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Journal of Physics B Atomic and Molecular Physics

Modifications to population rate equations resulting from correlations between collisional and radiative processes

R. J. Ballagh, J. Cooper

Published Nov 14, 1984• 5 citations
Publisher

Abstract

There are many systems of physical interest for which a set of rate equations for level populations can provide insight. If the system has two (or more) different mechanisms for effecting transition between levels, total rates of transfer are usually taken as the sum of rates that the individual mechanisms would cause acting alone. Using the example of a hydrogen atom subjected to (ionic and electronic) collisions and external radiation, it is shown that when these individual mechanisms overlap, the total transfer rates must be modified to account for correlations between collisional and radiative processes. For a broad-band radiation field the modified rates have a simple physical interpretation. In the case of a narrow-band field the overlapping events may cause new coherences to appear and interpretation of the modified 'rates' is more complicated.

DOI
10.1088/0022-3700/17/21/029

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There are many systems of physical interest for which a set of rate equations for level populations can provide insight. If the system has two (or more) different mechanisms for effecting transition between levels, total rates of transfer are usually taken as the sum of rates that the individual mechanisms would cause acting alone. Using the example of a hydrogen atom subjected to (ionic and electronic) collisions and external radiation, it is shown that when these individual mechanisms overlap, the total transfer rates must be modified to account for correlations between collisional and radiative processes. For a broad-band radiation field the modified rates have a simple physical interpretation. In the case of a narrow-band field the overlapping events may cause new coherences to appear and interpretation of the modified 'rates' is more complicated.
Perturbation theory for the interaction of light with a binary collisional system is applied to the calculation of line profiles, spontaneous emission rates and absorption or stimulated emission coefficients. For the usual radiation fields are perturber densities, this approximation is valid at all detunings Delta omega except the very narrow resonance region. Appropriate expansions of the total binary system wavefunctions give the emission and absorption profiles or rates, for atom-electron systems as well as for atom-atom systems, from the impact regime at small Delta omega up to the Born-Oppenheimer regime at very large Delta omega in collision-induced radiative transitions, the transitions being asymptotically allowed or forbidden. Emphasis is given to the proper averages and sums over initial and final states of the binary system which give rise to non-adiabatic effects. One shows that it is of crucial importance to include all dynamical effects during one collisional event before undertaking any calculation at small Delta omega where simultaneous collisions may occur.
As a first step in deriving accurate, calculable expressions for collisional redistribution of radiation, valid outside the impact regime, we present a method for calculating an accurate binary-collision density operator for an atom in the presence of a driving field and perturbers. We use projection-operator techniques to establish first the precise validity of the Markoff approximation for radiative relaxation. Using the same techniques, we show how the collisional-relaxation problem may be analyzed in a practical manner outside the Markoff approximation. The effect of correlations between radiative and collisional events is included in a consistent way, and the physical implications of these correlation effects are demonstrated for a simple example, the two-level atom with nondegenerate levels.
Abstract To describe the interaction of a two-level atom with radiation we shall use a semiclassical treatment, i.e. the radiation is treated as a classical electric field but we use quantum mechanics to treat the atom. From the theory of the interaction with radiation, we will be able to find the conditions for which the equations reduce to a set of rate equations that describe the populations of the atomic energy levels (with a steady-state solution). In particular, for an atom illuminated by broad band radiation, this approach allows us to make a connection with Einstein’s treatment of radiation that was presented in Chapter 1; we shall find the Einstein B coefficient in terms of the matrix element for the transition.
A quantum kinetic equation is obtained which includes scattering events beyond the binary collision process of the Boltzmann equation. The time rate of change of the Wigner function due to collisions is given as the difference of two scattering rates analogous to the collisional side of the Boltzmann equation. These total rates are given in terms of partial rates which are identified with probability amplitudes for scattering events in the medium. Thus, the method is inherently free of the divergence difficulties of theories which calculate the total rates in terms of scattering events in vacuum. Special attention is devoted to the question in what sense does some form of Fermi's golden rule apply to the calculation of the partial rates for a given event. It is shown that certain modifications to this picture are necessary, but the changes may be understood from a simple physical point of view.
We apply the analysis of paper I to degenerate atomic systems and calculate the effect of correlations between an atom (the absorber) and perturbers in the binary-collision approximation. The result of our calculation is a generalized absorption profile that specifies the final state of the atom an absorption event. This profile is then related to the total intensities of Rayleigh scattering and fluorescence (redistributed radiation) from the atom. The profile is expressed in terms of collision operators that depend on the index of the multipole of the atom that is created. Some of the operators represent correlations between radiative and collisional events and cannot, except in certain limits, be expressed in terms of the usual line-broadening operators. This, of course, implies that one needs extra dynamical information, on top of that obtainable from ordinary absorption experiments, to be able to describe redistribution. Conversely, redistribution offers a probe of collisional dynamics, not given by absorption experiments, via the polarization of fluorescent light. We give practical expressions for the calculation of these operators and discuss their implications.
The presence of coherent radiation fields can modify various atomic, ionic, and molecular processes. We examine cases where the changes can be effected without actual absorption or emission of photons. The radiation fields cause optical Stark shifts of the energies and dress the nonadiabatic couplings between states. Rates of processes such as dissociation, predissociation, atomic and molecular collisions can be radiatively controlled. Formation of new avoided crossings due to radiative interaction gives rise to new phenomena. We give both exact numerical results from which the influences of the radiation field on arbitrary configurations can be calculated, and simple but accurate analytic results, from which these effects can be conveniently assessed. It is shown that the inelastic transition probability is decreased at a true crossing and is increased at an avoided crossing as a result of the presence of the nonresonant radiation field. Furthermore, for two parallel levels with constant nonadiabatic coupling, it is shown that the amplitudes of the nonadiabatic transition are invariant, but its flopping frequency is reduced by the field. Depending on final measurements with the field on or off, general behaviors of the inelastic transition probability as a function of the field parameters and the charge-system parameters are predicted and the possibility of inversion (from a field-free value of less than \textonehalf{} to a value greater than \textonehalf{}) is demonstrated. Experimental investigation with iodine molecules is suggested. Comparison of our nonperturbative results with those of stationary perturbation theory shows that the latter is inadequate in an important parameter region where the field modification of processes is greater than 1%.