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Physical Review

General Impact Theory of Pressure Broadening

Michel Baranger

Published Nov 1, 1958• 645 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.112.855

Related papers

Simplified Quantum-Mechanical Theory of Pressure Broadening

Michel Baranger

Quantum mechanics is used to treat the motion of the perturbers broadening a line. Several simplifying assumptions are made, such as the Born-Oppenheimer approximation, and the assumption of no degeneracy. These assumptions can be removed. Also made is the "impact approximation," which is essential for the validity of the results and cannot be removed. With it, the line has a Lorentz shape, and its width and shift are expressed in terms of the two scattering amplitudes of the perturbers by the atom in its upper and lower state. The case where the perturbers do not interact appreciably with the lower state is particularly simple. Then, the width and shift are proportional to the imaginary part and the real part of the forward scattering amplitude, respectively. A quantity called the "collision volume" is defined. It is shown that the impact approximation is valid only if the collision volume is much smaller than the volume per perturber. There is a second validity condition, which has no classical analog.

Physical Review•Jul 15, 1958•584 citations

Broadening and Shift of Atomic Lines Produced by Fast Electron Collisions

José Maria Filardo Bassalo, M. Cattani

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.

Canadian Journal of Physics•Jan 15, 1972•8 citations

Unified Theory of Spectral Line Broadening in Gases

Robert P. Futrelle

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).

Physical Review A•May 1, 1972•40 citations

Overlapping Lines

Peter Joseph Rayer

The theory of Baranger is discussed, relating it to the approach taken by Anderson in the last chapter and that taken by Fano in the one to follow. Baranger is concerned to describe pressure broadening in a band of close, overlapping lines. His original concern was with line broadening by fast-moving electrons in a plasma, which allowed him to use the impact approximation, but not to assume that collisions may be associated with classical paths. For this reason, although matter here is in the form of neutral molecules, the use of Baranger’s theory in its most general form requires that collisions be treated in terms of quantum scattering theory. In forming the correlation function, Baranger sets the algebra in a product space where line vectors take the place of the energy states used by Anderson, and the optical cross-section that governs line broadening is replaced by the matrix of an operator in line space, with line widths and shifts on the diagonal and line coupling parameters for the other elements. In the case of isolated lines, Anderson’s theory may be regained, but the introduction of line space paved the way, later on, for a much more general viewpoint.

Cambridge University Press eBooks•Jun 30, 2020•0 citations

Quantum Oscillations in the Shape of Pressure-Broadened Atomic Lines

Frederick H. Mies

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.

The Journal of Chemical Physics•Jan 1, 1968•68 citations

Impact broadening of Rydberg atomic levels in inelastic collisions with alkali-metal atoms

Владимир Сергеевич Лебедев, V. S. Marchenko

The theory of impact broadening of the isolated spectral lines of the Rydberg atomic series perturbed by neutral particles is developed within the framework of the impulse approximation. The analysis of the broadening process is based on the consideration of the inelastic nl to n' transitions between highly excited states of the Rydberg atom. The role of the potential and resonance scattering of the quasifree electron by a perturbing particle in the impact broadening of the Rydberg nl levels is investigated. It is shown that the general expressions for the broadening cross sections presented in this paper contain the results of asymptotic Alekseev and Sobel'man theory (1965) as special cases. Both elastic and inelastic scattering contributions of the perturber by the parent core of Rydberg atom to the broadening cross sections are also considered within the framework of the 'shake' model. The results obtained are used for a quantitative explanation of experimental data on the broadening of Rydberg atomic levels perturbed by alkali-metal atoms. The energies and widths of the 3 P resonances on the quasidiscrete levels of the negative K - and Rb - ions are determined.

Journal of Physics B Atomic and Molecular Physics•Nov 28, 1987•24 citations
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.
Quantum mechanics is used to treat the motion of the perturbers broadening a line. Several simplifying assumptions are made, such as the Born-Oppenheimer approximation, and the assumption of no degeneracy. These assumptions can be removed. Also made is the "impact approximation," which is essential for the validity of the results and cannot be removed. With it, the line has a Lorentz shape, and its width and shift are expressed in terms of the two scattering amplitudes of the perturbers by the atom in its upper and lower state. The case where the perturbers do not interact appreciably with the lower state is particularly simple. Then, the width and shift are proportional to the imaginary part and the real part of the forward scattering amplitude, respectively. A quantity called the "collision volume" is defined. It is shown that the impact approximation is valid only if the collision volume is much smaller than the volume per perturber. There is a second validity condition, which has no classical analog.
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.
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 $\ensuremath{\varphi}(t)$ 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 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 theory of Baranger is discussed, relating it to the approach taken by Anderson in the last chapter and that taken by Fano in the one to follow. Baranger is concerned to describe pressure broadening in a band of close, overlapping lines. His original concern was with line broadening by fast-moving electrons in a plasma, which allowed him to use the impact approximation, but not to assume that collisions may be associated with classical paths. For this reason, although matter here is in the form of neutral molecules, the use of Baranger’s theory in its most general form requires that collisions be treated in terms of quantum scattering theory. In forming the correlation function, Baranger sets the algebra in a product space where line vectors take the place of the energy states used by Anderson, and the optical cross-section that governs line broadening is replaced by the matrix of an operator in line space, with line widths and shifts on the diagonal and line coupling parameters for the other elements. In the case of isolated lines, Anderson’s theory may be regained, but the introduction of line space paved the way, later on, for a much more general viewpoint.
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.
The theory of impact broadening of the isolated spectral lines of the Rydberg atomic series perturbed by neutral particles is developed within the framework of the impulse approximation. The analysis of the broadening process is based on the consideration of the inelastic nl to n' transitions between highly excited states of the Rydberg atom. The role of the potential and resonance scattering of the quasifree electron by a perturbing particle in the impact broadening of the Rydberg nl levels is investigated. It is shown that the general expressions for the broadening cross sections presented in this paper contain the results of asymptotic Alekseev and Sobel'man theory (1965) as special cases. Both elastic and inelastic scattering contributions of the perturber by the parent core of Rydberg atom to the broadening cross sections are also considered within the framework of the 'shake' model. The results obtained are used for a quantitative explanation of experimental data on the broadening of Rydberg atomic levels perturbed by alkali-metal atoms. The energies and widths of the 3 P resonances on the quasidiscrete levels of the negative K - and Rb - ions are determined.