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Physical Review A

Classical-Path Calculation of the Impact-Broadening Operator for the Stark Broadening of Hα

Michael E. Bacon

Published Feb 1, 1971• 42 citations
Publisher

Abstract

The impact-broadening operator $\ensuremath{\Phi}$ for the hydrogen line ${\mathrm{H}}_{\ensuremath{\alpha}}$ is calculated by solving numerically for the $S$ matrices using the straight-line classical-path approximation. All contributing multipoles in the perturber- (free-electron-) atom interaction [$V(t)$] are included; time ordering of the operators in the $S$ 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.

DOI
10.1103/physreva.3.825

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The impact-broadening operator \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ for the hydrogen line H\ensuremathα{\mathrm{H}}_{\ensuremath{\alpha}}H\ensuremathα​ 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 H\ensuremathα{\mathrm{H}}_{\ensuremath{\alpha}}H\ensuremathα​ 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.
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 broadening of spectral lines due to dipole-dipole interactions with ground-state atoms of the same element is calculated with the impact and classical path approximations. Summation over all contributing intermediate states accounts for the multiplicity of the levels involved and results in a width proportional to the geometrical mean of absorption and emission oscillator strengths. Higher multipole interactions are shown to be almost always negligible, and for Lyman-$\ensuremath{\alpha}$ the iterated solution of the time-dependent Schr\"odinger equation describing the collisions is extended to fourth order. Estimated errors of its calculated width and of the widths of other lines are about 5% in the validity regime of the impact approximation, apart from any uncertainties in oscillator strengths and statistical weight factors.
The classical-path approximation reduces the problem of pressure broadening of spectral lines to the evaluation of matrix elements of the scattering operator. If the intermolecular potential is long range and the interaction volume is large, the broadening is caused by distant or weak collisions. In this case, the scattering operator can be approximated by a second-order expansion, and the perturber trajectories can be taken to be straight paths. For neutral atoms or molecules, the intermolecular potential is short range and broadening arises from close or "strong" collisions. In this paper it is shown how classical trajectories, determined by a "scalar interaction" (i.e., one that does not depend upon the state of the radiator), can be used to expand the scattering operator in the sum of operators characteristic of the radiator's internal states.