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Physica Scripta

Semirelativistic description of Raman scattering of photons by hydrogenlike ions: comparison with relativistic and nonrelativistic approaches under the ionization threshold

Aline Djoussi, Falone Fotsing, Celsus Bouri, M.G. Kwato Njock

Published Apr 8, 2025• 0 citations
Publisher

Abstract

Abstract Recently we constructed a semirelativistic and nondipole approach (SRNDA) for Raman scattering of light by hydrogenlike ions, by means of the discrete semirelativistic Coulomb Sturmian basis sets, the direct Foldy-Wouthuysen photon-atom interaction Hamiltonian and the expansion of photon fields in electric and magnetic multipoles [Meyap Soh C et al 2018 Ann. Phys. 395 196]. We also proposed its nonrelativistic version which involves the paramagnetic and diamagnetic terms of the semirelativistic Hamiltonian. Following these treatments, the purpose of this paper is twofold. First, we have completed that previous work with the derivation of the fully relativistic Raman scattering based on the well-known Dirac-Coulomb Sturmians of the first order. The formulas obtained, convenient for computer calculations, make it possible to accurately evaluate cross sections. Second, in an effort to reliably judge the credibility of the SRNDA and to assess its scope of applicability as far as the nuclear charge of atomic systems and the incident photons energy are concerned, we have carried out for concreteness computations of these schemes under the ionization threshold. They allow us to compare the various numerical predictions available, including those existing in the literature.

DOI
10.1088/1402-4896/adca69

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Abstract Recently we constructed a semirelativistic and nondipole approach (SRNDA) for Raman scattering of light by hydrogenlike ions, by means of the discrete semirelativistic Coulomb Sturmian basis sets, the direct Foldy-Wouthuysen photon-atom interaction Hamiltonian and the expansion of photon fields in electric and magnetic multipoles [Meyap Soh C et al 2018 Ann. Phys. 395 196]. We also proposed its nonrelativistic version which involves the paramagnetic and diamagnetic terms of the semirelativistic Hamiltonian. Following these treatments, the purpose of this paper is twofold. First, we have completed that previous work with the derivation of the fully relativistic Raman scattering based on the well-known Dirac-Coulomb Sturmians of the first order. The formulas obtained, convenient for computer calculations, make it possible to accurately evaluate cross sections. Second, in an effort to reliably judge the credibility of the SRNDA and to assess its scope of applicability as far as the nuclear charge of atomic systems and the incident photons energy are concerned, we have carried out for concreteness computations of these schemes under the ionization threshold. They allow us to compare the various numerical predictions available, including those existing in the literature.
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.
The inelastic Raman scattering of light by hydrogenlike ions has been studied by means of second-order perturbation theory and the relativistic Coulomb Green's-function approach. In particular, we investigate the total and angle-differential Raman cross sections as well as the magnetic sublevel population of the residual (excited) ions. Detailed calculations are performed for the inelastic scattering of photons by neutral hydrogen as well as hydrogenlike xenon and uranium ions, accompanied by the $1{s}_{1/2}\ensuremath{\rightarrow}2{s}_{1/2}$, $1{s}_{1/2}\ensuremath{\rightarrow}2{p}_{1/2}$, and $1{s}_{1/2}\ensuremath{\rightarrow}2{p}_{3/2}$ transitions. Moreover, we discuss how the Raman scattering is affected by relativistic and resonance effects as well as the higher-multipole contributions to the electron-photon interaction.
The angular correlations in the two-photon decay of hydrogenlike ions are studied within the framework of second-order perturbation theory, based on Dirac's equation. Particular attention has been paid to the effects which arise from the higher (nondipole) terms in the expansion of the electron-photon interaction. It is shown that the photon-photon angular correlation function, which is found symmetric with respect to the angle $\ensuremath{\theta}=90\ifmmode^\circ\else\textdegree\fi{}$ in the electric dipole approximation, becomes asymmetric because of the nondipole contributions, and that this effect is enhanced as the nuclear charge ZZZ increases. Detailed computations on the photon-photon angular distribution have been carried out for the $2{s}_{1∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ and $3{d}_{5∕2}\ensuremath{\rightarrow}1{s}_{1∕2}$ transitions in neutral hydrogen (H) as well as for hydrogenlike xenon (Xe53+)({\mathrm{Xe}}^{53+})(Xe53+) and uranium (U91+)({\mathrm{U}}^{91+})(U91+) ions, and are compared with previous nonrelativistic results by Au [Phys. Rev. A 14, 531 (1976)]).
The hyperfine levels for the ground state of hydrogen are normally connected by magnetic dipole transitions. However, for astrophysical applications, it may be that two-photon electric dipole processes are also important over cosmological distances. This thesis sets up a general mathematical formalism for the calculation of two-photon processes within the hyperfine structure of an atom with spin 1/21/21/2, and applies it to the ground state of hydrogen. Rates are calculated for both spontaneous emission and absorption of radiation. Due to an extraordinary degree of cancellation in the matrix elements, the rates turn out to be too small to have a significant impact on astrophysical processes involving the absorption and emission of radiation. The results are nevertheless important in reducing the uncertainty due to possible contributions from these processes. Other two-photon processes involving Raman scattering are also considered and the relevant cross sections calculated. The calculations of the decay rate, absorption coefficient and Raman scattering cross section are extended to heavier hydrogenic ions as a function of the nuclear charge ZZZ, using an approximate interpolation formula for the hyperfine splitting in the ground state. Results are presented for the hydrogenlike ions from He+^++ to Pb81+^{81+}81+. Modifications for the case of isotopes with a nuclear spin greater than 1/21/21/2 are briefly considered
The angular distribution of the emitted electrons, following the two-photon ionization of hydrogen-like ions, is studied within the framework of second order perturbation theory and the Dirac equation. Using a density matrix approach, we have investigated the effects which arise from the polarization of the incoming light as well as from the higher multipoles in the expansion of the electron–photon interaction. For medium- and high- Z ions, in particular, the non-dipole contributions give rise to a significant change in the angular distribution of the emitted electrons, if compared with the electric dipole approximation. This includes a strong forward emission while, in the dipole approximation, the electron emission always occurs symmetrically with respect to the plane which is perpendicular to the photon beam. Detailed computations for the dependence of the photoelectron angular distributions on the polarization of the incident light are carried out for the ionization of H, and Xe 53+ and U 91+ (hydrogen-like) ions.
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.