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Physical Review A

Unified Theory of Spectral Line Broadening in Gases

Robert P. Futrelle

Published May 1, 1972• 40 citations
Publisher

Abstract

A quantum-mechanical unified theory of foreign-gas pressure broadening of atomic lines which includes the impact and statistical theories and duration-of-collision effects is developed. The model consists of an absorber atom with two internal states $u$ (upper) and $l$ (lower) interacting with a structureless perturber gas via two spherical pairwise additive potentials ${V}_{u}(\mathcal{r})$ and ${V}_{l}(\mathcal{r})$, e.g., a simple model of an alkali-atom line perturbed by a rare gas. The dipolemoment correlation function is analyzed in the time domain using Liouville (tetradic) techniques. The correlation function reduces exactly to a two-body form $\ensuremath{\varphi}(t)$ containing the proper thermal weight factors. Detailed balance is obeyed and no $|\frac{\ensuremath{\hbar}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}{\mathrm{kT}}|<1$ restriction is required. The impact theory follows from a long-time analysis of $\ensuremath{\varphi}(t)$. Sum rules or spectral moments are used to study short-time behavior revealing conventional statistical-theory effects (including satellite bands) and the important duration-of-collision effects which link the statistical and impact regimes. The sum rules are expressed as quadratures involving potential functions and the quantum-mechanical radial distribution function. Unified methods for calculating the total line shapes are suggested: (1) Pad\'e approximants (ratios of polynomials) can be used to interpolate from the known short-time behavior to the known long-time beahvior; this requires the least computational effort; (2) exact results for this model may be obtained numerically from the overlap integrals of radial wave functions (Franck-Condon factors) for the two potentials ${V}_{u}$ and ${V}_{l}$; (3) classical phase (trajectory) expressions are given which are reasonable approximations in both the impact and statistical limits. A number of related topics are discussed including diabatic effects, collision-induced absorption, non-Lorentzian wings of molecular vibration-rotation bands, and the Condon approximation for the dipole-moment transition strength vs $\mathcal{r}$. The use of line shapes to probe atomic interactions, especially in excited states, is emphasized and the suggestion made that experiments be done over as wide a temperature range as possible with the measured line shapes in computer-readable form. The entire line-shape problem is described as the study of the spectrum of an atom-perturber "quasimolecule" (with primarily unbound states).

