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Bulletin de la Classe des sciences

Derivation of kinetic equation for a bound electron interacting with radiation (up to order λ4)

Françoise Hénin

Published Jan 1, 1968• 0 citations
Publisher

Abstract

We consider an electron in a multilevel atom, interacting with radiation and discuss some aspects of the derivation of the kinetic equation, up to order λ4. The analogy with the problem of cubic interaction between phonons in anharmonic solids is stressed ; a detailed discussion shows how the results obtained in the latter case can be used in the present problem. The two photon scattering operator, the λ2 frequency and vertex renormalizations are then computed.

DOI
10.3406/barb.1968.62155

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We consider an electron in a multilevel atom, interacting with radiation and discuss some aspects of the derivation of the kinetic equation, up to order λ4. The analogy with the problem of cubic interaction between phonons in anharmonic solids is stressed ; a detailed discussion shows how the results obtained in the latter case can be used in the present problem. The two photon scattering operator, the λ2 frequency and vertex renormalizations are then computed.
The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.
A wide class of problems is considered involving the interaction of multilevel atoms with classical time-dependent fields switched on at a definite time. It is shown that the exact time-dependent Green's function can be written as the particular solution to an inhomogeneous differential equation involving derivatives with respect to the external field variables. Asymptotic expansions are discussed and analytic solutions found for several cases. The time and frequency distributions of radiation emitted by a field-perturbed atom are calculated by including the radiation field as a first-order perturbation.
Construction of a kinetic equation for a dynamical system interacting with a boson field in the case of spatial inhomogeneity is based on a method expounded in [1–3]. In the present article, it is shown that approaches considered in [1, 2] can be generalized for the case of spatial inhomogeneity. An arbitrary operator function that depends on the momentum and spatial variable is applied for derivation of a kinetic equation. We consider a method of studying an electron-phonon system by means of exclusion of boson operators from corresponding operator equations. In particular, interaction of an electron with a boson field is described by a kinetic equation for a polaron in the case of spatial inhomogeneity. In the relevant limit, a Boltzmann equation for the polaron is obtained.
In the preceding chapter electron rate equations of bimolecular form were derived. In this chapter these equations are generalized to include higher order non-linear terms. Such terms occur in rate equations for electrons, phonons, photons, etc., either if many-particle interaction operators are taken into account, or (and) if powers of oneparticle interaction operators are included. The latter case takes place if the transition probabilities (1.5.15) with respect to one-particle interaction operators are evaluated in higher orders. An estimate of the physical consequences of such transitions is, at present, not possible, since for higher order transitions the derivation of rate equations becomes more and more complicated. Hence a rigorous mathematical justification of higher order effects in rate equations starting from first principles seems at present to be extraordinarily hard and has not yet been undertaken. Thus in the following we consider only such interactions which already lead to higher order non-linear terms in the rate equations for first order (resonance) transition probabilities. In particular we concentrate on the electronic many-particle interactions. We will use the same model as in Chapter 3 and in principle the calculations will run along the same lines as in the previous chapter, but the calculation effort will become much greater.