Physical Review A

Expansion of the Spectrum in Powers of the Density in the Adiabatic Theory of Pressure Broadening

Antoine Royer

Published Jun 1, 1971• 30 citations

Abstract

Within the framework of the adiabatic theory of pressure broadening, an expansion of the spectrum in powers of the density is obtained by making use of a technique of local averaging in the time domain. The first term of the expansion, ${I}_{c}(\ensuremath{\omega})$, is essentially the impact distribution; the second term is $n{J}_{w}(\ensuremath{\omega})$ folded into ${I}_{c}(\ensuremath{\omega})$, where ${J}_{w}(\ensuremath{\omega})$ is the wing of the one-perturber distribution, that is, the intensity distribution obtained if there is only one perturbing atom interacting with the radiating atom. Higher-order terms are successive convolution powers of $n{J}_{w}(\ensuremath{\omega})$, all folded into ${I}_{c}(\ensuremath{\omega})$. The expansion should be useful mostly for understanding and predicting qualitative features of the spectrum. In particular, the results imply that structure in the wings, such as satellite bands, is broadened and shifted in the same way as the line center under increasing pressure, in agreement with experimental observations.

DOI: 10.1103/physreva.3.2044

Related papers

Cumulant Expansions and Pressure Broadening as an Example of Relaxation

Antoine Royer

Physical Review ANov 1, 197297 citations

Density Expansion of the Memory Operator in Pressure-Broadening Theory

Antoine Royer

Physical Review AMar 1, 197322 citations

Theory of Pressure Broadening in an Adiabatic Representation

Antoine Royer

Canadian Journal of PhysicsSep 15, 197428 citations

General Impact Theory of Pressure Broadening

Michel Baranger

Physical ReviewNov 1, 1958645 citations

Pressure dependence of Na resonance line broadening by Kr and Xe

W.P. West, Alan C. Gallagher

Physical Review AApr 1, 197835 citations

Unified Theory of Spectral Line Broadening in Gases

Robert P. Futrelle

Physical Review AMay 1, 197240 citations

Physical Review A

Expansion of the Spectrum in Powers of the Density in the Adiabatic Theory of Pressure Broadening

Antoine Royer

Published Jun 1, 1971• 30 citations

Publisher

Abstract

{I}_{c}(\ensuremath{\omega})

, is essentially the impact distribution; the second term is

n{J}_{w}(\ensuremath{\omega})

folded into

{I}_{c}(\ensuremath{\omega})

, where

{J}_{w}(\ensuremath{\omega})

is the wing of the one-perturber distribution, that is, the intensity distribution obtained if there is only one perturbing atom interacting with the radiating atom. Higher-order terms are successive convolution powers of

n{J}_{w}(\ensuremath{\omega})

, all folded into

{I}_{c}(\ensuremath{\omega})

. The expansion should be useful mostly for understanding and predicting qualitative features of the spectrum. In particular, the results imply that structure in the wings, such as satellite bands, is broadened and shifted in the same way as the line center under increasing pressure, in agreement with experimental observations.