DOI
10.1103/physreva.5.2162

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Canadian Journal of Physics•Sep 15, 1974•28 citations
A quantum-mechanical unified theory of foreign-gas pressure broadening of atomic lines which includes the impact and statistical theories and duration-of-collision effects is developed. The model consists of an absorber atom with two internal states uuu (upper) and lll (lower) interacting with a structureless perturber gas via two spherical pairwise additive potentials Vu(r){V}_{u}(\mathcal{r})Vu​(r) and Vl(r){V}_{l}(\mathcal{r})Vl​(r), e.g., a simple model of an alkali-atom line perturbed by a rare gas. The dipolemoment correlation function is analyzed in the time domain using Liouville (tetradic) techniques. The correlation function reduces exactly to a two-body form \ensuremathφ(t)\ensuremath{\varphi}(t)\ensuremathφ(t) containing the proper thermal weight factors. Detailed balance is obeyed and no ∣\ensuremathℏ(\ensuremathω\ensuremath−\ensuremathω0)kT∣<1|\frac{\ensuremath{\hbar}(\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{0})}{\mathrm{kT}}|<1∣kT\ensuremathℏ(\ensuremathω\ensuremath−\ensuremathω0​)​∣<1 restriction is required. The impact theory follows from a long-time analysis of \ensuremathφ(t)\ensuremath{\varphi}(t)\ensuremathφ(t). Sum rules or spectral moments are used to study short-time behavior revealing conventional statistical-theory effects (including satellite bands) and the important duration-of-collision effects which link the statistical and impact regimes. The sum rules are expressed as quadratures involving potential functions and the quantum-mechanical radial distribution function. Unified methods for calculating the total line shapes are suggested: (1) Pad'e approximants (ratios of polynomials) can be used to interpolate from the known short-time behavior to the known long-time beahvior; this requires the least computational effort; (2) exact results for this model may be obtained numerically from the overlap integrals of radial wave functions (Franck-Condon factors) for the two potentials Vu{V}_{u}Vu​ and Vl{V}_{l}Vl​; (3) classical phase (trajectory) expressions are given which are reasonable approximations in both the impact and statistical limits. A number of related topics are discussed including diabatic effects, collision-induced absorption, non-Lorentzian wings of molecular vibration-rotation bands, and the Condon approximation for the dipole-moment transition strength vs r\mathcal{r}r. The use of line shapes to probe atomic interactions, especially in excited states, is emphasized and the suggestion made that experiments be done over as wide a temperature range as possible with the measured line shapes in computer-readable form. The entire line-shape problem is described as the study of the spectrum of an atom-perturber "quasimolecule" (with primarily unbound states).
The work of two previous papers is extended and a theory of pressure broadening is developed which treats the perturbers quantum mechanically and allows for inelastic collisions, degeneracy, and overlapping lines. The impact approximation is used. It consists in assuming that it takes, on the average, many collisions to produce an appreciable disturbance in the wave function of the atom, and it results in an isolated line having a Lorentz shape. Validity criteria are given. When the approximation is valid, it is allowable to replace the exact, fluctuating interaction of the perturbers with the atom by a constant effective interaction. The effective interaction is expressed in terms of the one-perturber quantum mechanical transition amplitudes on and near the energy shell and its close relationship to the scattering matrix is stressed. The calculation of the line shape in terms of the effective interaction is the same as when the perturbers move on classical paths. Results are written explicitly for isolated lines. If the interaction of the perturbers with the final state can be neglected, the shift and width are proportional to the real and imaginary part of the forward elastic scattering amplitude, respectively. By the optical theorem, the width can also be written in terms of the total cross section. When the interaction in the final state cannot be neglected, the shift and width are still given in terms of the elastic scattering amplitudes, in a slightly more complicated fashion. Finally, rules are given for taking into account rotational degeneracy of the radiating states.
Oscillatory or bandlike structure is predicted to occur in the wings of pressure-broadened atomic lines when the interaction potential between the emitting atom M* and the perturbing atom A is deeply attractive compared to kT and the interaction in the final state is repulsive. The oscillations in intensity arise from quantum effects and are associated with the nodal structure of the continuum wavefunction which describes the dissociated state of the diatomic molecule (A + M*). The number of maxima in the line shape is a measure of the number of bound states which can be supported in the potential while the amplitude and resolution of the oscillations depend on the long-range behavior of the potential. The bandlike structure, which may extend over thousands of cm−1 on the long-wavelength side of the atomic emission line, is resolvable as long as kT < ∼ 1 / 10 the vibrational quantum ℏωe of the AM* molecule. This is a generalization of similar bandlike structure which has been calculated and observed to occur in the collision-induced radiation of helium metastable atoms and which is merely a special case of pressure broadening involving forbidden lines. Specific calculations are presented for the broadening of the hydrogen Lyman-α line by H(1s), and the visible absorption spectra of H(1s) in the presence of free protons. The quantum theory of line shape which is employed here was first developed by Jablonski but then neglected due to difficulties involved in evaluating essentially the band strength for continuum–continuum transitions of diatomic molecules. Jablonski introduced the WKB and other approximations into his theory which reduced it to the classical statistical theory. These approximations are compared to exact quantum-mechanical calculations. The WKB approximation is shown to be quite valid in the special case of attractive–replusive transitions, but Jablonski then improperly averaged over those very quantum oscillations of interest here. This structure offers a unique and extremely sensitive method of studying interaction potentials involving excited states of atoms.
