In a recent paper (Lewis, 2008) a class of models suitable for application to collision-sequence interference was introduced. In these models velocities are assumed to be completely
randomized in each collision. The distribution of velocities was assumed to be Gaussian. The
integrated induced dipole moment
Spectra resulting from dipole moments induced in molecular collisions typically have the form of broad bands with widths determined by the durations of those collisions [
The present work is based on that of paper I, which will henceforth be referred to as paper I. In paper it was assumed that the collisions suffered by a molecule occur at equally spaced times. This drastic Ansatz allowed the use of the apparatus of discrete Fourier transforms. In this present work it is assumed that the collision times of a given molecule are distributed exactly as a Poisson process, which is in fact an excellent approximation to reality (see [
At sufficiently low densities and for the study of interference phenomena collisions can be assumed to be instantaneous; the dipole moment induced in one atom or molecule by interaction with a bath of dissimilar atoms or molecules can be represented as
In general, as stated above, the collision times
Equation (
It will be assumed initially that
A principal assumption of the present model, and the feature in which it differs from the class of models discussed in paper I, is that the intervals
If the random variables
From (
The dipole moment or transition moment induced in a collision is roughly but not exactly proportional to the intermolecular force; the overlap parts differ in range by about 25%. For purposes of calculating the intercollisional interference the integrated induced dipole moment
For two dimensions it was found in paper I that
Because
For the power-law model, which is a limiting case for
The necessary integrals to evaluate
For the power-law model given in (
In this paper, we have extended a class of model developed in paper I for the study of collision-sequence interference effects in collision-induced absorption, to include realistic distributions of collision times. In these models, a single particle is followed. Its collisions are supposed to be instantaneous. In paper I the collisions were assumed to occur at equally spaced times, whereas in the present work the collision times are distributed according to a Poisson process. Velocities are supposed to be completely randomized at each collision. It is supposed that the dipole moment
It is important that the model spectra can be determined analytically, or at worst, reduced to straightforward numerical integrations. The models of paper I, of [
The extension of the induced dipole moment model to dipole moments which are proportional to an arbitrary power of the integrated intermolecular force shows that the interference dip is partially filled in for any disproportionality between induced dipole moment and integrated induced dipole moment. In this paper, the calculation is given for the three-dimensional case. For a realistic value of the power the infilling is slight, being about 0.6% of spectral maximum for the three-dimensional case.
In three dimensions, the Gaussian distribution of velocities is given by
For these vectorial cross terms we have
The author thanks the Department of Physics of the Pennsylvania State University for its hospitality in 1999, in 2000, in 2005, and in 2008; he gratefully acknowledges many useful discussions on collision-induced absorption with Roger Herman; and he thanks Eugene Oks for providing the opportunity to develop this paper. The support from the Natural Sciences and Engineering Research Council of Canada is acknowledged.