DOI: 10.1103/physreva.3.2044

Related papers

Cumulant Expansions and Pressure Broadening as an Example of Relaxation

Antoine Royer

Pressure broadening by neutral atoms is treated in a time-dependent formalism making use of generalized cumulants due to Kubo. A thermal-equilibrium initial density matrix is assumed, unlike in previous theories by Baranger and Fano who neglect initial correlations between atoms; it is pointed out that the wings of the spectrum depend in an essential manner on these initial correlations. The treatment centers on a time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\mathrm{x}}}$ operating in the Liouville space of the radiating atom and governing its motion under the influence of the perturbing gas or bath { ${\mathrm{Tr}}_{B}$ is the trace over bath coordinates, $\ensuremath{\rho}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is the density matrix, and ${H}^{\mathrm{x}}$ is the quantum-mechanical Liouvillian: ${H}^{\mathrm{x}}( )=[H, ( )]$ ; $H$ is the total Hamiltonian for the system radiator plus bath}; $U(t)$ is written as $U(t)={T}_{\ensuremath{\rightarrow}}\mathrm{exp}[i\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}L({t}^{\ensuremath{'}})]$, $L(t)={H}_{s}^{\mathrm{x}}+R(t)$ (${T}_{\ensuremath{\rightarrow}}$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation $R(t)$ added to the Liouvillian ${H}_{s}^{\mathrm{x}}$ of the unperturbed radiator. The operator $R(t)$ is expanded in powers of a "reduced" density equal to the perturbing-gas density multiplied by the ratio of the fugacities corresponding to mutually interacting and noninteracting perturbing atoms, respectively; the terms of the expansion are expressed by means of generalized cumulants, and describe interactions of the radiator with clusters of perturbers. By setting $R(t)=\overline{R}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, where $\overline{R}\ensuremath{\equiv}{\mathrm{lim}}_{T\ensuremath{\rightarrow}\ensuremath{\infty}}[(\frac{1}{T})\ensuremath{\int}{0}^{T}\mathrm{dtR}(t)]$, the spectrum is written as the sum of its impact approximation, determined by $\overline{R}$ , plus a correction expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.Pressure broadening by neutral atoms is treated in a time-dependent formalism making use of generalized cumulants due to Kubo. A thermal-equilibrium initial density matrix is assumed, unlike in previous theories by Baranger and Fano who neglect initial correlations between atoms; it is pointed out that the wings of the spectrum depend in an essential manner on these initial correlations. The treatment centers on a time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}\ifmmode\times\else\texttimes\fi{}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\mathrm{x}}}$ operating in the Liouville space of the radiating atom and governing its motion under the influence of the perturbing gas or bath {${\mathrm{Tr}}_{B}$ is the trace over bath coordinates, $\ensuremath{\rho}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is the density matrix, and ${H}^{\mathrm{x}}$ is the quantum-mechanical Liouvillian: ${H}^{\mathrm{x}}( )=[H, ( )]$; $H$ is the total Hamiltonian for the system radiator plus bath}; $U(t)$ is written as $U(t)={T}_{\ensuremath{\rightarrow}}\mathrm{exp}[i\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}L({t}^{\ensuremath{'}})]$, $L(t)={H}_{s}^{\mathrm{x}}+R(t)$ (${T}_{\ensuremath{\rightarrow}}$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation $R(t)$ added to the Liouvillian ${H}_{s}^{\mathrm{x}}$ of the unperturbed radiator. The operator $R(t)$ is expanded in powers of a "reduced" density equal to the perturbing-gas density multiplied by the ratio of the fugacities corresponding to mutually interacting and noninteracting perturbing atoms, respectively; the terms of the expansion are expressed by means of generalized cumulants, and describe interactions of the radiator with clusters of perturbers. By setting $R(t)=\overline{R}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, where $\overline{R}\ensuremath{\equiv}{\mathrm{lim}}_{T\ensuremath{\rightarrow}\ensuremath{\infty}}[(\frac{1}{T})\ensuremath{\int}{0}^{T}\mathrm{dtR}(t)]$, the spectrum is written as the sum of its impact approximation, determined by $\overline{R}$, plus a correction expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.

