Shift, width, and asymmetry of pressure-broadened spectral lines at intermediate densities

Abstract

Our main objective is to understand the behavior of the shift, width, and asymmetry of spectral lines, as a function of the perturbing gas density, at densities above those at which the impact approximation is applicable. We first set up a general theory of pressure broadening of an isolated line by a foreign gas, by making use of an adiabatic representation. We then construct a cluster or cumulant expansion, which translates into the density expansion of the logarithm of the dipole autocorrelation function. This expansion is expected to converge rapidly, for each term does not measure just the probability for multiple collisions of a given order (as, e.g., in the density expansion of the memory function), but rather the correlations existing between the effects of different perturbers on the line shape. In lowest order, we get a nonadiabatic generalization of the Anderson-Talman-Baranger theory, which we term Anderson-Talman (AT) approximation. This seems applicable at densities of up to a few tens of atmospheres in certain cases. By retaining the next-order term, we apparently extend the applicability to several tens of atmospheres, at least near the line center. To analyze the shift, width, and asymmetry at relatively low densities, but still above the impact regime, we obtain for them expansions in powers of density, the first terms of which are the impact values. At the densities at which these expansions, relevant, the AT approximation is expected to be valid; then, $shift=nd+\frac{n^2\mathrm{ab}}{2}+\cdots$, $width=2nb+O\left(n^3\right)$; $asym=-na+\frac{n^2{a}^2}{2}\dots$, where $n$ is the density, $a,b,d$ constants. The initial curvature of the shift versus density curve, $\mathrm{ab}$, is seen to equal minus one-half of the product of the initial slopes of width and asymmetry; this can be verified with experimental data, and seems well obeyed. The quadratic term in the shift can cause it to reverse sign as the density increases, as is sometimes observed experimentally. To deal with much higher densities, at which the line shape approaches a Gaussian, we obtain for it and its shift, width, and asymmetry, expansions in inverse powers of roughly the width. In the AT approximation, these expansions become in powers of $n^{\frac{-1\mathrm{}}{2}}$, $shift=n\Delta +\frac{{\Gamma }_1^{\mathrm{AT}}}{2}+O\left(n^{-1\mathrm{}}\right)$, $width=\kappa {\left(n\gamma \right)}^{\frac{1}{2}}+O\left(n^{\frac{-1\mathrm{}}{2}}\right)$, $asym=\left(\frac{\kappa }{6}\right){\left(n\gamma \right)}^{\frac{-1\mathrm{}}{2}}{\Gamma }_1^{\mathrm{AT}}+O\left(n^{-1\mathrm{}}\right)$, where $\kappa =2\mathrm{}{\left(2\mathrm{}\ln 2\right)}^{\frac{1}{2}}$ and $\Delta ,\gamma ,{\Gamma }_1^{\mathrm{AT}}$ are constants. One experimental case, $\frac{\mathrm{Cs}(6\mathrm{}{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}\to 6{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{})}{\mathrm{Xe}}$, is found to obey these relations closely. But to explain most of the high-density data available, we must add in the first density correction to the AT approximation; the density dependence is then rather more complicated than the above, except under certain conditions for which we get a high-density region of linear shift, width, and inverse asymmetry, as is sometimes observed experimentally. To illustrate the theory, we include numerical calculations of the classical AT line shape for a square-well frequency perturbation.