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Physical Review

Raman Scattering by Hydrogenic Systems

Wayne M. Saslow, D. L. Mills

Published Nov 15, 1969• 35 citations
Publisher

Abstract

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 $2s$ 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 $S$ 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 $S$ assumes an appearance identical to the expression encountered in the phenomenological theory of the first-order Raman effect in solids.

DOI
10.1103/physrev.187.1025

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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.
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 temperature effect on the resonance Raman scattering from molecules is investigated. Considering a total system consisting of a molecular system, a heat bath, and the weak radiation field, a general expression for the relaxation processes is derived utilizing the Liouville operator technique for the time evolution of the density matrix of the total system. By using the perturbation method with respect to the radiation and matter up to second order, rate constants for zero photon (radiationless processes), one-photon and two-photon processes are derived after taking the long time limit in the Markoff approximation. Assuming a weak interaction between the molecular system and heat bath, a temperature-dependent resonance Raman scattering cross section is formulated in the adiabatic basis set for the molecular systems. In the displaced harmonic oscillator model, we obtain an analytical expression for the temperature-dependent Raman scattering cross section for the multimode case, which is suitable for both weak coupling and strong coupling cases. Analytical expressions for the Stokes and anti-Stokes bands of the jth order vibrational transition in the one oscillator model can easily be obtained from the general expression for the resonance Raman scattering cross section derived. An approximate Stokes to anti-Stokes ratio is given up to the second order of the dimensionless displacement Δ. The resonance Raman scattering cross section and the excitation energy profiles are numerically calculated as a function of physical constant such as temperature, molecular vibrational frequency and Δ. It is shown that the excitation profiles of the anti-Stokes band of the jth order vibrational transition have the similar envelope of the corresponding Stokes band shifted by jh/ω. The inverse Raman effect which is closely related to the ordinary Raman effect is also discussed.
Quantum-mechanical expressions for nonresonance-dipole light-scattering cross sections are related to corresponding expressions for oscillator strengths and refractive indices. In particular, Rayleigh and Raman cross sections for atoms are expressed in terms of oscillator strengths and vector-coupling coefficients. Several of our results, including the relationship between Rayleigh scattering and refractive index, differ in general from the corresponding results of classical dispersion theory. We show that these differences arise from antisymmetric components Ckja = (Ckj − Cjk)/2 of the polarizability tensor, which have been neglected in the classical analyses. Calculated Rayleigh cross sections for cesium and aluminum atoms, and Raman cross sections for aluminum atoms are presented to illustrate our results. The antisymmetric contribution is found to be substantial in all of these cross sections; its most obvious effect is to cause the depolarization (for linearly polarized incident light) to exceed 34 over extended wavelength ranges away from resonance. On the other hand, for atoms initially in states of zero angular momentum, and molecules under conditions which allow the use of Placzek’s polarizability approximation, our results agree with those of classical dispersion theory.
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
It has been shown by Schr\"odinger and O. Klein that the Kramers-Heisenberg formulas for dispersion and incoherent scattering can both be obtained by an extension of Schr\"odinger's method for treating dispersion, the terms in the Hamiltonian involving squares of the potentials being neglected. The resulting formulas agree with those of Dirac. They are, however, inconvenient for calculations, as they contain summations with respect to all energy levels combining with the pair of levels under consideration, and thus imply complicated integrations when a continuous spectrum is present. This paper treats the problem of a hydrogen-like atom, acted on by light the wave-length of which is large compared to the size of the atom and the frequency of which is not too near a resonance frequency, and develops a method which obviates the necessity of integrating over the continuous range. It is an extension of the method used by one of us in treating the dispersion by atomic hydrogen. It is applied in detail to the first two levels of H. Formulas are derived for the intensities of the Smekal-Raman lines with a shift corresponding to the first absorption line.