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Journal of Physics B Atomic and Molecular Physics

A simple quantum theory of Stark broadening of hydrogen lines including short-range interactions. Application to Ly-α

N. Feautrier, N. Tran Minh

Published Dec 1, 1977• 10 citations
Publisher

Abstract

Starting from the exact unified expression of the line profile, a quantum theory is developed making exact allowance for the long-range dipole interaction. The knowledge of the positions of the bound states below the n=2 threshold enables approximation of the contribution of short-range interactions. Good agreement with exact quantum calculations is obtained.

DOI
10.1088/0022-3700/10/17/017

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Physical Review•Apr 15, 1939•20 citations
Starting from the exact unified expression of the line profile, a quantum theory is developed making exact allowance for the long-range dipole interaction. The knowledge of the positions of the bound states below the n=2 threshold enables approximation of the contribution of short-range interactions. Good agreement with exact quantum calculations is obtained.
The impact-broadening operator $\ensuremath{\Phi}$ for the hydrogen line ${\mathrm{H}}_{\ensuremath{\alpha}}$ is calculated by solving numerically for the SSS matrices using the straight-line classical-path approximation. All contributing multipoles in the perturber- (free-electron-) atom interaction [V(t)V(t)V(t)] are included; time ordering of the operators in the SSS matrices is retained. The matrix elements are compared with previous approximations based on the same model; the effects on the ${\mathrm{H}}_{\ensuremath{\alpha}}$ line profile are considered.
The SSS matrices for Stark broadening of the n=1n=1n=1 and n=2n=2n=2 levels of hydrogen based on the straight-line classical-path model are calculated numerically. The interaction terms (excluding exchange) are included to all orders, and time ordering of the operators in the SSS matrix is retained. The resultant values for the impact-broadening operator $\ensuremath{\Phi}$ are presented. These results are compared with previous approximations based on the same model. The effect of the present calculations on the center and near wings of the Lyman-$\ensuremath{\alpha}$ line profile is considered.
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.