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Journal of Mathematical Physics

The effect of configuration mixing on the first-order moments of polarized hydrogen lines

R. Casini

Published Sep 1, 1998• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.532512

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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 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.
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.
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.
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
We present experimental and theoretical investigations of the He spectral series 2S1−nQ1 (n=3÷9, Q=S,P,D,…,n−1) and 2P1−nQ1 (n=3÷9, Q=S,P,D,…,n−1) in electric fields up to 1635 kV/cm. Apart from the allowed transitions with |ΔL|=1, the transitions with |ΔL|=0,2,3,…—without field strictly forbidden—were observed. Several He patterns become similar to hydrogen patterns, which means they are nearly symmetric and show in higher fields nearly linear Stark shifts. The applied fields are high enough that patterns belonging to neighboring principal quantum numbers (for n>6) begin to overlap, which leads to interesting level-anticrossing effects. The experimental results are compared with numerical calculations, taking into account mixing between states of different principal quantum numbers and between singlet and triplet states. The agreement between experimental and theoretical results is quite good.
The prediction of the energy level shifts and intensity transfer in atomic hydrogen and helium spectra caused by an external electric field was one of the first applications of wave mechanics. Early measurements relied on electrical discharge sources for excited atoms; since discharge processes will not produce excited states with n > 7 in observable densities, very large (10s to 100s kV cm) electric fields were required to resolve spectral modifications. The development of high peak power lasers enabled excitation to high n (Rydberg) states, increasing field sensitivity down to millivolts/cm and allowing comparison of experiment and theory in well-defined conditions. Field modified selection rules can be used to measure the field direction as well as magnitude. Analysis of line shape simulations of Stark modified Rydberg spectra reveals pitfalls in commonly used field measurement techniques. An ambiguity in defining series termination results in a factor of 2 discrepancy in field measurements based on Inglis Teller line merging. Field-induced line broadening (unresolved manifold splitting) in triplet and singlet helium are shown to differ significantly, with triplet broadening nonlinear with field.