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Physical Review

Stark-Induced Temporal Intensity Variations in Spectral Lines

Roald K. Wangsness

Published Sep 9, 1966• 30 citations
Publisher

Abstract

The effect of externally produced Stark splitting upon fine-structure-level probabilities is investigated in order to describe periodic intensity variations previously found in hydrogen lines. Coupled differential equations for the probability amplitudes of mixed levels are solved exactly in terms of initial amplitudes; effects due to spontaneous transition probabilities and Lamb shift are included. In general, the time-dependent line intensity is a combination of exponential decays and sinusoidal oscillations. The frequencies of oscillations are discussed in detail, and the effects of different initial conditions upon the intensities are considered for several special cases.

DOI
10.1103/physrev.149.60

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The effect of externally produced Stark splitting upon fine-structure-level probabilities is investigated in order to describe periodic intensity variations previously found in hydrogen lines. Coupled differential equations for the probability amplitudes of mixed levels are solved exactly in terms of initial amplitudes; effects due to spontaneous transition probabilities and Lamb shift are included. In general, the time-dependent line intensity is a combination of exponential decays and sinusoidal oscillations. The frequencies of oscillations are discussed in detail, and the effects of different initial conditions upon the intensities are considered for several special cases.
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.
Protons with velocities ranging from 3.2 \ifmmode\times\else\texttimes\fi{} 108{10}^{8}108 to 6.3 \ifmmode\times\else\texttimes\fi{} 108{10}^{8}108 cm/sec were passed through a carbon foil producing excited hydrogen atoms which decayed in flight after emergence. With the foil in magnetic fields from 2 to 25 G, the lines ${\mathrm{H}}_{\ensuremath{\alpha}}$ through ${\mathrm{H}}_{\ensuremath{\epsilon}}$ showed periodic intensity fluctuations due to Stark mixings of fine-structure levels caused by motional electric fields (6-158 V/cm). An extension of the Bethe-Lamb theory of Stark quenching provides good agreement between a small number of theoretically dominant frequencies and the measured fluctuation frequencies. Apparent discrepancies between theory and experiment noted by other experimenters are shown to be removed by our calculations except in one measurement on ${\mathrm{Ly}}_{\ensuremath{\alpha}}$. The necessity of using double foils postulated in the previous work is discussed. Our analysis suggests the preferential alignment of the final-state angular momenta perpendicular to the beam direction. Reasons for this alignment are given.
The strong field-induced quantum beats observed in beam-foil studies of Ly- alpha radiation are obtained in a conventional metastable-hydrogen quenching experiment. The phase relation between the Stark shifted 2s 1/2 -2p 1/2 Lamb-shift oscillations and the much more rapid 2s 1 -2p 3/2 fine-structure oscillations depends on the detailed way in which the quenching field is switched on. Apart from a phaseshift, the results agree with a non-perturbative theoretical calculation which assumes that the field is applied suddenly. Various frequency components of the time-dependent radiation intensity are identified with specific hyperfine transitions or groups of transitions. No adjustable parameters are used for the initial state amplitudes.
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.
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.