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Journal of Physics B Atomic and Molecular Physics

Quantum theory of Stark broadening by electrons

N. Tran-Minh, Henri Van Regemorter

Published May 1, 1972• 22 citations
Publisher

Abstract

The one electron approximation is used to determine the exact form of the wings of a line broadened by electron-atom collisions. The validity of this model is examined. The profile being proportional to the square of the scalar product of scattered waves, the impact approximation is obtained by replacing the exact form of the scattered waves by their asymptotic form. In this case of hydrogen the quasistatic approximation is obtained by using the JWKB method together with the usual assumptions.

DOI
10.1088/0022-3700/5/5/010

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Physical Review•Jul 5, 1967•37 citations
The one electron approximation is used to determine the exact form of the wings of a line broadened by electron-atom collisions. The validity of this model is examined. The profile being proportional to the square of the scalar product of scattered waves, the impact approximation is obtained by replacing the exact form of the scattered waves by their asymptotic form. In this case of hydrogen the quasistatic approximation is obtained by using the JWKB method together with the usual assumptions.
A simple and tractable quantum expression of the line wings is given when the upper and lower levels of the spectral lines are both perturbed. This quantum profile converges towards the impact regime at small distances from the line centre and towards the quasistatic limit when the usual semiclassical approximations are made. The correspondence between quantum and semiclassical expressions is examined, and the passage from the former to the latter clarifies the implicit assumptions which are always made in any unified semiclassical approach.
The profile of a hydrogen emission line is investigated on the assumption that the hydrogen atom is perturbed by passing ions. Provided that the ions do not approach to within ten times the average atomic radius the electric field of the ion may be assumed homogeneous and the matrix elements calculated for the usual Stark effect may be used. For a collision at low velocity or with close approach, the adiabatic approximation is valid, and the profile depends solely on the total phase shift produced by the collision. The Fourier analysis of the wave train may be carried through exactly for certain values of this phase shift. Nonadiabatic transitions are examined for the lowest lines, and an exact solution given for this case also. For higher lines a modified Born approximation is used. The Weisskopf collision broadening formula is shown to hold only in certain cases, and the dispersion curve smearing function for impact broadening is shown to lead in some instances to incorrect results.
The work of two previous papers is extended and a theory of pressure broadening is developed which treats the perturbers quantum mechanically and allows for inelastic collisions, degeneracy, and overlapping lines. The impact approximation is used. It consists in assuming that it takes, on the average, many collisions to produce an appreciable disturbance in the wave function of the atom, and it results in an isolated line having a Lorentz shape. Validity criteria are given. When the approximation is valid, it is allowable to replace the exact, fluctuating interaction of the perturbers with the atom by a constant effective interaction. The effective interaction is expressed in terms of the one-perturber quantum mechanical transition amplitudes on and near the energy shell and its close relationship to the scattering matrix is stressed. The calculation of the line shape in terms of the effective interaction is the same as when the perturbers move on classical paths. Results are written explicitly for isolated lines. If the interaction of the perturbers with the final state can be neglected, the shift and width are proportional to the real and imaginary part of the forward elastic scattering amplitude, respectively. By the optical theorem, the width can also be written in terms of the total cross section. When the interaction in the final state cannot be neglected, the shift and width are still given in terms of the elastic scattering amplitudes, in a slightly more complicated fashion. Finally, rules are given for taking into account rotational degeneracy of the radiating states.
Quantum mechanics is used to treat the motion of the perturbers broadening a line. Several simplifying assumptions are made, such as the Born-Oppenheimer approximation, and the assumption of no degeneracy. These assumptions can be removed. Also made is the "impact approximation," which is essential for the validity of the results and cannot be removed. With it, the line has a Lorentz shape, and its width and shift are expressed in terms of the two scattering amplitudes of the perturbers by the atom in its upper and lower state. The case where the perturbers do not interact appreciably with the lower state is particularly simple. Then, the width and shift are proportional to the imaginary part and the real part of the forward scattering amplitude, respectively. A quantity called the "collision volume" is defined. It is shown that the impact approximation is valid only if the collision volume is much smaller than the volume per perturber. There is a second validity condition, which has no classical analog.
A general theory of line broadening is used to study the effects of collisions on absorption line shapes when self-broadening is important. Special attention is focused on the role of processes in which an excited atom transfers its electronic excitation to an unexcited atom; it is this process which distinguishes self-broadening from foreign-gas broadening. Several forms for the absorption rate are presented corresponding to particular assumptions about the amplitudes associated with this phenomenon. For example, one approximation leads to a generalized Lorentz-Lorenz law in which the one-atom polarizability is replaced by one that accounts for line broadening. Finally, the importance of many-body effects in the calculation of the linewidth is briefly discussed to determine when simple impact theories are applicable.