Physical Review ANov 1, 197297 citations

Density Expansion of the Memory Operator in Pressure-Broadening Theory

Antoine Royer

The motion of a radiating atom immersed in a gas of other atoms, the bath, may be described by means of a nonunitary time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\ifmmode\times\else\texttimes\fi{}}}$ acting in its Liouville space ( ${\mathrm{Tr}}_{B}$ is the trace over bath variables, $\ensuremath{\rho}$ is the density matrix, and ${H}^{\ifmmode\times\else\texttimes\fi{}}$ is the quantum-mechanical Liouvillian). In a previous paper, $U(t)$ was written in the form $U(t)={\mathrm{exp}}_{\ensuremath{\rightarrow}}[i\ensuremath{\int}{0}^{t}{\mathrm{dt}}^{\ensuremath{'}}{L}_{1}({t}^{\ensuremath{'}})]$ (${\mathrm{exp}}_{\ensuremath{\rightarrow}}$ denotes a time-ordered exponential), and the time-dependent effective Liouvillian ${L}_{1}(t)$ was expanded in powers of a "reduced density," or activity. In this paper the Fourier transform $U(\ensuremath{\omega})=\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dt}{e}^{\ensuremath{-}i\ensuremath{\omega}t}U(t)$ is written in the form $iU(\ensuremath{\omega})={[\ensuremath{\omega}\ensuremath{-}{L}_{2}(\ensuremath{\omega})]}^{\ensuremath{-}1}$, and the frequency-dependent effective Liouvillian ${L}_{2}(\ensuremath{\omega})$, or "memory operator," is expanded in powers of the reduced density. It is argued that in treating ${L}_{2}(\ensuremath{\omega})$ to first order in the reduced density, one effectively allows the radiator to interact with only one perturber at a time, thus neglecting all multiple-collision effects. This is in contrast to performing the same-order approximation on ${L}_{1}(t)$ , which is equivalent to treating different perturbers as uncorrelated, but still allows for multiple-collision effects. By adding the terms of higher order in the expansion of ${L}_{2}(\ensuremath{\omega})$, one allows the radiator to interact simultaneously with two, three,... perturbers.The motion of a radiating atom immersed in a gas of other atoms, the bath, may be described by means of a nonunitary time-evolution operator $U(t)={({\mathrm{Tr}}_{B}\ensuremath{\rho})}^{\ensuremath{-}1}{\mathrm{Tr}}_{B}\ensuremath{\rho}{e}^{\mathrm{it}{H}^{\ifmmode\times\else\texttimes\fi{}}}$ acting in its Liouville space (${\mathrm{Tr}}_{B}$ is the trace over bath variables, $\ensuremath{\rho}$ is the density matrix, and ${H}^{\ifmmode\times\else\texttimes\fi{}}$ is the quantum-mechanical Liouvillian). In a previous paper, $U(t)$ was written in the form $U(t)={\mathrm{exp}}_{\ensuremath{\rightarrow}}[i\ensuremath{\int}{0}^{t}{\mathrm{dt}}^{\ensuremath{'}}{L}_{1}({t}^{\ensuremath{'}})]$ (${\mathrm{exp}}_{\ensuremath{\rightarrow}}$ denotes a time-ordered exponential), and the time-dependent effective Liouvillian ${L}_{1}(t)$ was expanded in powers of a "reduced density," or activity. In this paper the Fourier transform $U(\ensuremath{\omega})=\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dt}{e}^{\ensuremath{-}i\ensuremath{\omega}t}U(t)$ is written in the form $iU(\ensuremath{\omega})={[\ensuremath{\omega}\ensuremath{-}{L}_{2}(\ensuremath{\omega})]}^{\ensuremath{-}1}$, and the frequency-dependent effective Liouvillian ${L}_{2}(\ensuremath{\omega})$, or "memory operator," is expanded in powers of the reduced density. It is argued that in treating ${L}_{2}(\ensuremath{\omega})$ to first order in the reduced density, one effectively allows the radiator to interact with only one perturber at a time, thus neglecting all multiple-collision effects. This is in contrast to performing the same-order approximation on ${L}_{1}(t)$, which is equivalent to treating different perturbers as uncorrelated, but still allows for multiple-collision effects. By adding the terms of higher order in the expansion of ${L}_{2}(\ensuremath{\omega})$, one allows the radiator to interact simultaneously with two, three,... perturbers.

Physical Review AMar 1, 197322 citations

Theory of Pressure Broadening in an Adiabatic Representation

Antoine Royer

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.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.

Canadian Journal of PhysicsSep 15, 197428 citations

General Impact Theory of Pressure Broadening

Michel Baranger

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.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.

Physical ReviewNov 1, 1958645 citations

Pressure dependence of Na resonance line broadening by Kr and Xe

W.P. West, Alan C. Gallagher

The fluorescent spectrum of the Na $D$ lines, pressure broadened by Xe and Kr, has been measured for noble-gas densities of 2\ifmmode\times\else\texttimes\fi{} ${10}^{19}$ -3\ifmmode\times\else\texttimes\fi{} ${10}^{20}$ ${\mathrm{cm}}^{\ensuremath{-}3}$; at the lower density, the lines are isolated while, at the higher, they are severely blended. The spectra are obtained in normalized intensity units allowing the nonbinary behavior of the line wing intensity to be clearly observed. At the lower density the broadening is well characterized by isolated binary interactions; at the higher density multiple-perturber interactions dominate. Nonlinearities in the pressure dependence of shifts, widths, and satellite shape are reported.The fluorescent spectrum of the Na $D$ lines, pressure broadened by Xe and Kr, has been measured for noble-gas densities of 2\ifmmode\times\else\texttimes\fi{}${10}^{19}$-3\ifmmode\times\else\texttimes\fi{}${10}^{20}$ ${\mathrm{cm}}^{\ensuremath{-}3}$; at the lower density, the lines are isolated while, at the higher, they are severely blended. The spectra are obtained in normalized intensity units allowing the nonbinary behavior of the line wing intensity to be clearly observed. At the lower density the broadening is well characterized by isolated binary interactions; at the higher density multiple-perturber interactions dominate. Nonlinearities in the pressure dependence of shifts, widths, and satellite shape are reported.

