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Physical Review

Stark-Effect Broadening of Hydrogen Lines

Lyman Spitzer

Published Apr 15, 1939• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.55.699

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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.

Physical Review•Dec 5, 1969•28 citations
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 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.
We extend previous work on the algebraic first-order moments of polarization profiles of hydrogen lines in the presence of external fields, to include the effect of configuration mixing induced by the applied fields. An algebraic formula for hydrogen line shifts due to the quadratic Stark effect is derived, under the assumption that the applied electric field is weak enough that line quenching induced by field ionization can be neglected. The derivation of such a formula is made possible by an aimed use of standard techniques of projection-operator algebra and of the fundamental implications of the spectral theorem. The results of this paper should help in decreasing the amount of numerical work needed in the calculation of line shifts of hydrogen lines induced by the presence of electric fields, as the preliminary numerical calculation of the Hamiltonian eigenvalues and eigenvectors is avoided, thus making calculations tractable for lines of higher hydrogen series than considered so far, e.g., in applications of pressure-broadening theory.
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.
Wave functions of a hydrogen atom perturbed by an electric field are calculated under the assumption of a first-order Stark effect. It is assumed that the strength of the perturbation field is constant and its direction rotates around a fixed axis with a constant angular velocity. The ground state of the atom is not affected by the perturbation field. The energy level of the first excited state of the perturbed atom splits into nine sub-levels. The Lymann alpha line emitted from the perturbed atom consists of seven components. In the case where the angular velocity of the field rotation is small, the structure of the line is similar to that of the line emitted from an atom perturbed by a static field. In the case where, the angular velocity is large, it is similar to that of the line emitted from an unperturbed atom.
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.