the relative importance of various classes of intermediate states in a given frequency range. We also point out that the Raman efficiency SSS may be expressed in terms of the derivative of the electronic polarizability tensor with respect to a parameter that measures the degree of excitation of the atom. Thus, the formal expression for SSS assumes an appearance identical to the expression encountered in the phenomenological theory of the first-order Raman effect in solids.
An analysis of spin-polarization in low energy elastic scattering electrons by one electron s-state atoms is carried out allowing both for electron exchange and relativistic effects. First the collision matrix is written in terms of the asymptotic form of the solutions of the radial equations. Then arguments, based on the invariance of the Hamiltonian under time reversal, space rotation and space reflection are used to write down the most general form of this matrix. It is found that it can be expressed in terms of six independent amplitudes. Then expressions for the observable quantities, including the spin polarization of the electron and of the atom after the collision are derived. Finally, the analysis is specialized to the case where spin-polarization effects are caused by a spin-orbit interaction and these effects are considered in the neighbourhood of cusps and resonances in electron alkali atom scattering.
The standard theoretical treatment of the Kapitza-Dirac effect---that is, the scattering of an electron passing through a standing-wave laser field---is extended here through the use of the Pauli equation to account for the interaction of the electron spin with the magnetic field of the standing wave. Prescriptions for determining unitarity-preserving approximations for the transition probabilities for scattering both with and without rotation of the electron spin direction are provided. This formalism is used to develop a perturbation theory for the spin-flip probability which, in the strong-field limit of interest here, reduces to a fairly simple relation between SSS-matrix elements for scattering with and without change in spin orientation, each expressed in terms of a Bessel function. A similar perturbative procedure is applied to estimate corrections to the standard theory for scattering in the absence of spin-flip processes, in which interactions that change the net number of photons in the field are ignored.
In this paper I calculate the leading transition amplitudes for a single-photon-double-excitation process which is dominated by long-range correlation effects. The process examined is an excitation of a Rydberg atom with the inner core in an SSS state and the outer electron in essentially a hydrogenic state nnn to another SSS inner-core state and the outer electron in another hydrogenic state ${n}^{\ensuremath{'}}$. We find the leading contribution comes from terms in which the photon is received by the inner core. The excitation of the outer electron is done in intermediate states. The transition amplitude factors into a part entirely dependent on the inner core and a part which is a simple hydrogenic matrix element for the outer electron. The form of the amplitudes suggest that the coefficient to the hydrogenic matrix element be called a "transition polarizability."
In this paper, we extend to three-photon radiative transitions in hydrogenic ions our fully relativistic multipole approach [Z. Bona et al., Phys. Rev. A 89, 022514 (2014)], which has proven so far to yield extremely accurate results for two-photon emission processes. Closed-form formulas of double and single differential frequency distributions as well as total emission probabilities are derived for arbitrary multipole channels by using the Dirac-Coulomb Sturmian functions of the first order. Two cases of atomic systems are considered in the study, namely, ions with a spinless nucleus and those with a nonzero nuclear spin to check the validity of the Bose-Einstein statistics for three-photon emission investigated by Zalialiutdinov et al. [J. Phys. B: At. Mol. Opt. Phys. 49, 055001 (2016)]. In an effort to assess relativistic effects and the influence of the negative spectrum of the set of Dirac-Coulomb Sturmians, we formulate two nonrelativistic schemes that involve Schr\"odinger-Coulomb Sturmians, a transition operator with retardation, on one hand, and a transition operator in the long wavelength approximation, on the other hand. An application of these theories is made for the relativistic and nonrelativistic atomic transitions $2{s}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}, 2{p}_{1/2}\ensuremath{\rightarrow}1{s}_{1/2}, 2{p}_{3/2}\ensuremath{\rightarrow}2{s}_{1/2}$ and $2s\ensuremath{\rightarrow}1s, 2p\ensuremath{\rightarrow}1s$, respectively, with nuclear charge ranging from 1 to 100. Some of our numerical values are compared with the scarce data available in the literature and they enable us to draw inferences as to the improvements obtained with our approach, while the others are additional results that we provide as a further step for more investigations of three-photon decays.