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The Journal of Chemical Physics

Forbidden Raman scattering processes. I. General considerations and E1–M1 scattering

R. C. Harney

Published Mar 1, 1979• 0 citations
Publisher

Abstract

The generalized theory of forbidden Raman scattering processes is developed in terms of the multipole expansion of the electromagnetic interaction Hamiltonian. Using the general expressions, the theory of electric dipole–magnetic dipole (E1–M1) Raman scattering is derived in detail. The 1S0→3P1 E1–M1 Raman scattering cross section in atomic magnesium is calculated for two applicable laser wavelengths using published f-value data. Since resonantly enhanced cross sections larger than 10−29 cm2/sr are predicted it should be possible to experimentally observe this scattering phenomenon. In addition, by measuring the frequency dependence of the cross section near resonance, it may be possible to directly determine the relative magnitudes of the A⋅p and A⋅A contributions to the scattering cross section. Finally, possible applications of the effect in atomic and molecular physics are discussed.

DOI
10.1063/1.437760

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The generalized theory of forbidden Raman scattering processes is developed in terms of the multipole expansion of the electromagnetic interaction Hamiltonian. Using the general expressions, the theory of electric dipole–magnetic dipole (E1–M1) Raman scattering is derived in detail. The 1S0→3P1 E1–M1 Raman scattering cross section in atomic magnesium is calculated for two applicable laser wavelengths using published f-value data. Since resonantly enhanced cross sections larger than 10−29 cm2/sr are predicted it should be possible to experimentally observe this scattering phenomenon. In addition, by measuring the frequency dependence of the cross section near resonance, it may be possible to directly determine the relative magnitudes of the A⋅p and A⋅A contributions to the scattering cross section. Finally, possible applications of the effect in atomic and molecular physics are discussed.
Introduction The selection rules discussed in Sec. 3.5 forbid many kinds of transitions that are frequently observed or exploited. It must be remembered that these “rules” are based on the approximations of Sec. 3.2, but of course, those can be extended. Such expansions to higher orders of approximation may seem to be little more than a tedious exercise, but this chapter will show that the results are extremely important, because they cause qualitatively different kinds of transitions. Since they involve both the electric and magnetic components of the light, their atomic dipole moments will be identified with the superscript E and B for the remainder of this chapter. The magnetic dipole transitions, called M1, discussed in Sec. 4.2.1 that arise from extension of the electric dipole approximation encompass all of magnetic resonance, including atomic clocks, forced evaporative cooling in Bose–Einstein Condensation (BEC), MRI, and astrophysical processes such as generation of the all-important 21 cm radio frequency line. The electric quadrupole transitions, called E2, discussed in Sec. 4.2.2 describe myriad stellar spectral features as well as the vast majority of nuclear γ-ray transitions. Even with these extensions there are many cases where the relaxation of the electric dipole (E1) selection rules to include M1 and E2 transitions as in Sec. 4.2 still do not account for observed processes. Thus other approximations have to be extended so that their limitations can be relaxed as well. Extension of the perturbation approximation (see Sec. 4.3.1) provides an analytical description of most of non-linear optics, including second harmonic generation, four wave mixing, Raman spectroscopy, and the fabled decay of the hydrogen 2S state. A semi-classical description of the interaction between radiation and atoms can begin with a multipole expansion of the Hamiltonian. The lowest-order term is just from the point charges α located at, and vanishes for neutral atoms. The next-order term is the electric dipole interaction that arises from the non-isotropic distribution of charges induced by the electric field of the light. This term produces the electric dipole transitions (E1) that have been described in some detail in Chap. 3.
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.
A theory for the resonance light scattering in a molecular system interacting with the heat bath is developed, taking into account the multilevel structure of the molecular system. An expression for the cross section of the resonance light scattering process is derived by using the Liouville operator method in the Markoff approximation; the cross section consists of three terms: resonance Raman scattering, resonance fluorescence, and their mixing. The mixing term vanishes in the limit of the zero vibrational dephasing constant. Analytical expressions for the resonance Raman scattering and resonance fluorescence cross sections are derived in the displaced harmonic oscillator model. Model calculations of the vibrational band structure in the resonance light scattering cross section are performed by using the expressions derived. Differences of the band intensity distribution in the resonance fluorescence from that in the resonance Raman scattering are illustrated.
The relative contributions of the magnetic dipole interaction ( M 1) and the electric quadrupole interaction ( E 2) in the dipole-forbidden three-wave process of mixing two-colour radiation in free atoms may be determined from the dependence of the frequency-conversion efficiency on the incident-wave polarization. Analytical expressions are derived for the polarization dependence of amplitudes and cross sections of a double-resonant scattering of the sum- and difference-frequency photons. Numerical estimates are presented for the ratio of E 2 and M 1 three-photon amplitudes in ground-state indium and thallium atoms.
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.
An inhomogeneous differential equation method is used to calculate the Rayleigh and Raman scattering cross sections for atomic hydrogen and atomic caesium as functions of wavelengths below and above the ionization limits. The profile of the scattering cross section near the 6s-6p resonance of atomic caesium is determined by including an energy-independent radiative width in the scattering formula. The effect of replacing the electric dipole operation by a modified form that takes account of the polarization of the core by the valence electron is explored.