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Physical Review

Classical-Path Calculation of the S Matrices for Stark Broadening of the Lyman- α Line

Michael E. Bacon, K. Y. Shen, J. Cooper

Published Dec 5, 1969• 28 citations
Publisher

Abstract

The $S$ matrices for Stark broadening of the $n=1$ and $n=2$ 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 $S$ 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.

DOI
10.1103/physrev.188.50

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Journal of Physics B Atomic Molecular and Optical Physics•Dec 4, 2000•6 citations
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 \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ 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-\ensuremathα\ensuremath{\alpha}\ensuremathα line profile is considered.
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 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 intensity distribution in the core region of the Lyman-$\ensuremath{\alpha}$ line of hydrogen perturbed by neutral hydrogen atoms in the presence of an external cone-rotating electric field has been evaluated using an impact approximation. It was found that the line shape in this region can be described as the sum of four Lorentzian profiles representing contributions coming from transitions between various Stark and Berry sublevels of the resonance and ground states of the hydrogen atom. The effects due to Berry's phase are shown to produce a Lorentzian-shaped feature on each side of the Lyman-$\ensuremath{\alpha}$ line, with maxima located at frequencies linearly dependent on the angular frequency with which the electric field rotates about the laboratory fixed zzz axis, as well as on the cosine of the polar angle that the field makes with respect to this axis. The remaining two Lorentzian profiles are due to the Stark effect, and their maxima are located on both sides of the Lyman-$\ensuremath{\alpha}$ line at frequencies corresponding to the Stark shift.
A general method is presented for calculating the higher-order terms of series in powers of the DC field for the hydrogen Stark-state wavefunctions, dipole transition matrix elements and corresponding line intensities. Analytical expressions are derived for the second-order Stark effect on the matrix elements and intensities of the Stark lines. Simple formulae for transitions to states with zero parabolic quantum numbers are given. The numerical data for the Lyman and Balmer series and for some transitions between Rydberg levels are analysed in detail.