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Physical Review

On the Theory of the Smekal-Raman Effect in Hydrogen-Like Atoms

Boris Podolsky, Vladimir Rojansky

Published Nov 15, 1929• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.34.1367

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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.
In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.
In view of the recent experimental determination of the dispersion by atomic hydrogen it seems interesting to apply the theory of dispersion developed by Schrodinger to this case. In this paper we restrict ourselves to an approximation in which terms of the order of relativistic correction are neglected. For this purpose it is simpler to obtain our wave equation by the operational method of Schrodinger and Eckart, as extended by Epstein, for in this way we immediately obtain an equation free of relativistic terms. In what follows, in order to preserve a continuity of the discussion, details of calculations are omitted from the main text and are given as Supplementary Notes at the end of the paper. References will be found immediately preceding the Supplementary Notes.
The effect of spontaneous decay on the linear polarizability of an atom is typically included by adding imaginary parts to the frequency denominators that appear in the Kramers-Heisenberg formula. It has been shown for a two-level atom with radiative broadening that these (frequency-dependent) imaginary parts must be included in both the resonant and antiresonant frequency denominators [P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 (2004)]; however, the expression obtained by Milonni and Boyd for the polarizability does not satisfy the optical theorem, if contributions from non-rotating-wave terms are included. In this paper, we derive a more accurate expression for the polarizability. The calculations are rather complicated and require that we go beyond the standard Weisskopf-Wigner approximation. We present calculations carried out in both the Heisenberg and Schr\"odinger pictures, since they offer complementary methods for understanding the dynamics of the Rayleigh scattering associated with the atomic polarizability. Moreover, it is shown that the shifts associated with the excited state are not the Lamb shifts of an isolated atom, but depend on the dynamics of the atom-field interaction. Our results for the polarizability are consistent with those obtained recently by Loudon and Barnett using a Green's-function approach.
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.
We describe a unified approach for the determination of ac-Stark correction and Kramers-Heisenberg dispersion formula. In both cases the contribution from infinite intermediate summation appearing in the expression for the corresponding matrix elements are evaluated exactly in the dipole approximation for the ground state of hydrogen atom using a variation of the Dalgarno-Lewis method. The analytical expressions obtained can be efficiently used for the numerical evaluation of matrix element for all values of incident photon energy and comparison is made with results obtained by different methods.
A generalization of Dirac's transformation of the off-resonance single-photon scattering amplitude that leads to the Kramers-Heisenberg dispersion formula is described. The amplitude is first calculated to order e2{e}^{2}e2 in perturbation theory with the minimal-coupling interaction Hamiltonian as perturbing energy, and is then transformed so that the effects of multipole transition moments of all orders and types (electric, magnetic, and diamagnetic) readily become manifest. No appeal is made to any multipolar expansions, however, as exact integral expressions are used for the summed series. It is shown that the same dispersion formula is obtained by using the multipolar Hamiltonian partitioned in the usual way, and this despite the fact that the unperturbed Hamiltonians are different in the two calculations.