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Journal of Physics B Atomic Molecular and Optical Physics

Spectral line narrowing in the Keilson-Storer model

David A. Shapiro

Published Dec 23, 1999• 12 citations
Publisher

Abstract

The spectral lineshape is analysed in a gas where collisions are described by the Keilson-Storer kernel. The model takes into account the correlation of the velocities before and after the collision, but ignores the velocity dependence of the collision rate. The expression for the line profile is derived in the form of an infinite continued fraction. It is shown that the high-pressure Dicke asymptotics of the linewidth is independent of the correlation. Under lower pressure a significant departure is found from both soft- and hard-collision limits.

DOI
10.1088/0953-4075/33/1/106

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The spectral lineshape is analysed in a gas where collisions are described by the Keilson-Storer kernel. The model takes into account the correlation of the velocities before and after the collision, but ignores the velocity dependence of the collision rate. The expression for the line profile is derived in the form of an infinite continued fraction. It is shown that the high-pressure Dicke asymptotics of the linewidth is independent of the correlation. Under lower pressure a significant departure is found from both soft- and hard-collision limits.
The Rautian-Sobelman approach describing the collision-Doppler shape of spectral line that takes into account both soft and hard velocity-changing collisions is extended to the case when correlation between Doppler and collision broadening as well as the dispersion line asymmetry should be considered. Numerical calculations show that speed-dependent effects can produce the additional line narrowing as well as broadening and also the line asymmetry. It is shown that speed-dependent profiles based on the soft and hard collision models differ much more one from another than those which omit the speed-dependent effects.
Line-shape expressions for two-step resonant excitation in the presence of low-pressure perturbers are derived. These expressions allow for velocity-dependent broadening rates and are well suited for studying the line-shape contributions due to coherent excitation, incoherent excitation (due to dephasing collisions), interference between coherent and incoherent excitation, and velocity-changing collisions. Using the sodium 3S-3P-4D system perturbed by noble-gas atoms as an example and assuming C6{C}_{6}C6​ interaction potentials, line shapes at a variety of laser detunings (Doppler selected atomic velocities) are numerically calculated to study variation in the line shape due to competing excitation paths and speed-dependent collision rates. In order to deal with both the line core and the line wing a composite collision kernel consisting of a Lorentzian kernel for small \ensuremath{\Delta}vz{v}_{z}vz​ collisions and a Keilson-Storer kernel for large \ensuremath{\Delta}vz{v}_{z}vz​ collisions is used. It is also demonstrated how the inelastic (3P3/2{P}_{3/2}P3/2​-3P1/2{P}_{1/2}P1/2​) velocity-changing collision kernel may be obtained from experimental data at low pressures. Theoretical line shapes and broadening rates are compared with experiment.
A theory of saturation spectroscopy in three-level gas-vapor systems, including collisional effects, is presented. Using a model of collisions in which they are phase interrupting in their effect on level coherences and velocity changing in their effect on level population densities, the authors calculate the absorption profile of a weak probe field on a given transition when an arbitrarily strong pump field acts on a coupled transition. Line shapes are derived for an arbitrary collision kernel describing the velocity-changing collisions, and these line shapes are evaluated for the collision kernel proposed by Keilson and Storer. Various limiting forms for the line shapes are derived and representative line shapes are displayed. Several new features of the line shapes, including collisionally induced increased probe absorption are discussed. The line shapes are seen to reflect the various collisional processes that may occur in atomic or molecular systems.
We report on a quantitative experimental investigation of velocity-changing collisions by means of velocity-selective optical pumping (VSOP). We have calculated the VSOP line shape for an atom with hyperfine structure with the use of two phenomenological kernels for the collision effects: the Keilson-Storer kernel, and a two-term kernel consisting of a broad Keilson-Storer part and a narrower Gaussian component. Corrections were included to account for the finite absorption in the sample and the backward reflection of the pumping beam. The experiments were carried out in sodium vapor with neon as the perturber gas. The D1{D}_{1}D1​ line of sodium was used for optical pumping, and the orientation of the ground state was detected. Free parameters of the theory were determined by fitting the predicted line shapes to experimental curves. The Keilson-Storer kernel proved unsatisfactory, but the two-term kernel reproduced well the observed line shapes over the entire collision profiles in the neon pressure range 0-57 mtorr. In an independent experiment using rapidly modulated VSOP we also measured directly the cross section of velocity-changing collisions: $\ensuremath{\sigma}=(1.13\ifmmode\pm\else\textpm\fi{}0.10)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ cm2{\mathrm{cm}}^{2}cm2. The large weight obtained for the narrow Gaussian from the fits, as well as the collision cross section which is three times as large as the cross section deduced from tabulated gas kinetic radii, may indicate the presence of collisions with relatively small velocity changes in addition to hard-sphere encounters.
Spectral lineshape narrowing of atomic vapour emission in a bufferless, wall-coated absorption cell has been analysed using the same formalism developed to describe sub-Doppler linewidths, Dicke narrowing, in a buffered cell. Within this formalism lineshapes are generated using Monte Carlo techniques. Dramatic linewidth reductions in the bufferless cell, compared with the buffered cell, occur because of the high probability that Doppler shifts before and after a particular wall collision will cancel each other. The formalism described allows more physical insight than the classic one-dimensional box model previously invoked to explain spectra narrowing in bufferless, wall-coated cells. Additionally, this formalism reveals a correlation between post-collision velocities and intercollision times in the buffered cell resulting in modifications of the standard Dicke analysis of these cells.
The effect of the velocity ( v ) dependence of the transport collision frequency ν tr v on the Dicke line narrowing is analyzed in terms of the strong-collision model generalized to velocity-dependent collision frequencies (the so-called kangaroo model). This effect has been found to depend on the mass ratio of the resonance ( M ) and buffer ( M b ) particles, β = M b / M : it is at a minimum for β ≪ 1 and reaches a maximum for β ≳ 3. A power-law particle interaction potential, U ( r ) ∝ r − n , is used as an example to show that, compared to ν tr v ( v ) = const ( n = 4), the line narrows if ν tr v ( v ) decreases with increasing v ( n < 4) and broadens if ν tr v ( v ) increases with v ( n > 4). At β ≳ 3, the line width can increase [compared to ν tr v ( v ) = const] by 5 and 12% for the potentials with n = 6 and n ≳ 10, respectively; for the potentials with n = 1 (Coulomb potential) and n = 3, it can decrease by more than half and 6%, respectively. The line profile I (Ω) has been found to be weakly sensitive to ν tr v ( v ) at some detuning Ω c of the radiation frequency Ω. Dicke line narrowing is used as an example to analyze the collisional transport of nonequilibrium in the resonance-particle velocity distribution in a laser field. The transport effect is numerically shown to be weak. This allows simpler approximate one-dimensional quantum kinetic equations to be used instead of the three-dimensional ones to solve spectroscopic problems in which it is important to take into account the velocity dependence of the collision frequency when the phase memory is preserved during collisions.