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Physical Review A

Shift and width of the Balmer-α and Lyman-β lines of neutral hydrogen due to electron collisions

J. Callaway, K. Unnikrishnan

Published Sep 1, 1991• 6 citations
Publisher

Abstract

The contribution from electron scattering to the width and shift of the ${\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}$ and ${\mathrm{Ly}}_{\mathrm{\ensuremath{\beta}}}$ lines of neutral hydrogen is evaluated. Elements of the S matrix for electron-atom scattering calculations are obtained from close-coupling calculations including pseudostates, either of the variational or the optical potential type. The convergence of sums over angular-momentum variables is discussed. Results are compared with some other calculations and with experiment where possible.

DOI
10.1103/physreva.44.3001

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The contribution from electron scattering to the width and shift of the H\ensuremathα{\mathrm{H}}_{\mathrm{\ensuremath{\alpha}}}H\ensuremathα​ and Ly\ensuremathβ{\mathrm{Ly}}_{\mathrm{\ensuremath{\beta}}}Ly\ensuremathβ​ lines of neutral hydrogen is evaluated. Elements of the S matrix for electron-atom scattering calculations are obtained from close-coupling calculations including pseudostates, either of the variational or the optical potential type. The convergence of sums over angular-momentum variables is discussed. Results are compared with some other calculations and with experiment where possible.
In this paper general expressions for the width and shift of atomic lines produced by fast electron collisions are obtained. We use the recent quantum-mechanical theories developed to explain the pressure line shape in the impact approximation. Only isolated and Lorentzian lines are analyzed. The calculations are performed by expanding the scattering matrix up to the second Born approximation. The contributions of both elastic and inelastic virtual collisions are calculated. Degenerate and nondegenerate atomic states are considered.
We calculate the cross section for the resonance scattering of Lyman-$\ensuremath{\alpha}$ radiation by spinless nonrelativistic hydrogen atoms in the ground state using a two-level model. A generalization of Dirac's resonance scattering theory is used together with the exact matrix elements for the electromagnetic interaction. In contrast to the usual treatments in which only the dipole approximation for the matrix elements are taken, the shift in position of the resonance is finite and has a value of 75% of the Lamb shift of the n=1n=1n=1 state. Whether this latter fact is significant in renormalization calculations is left open.
General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.
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.
This paper is a theoretical discussion of a "coherence effect" in the excitation of hydrogen atoms to the n=3n=3n=3 state by electron bombardment. By coherence is meant that the density matrix describing the atom after the collision has off-diagonal matrix elements in the field-free basis. The differential cross section for producing Balmer-$\ensuremath{\alpha}$ photons at 90\ifmmode^\circ\else\textdegree\fi{} to the electron beam is calculated as a function of an applied longitudinal electric field. The off-diagonal elements show up as a dependence of the observed light on the directions of the electric field. Theoretical results based on the Born approximation are compared with an experiment at 200 and 500 eV. It is pointed out that this method for measuring relative phases of scattering amplitudes can be generally applied and is not restricted to n=3n=3n=3 excitation of hydrogen.