Physical Review AApr 1, 197835 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).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 AMay 1, 197240 citations

{I}_{c}(\ensuremath{\omega})

, is essentially the impact distribution; the second term is

n{J}_{w}(\ensuremath{\omega})

folded into

{I}_{c}(\ensuremath{\omega})

, where

{J}_{w}(\ensuremath{\omega})

n{J}_{w}(\ensuremath{\omega})

, all folded into

{I}_{c}(\ensuremath{\omega})

{\mathrm{Tr}}_{B}

is the trace over bath coordinates, $\ensuremath{\rho}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}H}$ is the density matrix, and

{H}^{\mathrm{x}}

is the quantum-mechanical Liouvillian:

{H}^{\mathrm{x}}( )=[H, ( )]

;

H

is the total Hamiltonian for the system radiator plus bath};

U(t)

is written as $U(t)={T}_{\ensuremath{\rightarrow}}\mathrm{exp}[i\ensuremath{\int}{0}^{t}d{t}^{\ensuremath{'}}L({t}^{\ensuremath{'}})]$,

L(t)={H}_{s}^{\mathrm{x}}+R(t)

(${T}_{\ensuremath{\rightarrow}}$ is a time-ordering operator), where the effect of the bath is contained in the time-dependent non-Hermitian perturbation

R(t)

added to the Liouvillian

{H}_{s}^{\mathrm{x}}

of the unperturbed radiator. The operator

R(t)

is expanded in powers of a "reduced" density equal to the perturbing-gas density multiplied by the ratio of the fugacities corresponding to mutually interacting and noninteracting perturbing atoms, respectively; the terms of the expansion are expressed by means of generalized cumulants, and describe interactions of the radiator with clusters of perturbers. By setting $R(t)=\overline{R}+\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, where $\overline{R}\ensuremath{\equiv}{\mathrm{lim}}_{T\ensuremath{\rightarrow}\ensuremath{\infty}}[(\frac{1}{T})\ensuremath{\int}{0}^{T}\mathrm{dtR}(t)]$, the spectrum is written as the sum of its impact approximation, determined by

\overline{R}

, plus a correction expanded in powers of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{R}(t)$, which to first order in the gas density equals the one-perturber spectrum minus its singularities at the resonance frequencies.

{\mathrm{Tr}}_{B}

is the trace over bath variables, $\ensuremath{\rho}$ is the density matrix, and ${H}^{\ifmmode\times\else\texttimes\fi{}}$ is the quantum-mechanical Liouvillian). In a previous paper,

U(t)

was written in the form $U(t)={\mathrm{exp}}_{\ensuremath{\rightarrow}}[i\ensuremath{\int}{0}^{t}{\mathrm{dt}}^{\ensuremath{'}}{L}_{1}({t}^{\ensuremath{'}})]$ (${\mathrm{exp}}_{\ensuremath{\rightarrow}}$ denotes a time-ordered exponential), and the time-dependent effective Liouvillian

{L}_{1}(t)

was expanded in powers of a "reduced density," or activity. In this paper the Fourier transform $U(\ensuremath{\omega})=\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dt}{e}^{\ensuremath{-}i\ensuremath{\omega}t}U(t)$ is written in the form $iU(\ensuremath{\omega})={[\ensuremath{\omega}\ensuremath{-}{L}_{2}(\ensuremath{\omega})]}^{\ensuremath{-}1}$, and the frequency-dependent effective Liouvillian ${L}_{2}(\ensuremath{\omega})$, or "memory operator," is expanded in powers of the reduced density. It is argued that in treating ${L}_{2}(\ensuremath{\omega})$ to first order in the reduced density, one effectively allows the radiator to interact with only one perturber at a time, thus neglecting all multiple-collision effects. This is in contrast to performing the same-order approximation on

{L}_{1}(t)

, which is equivalent to treating different perturbers as uncorrelated, but still allows for multiple-collision effects. By adding the terms of higher order in the expansion of ${L}_{2}(\ensuremath{\omega})$, one allows the radiator to interact simultaneously with two, three,... perturbers.

The fluorescent spectrum of the Na

D

lines, pressure broadened by Xe and Kr, has been measured for noble-gas densities of 2\ifmmode\times\else\texttimes\fi{}

{10}^{19}

-3\ifmmode\times\else\texttimes\fi{}

{10}^{20}

${\mathrm{cm}}^{\ensuremath{-}3}$; at the lower density, the lines are isolated while, at the higher, they are severely blended. The spectra are obtained in normalized intensity units allowing the nonbinary behavior of the line wing intensity to be clearly observed. At the lower density the broadening is well characterized by isolated binary interactions; at the higher density multiple-perturber interactions dominate. Nonlinearities in the pressure dependence of shifts, widths, and satellite shape are reported.

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).