A new approach is suggested for dealing with the many-body aspects of resonance broadening of spectral lines in gases. A Liouville-space formalism is used in which the absorption by NNN identical molecules undergoing resonance-exchange collisions is described by an NNN-dimensional projection operator in Liouville space. In the impact approximation, under conditions in which the Doppler shift of resonance frequencies can be neglected, it is shown that the effect of resonance collisions is to modify the linewidth expression obtained by ordinary foreign-gas-type theories. The shape of isolated lines remains, however, Lorentzian. Only a combined effect of Doppler and collision broadening causes appreciable deviations from the Lorentzian shape near the line center, but these deviations are not peculiar to resonance broadening.
In the limit of a dipole-dipole interaction between neutral atoms and the binary-collision approximation, we present a theory of pressure broadening in which the multiplet structure of the atomic levels is considered. The appropriate Schr\"odinger equations for the collision problem are numerically integrated to obtain solutions for both resonant and foreign gas broadening. The presence of radiation fields is neglected in the first half of the paper, where the collisional time rate of change of the density matrix is found. The pertinent collisional decay parameters are extracted from these equations. The ratios of the parameters of the average magnetic dipole to electric quadrupole moment obtained were 1.21\ifmmode\pm\else\textpm\fi{}0.02 for the relaxation of the entire ensemble in resonant broadening, 0.966\ifmmode\pm\else\textpm\fi{}0.012 for the relaxation of the initially excited atoms in resonant broadening, and 1.12\ifmmode\pm\else\textpm\fi{}0.02 for the relaxation of the initially excited atoms in foreign gas broadening. In the second half of the paper, the collisional broadening results are incorporated in a systematic evaluation of (i) spectral profiels (effects arising from the velocity and recoil of the emitting atom are neglected), (ii) Hanle-effect line shapes, and (iii) laser phenomena. For the case of resonant broadening, the radiation trapping process is reviewed and its influence on line shapes discussed. Finally, a detailed comparison of the theory with experimental findings as well as with previous theoretical results is made. We conclude that the dipole-dipole interaction is sufficient to explain most cases of resonant and some cases of foreign gas broadening.
A general method for analysing line broadening caused by Doppler, phaseshift, inelastic and velocity-changing collisional effects is proposed for transitions at optical wavelengths. In a unified approach Poisson statistics are used to describe collision probabilities randomly distributed in a time interval which can terminate the optical coherence in a combination of the above effects; markoffian independence is assumed for each collision. Then the dipole moment correlation function is obtained in a form which is virtually exact in accounting for all effects of isotropic forces, and which is correct to second-order terms in the anisotropic interaction potentials. Phenomena as diverse as standard optical pressure broadening, broadening due to velocity-changing collisions in nonlinear spectroscopies, and Dicke narrowing of the Doppler contour are described. The implications of the theory are discussed in detail and applied to the case of the NaD lines perturbed by He buffer gas. Numerical calculations reveal that the substantially different lineshift and broadening parameters in nonlinear and in standard optical spectroscopies can be reasonably well understood within the limitations posed by present uncertainties in the Na 3 2 P-He interaction potentials. However, the large differences in shift and width between the D 2 and D 1 lines in saturation spectroscopy cannot be explained within the present theory.
The theory of pressure broadening by foreign atoms is formulated within the framework of an adiabatic representation (which we distinguish from the adiabatic approximation). This allows us to treat all the electrons in the gas as indistinguishable, and thus include the possibility of electron exchange between atoms; this effect, which is responsible for much of the interatomic interaction, is neglected in all previous theories, apart from the adiabatic and 'nearest neighbor' theories which are of limited applicability. By means of projection operators, we define a spectral matrix, whose dimension is equal to the number of distinct frequencies characteristic of the isolated radiator. The spectrum is equal to the sum of all the elements of the spectral matrix; the diagonal elements correspond to the different lines of the spectrum, and the off diagonal elements represent quantum interference between overlapping lines. The Fourier transform of the spectral matrix is the correlation matrix, whose elements time correlate components of the dipole moment operator responsible for different lines of the spectrum. The time evolution of the zero perturber correlation matrix is given by eiΩτ, where Ω is a diagonal matrix whose elements are the different frequencies characteristic of the radiator. The perturbing gas causes the 'reduced Liouvillian' Ω to acquire a time dependent or frequency dependent non-Hermitian nondiagonal part. To obtain low density approximations, we treat that non-Hermitian 'perturbation' to first order in the gas density.