A dipole correlation function which incorporates velocity-changing (motional narrowing) effects and the effects of speed-dependent Lorentz relaxation rates into otherwise Voigt profile correlation functions is developed, based partly upon previous work by the author. For the first time simple closed expressions, which lend themselves to elementary calculation beginning only with the relevant parts of intermolecular interaction energies, are developed for the cubic time-dependent term within the exponent describing the decay of the correlation function. This term is of first order in perturber number density, as are the Lorentz parameters, and is complex, thereby allowing for narrowing, changing in shape and asymmetry in the line profile. “Soft” and “hard” collisions play no explicit role, though both are variously present for each line. Quartic time dependencies are also discussed, though they are thought to be negligible in nonhydrogen molecular spectroscopy. Finally, some comments are added about a relevant technique for hydrogen spectra.

For several decades, now, neutral species coherence transport and relaxation has followed a number major thrusts, above and beyond the simplest convolution of the optical coherence interruption (Lorentz) and Gaussian (Doppler) contours, as embodied in the Voigt profile for isolated spectral lines. To mention a few, five such developments come immediately to mind: (1) the incorporation of the speed dependence of coherence destruction (with speed dependent Lorentz lineshape parameters) within the Voigt convolution (SDVP effect) [

Of special interest in the present paper are the highly detailed experimental studies of individual spectral lines, such as those carried out by Pine et al. [

One of the difficulties of the descriptions of Dicke narrowing is that there seem not to be any reasonably transparent first principles expressions, or series of expressions giving the relevant parameters associated with these phenomena. In the present paper we shall, for the first time, derive intuitively appealing closed expressions for the relevant quantities, in terms of relevant parts of intermolecular potential functions and intrinsic parameters such as number density and temperature. The present work follows a partial description which was first published in 2000 [

The dipole-dipole (or other optically active operators, depending upon the spectroscopy being described) correlation function for an isolated transition can be described as having an exponential behavior of type [

We will assume that classical paths are sufficient for the description of collisional processes. As indicated, one can presumably make the entire calculation quantum mechanically as has been done in the past for the calculation of Lorentz lineshape parameters [

A note of caution should be expressed at the present time. This is, that obtaining agreement between theory and experiment on a single molecular line at a single density can lead to deceptive satisfaction. One must strive to analyze all lines that have been observed at all possible pressures, while rigorously adhering to the correct pressure dependencies of the real and imaginary parts of

The development of a correlation function describing isolated spectral lines, omitting duration-of-collision effects, essentially follows that of paper

The spectral line profile for absorption from a lower state

In the present work, we will regard the calculation of the collisional interruption quantities

Following the development of paper

The integrations over

Over the years, as mentioned above, the collision efficiency function

The beauties of (

It is not so easy to proceed further from this point without (elementary) computer analysis. Nonetheless, we can make arguments which may indicate some of the basic properties. Beginning with the first Taylor term we note that integrating only over the angles for

At this stage, therefore,

The second term in (

The overall correlation function

Comparison of small angle approximations reveals that

For many spectra the fourth order,

Finally, as previously noted, hydrogen spectra are extremely collisionally narrowed, such that an expansion in terms of individually computed powers of

For some time now it has been known that the speed-dependence of internal coherence relaxation has had importance in determining the shape of spectral lines, shifting them away from the simple Voigt profile [

The same steps can be followed for the

Note that

As described by Looney [

As a result of the above analysis the

To summarize, a formalism has been presented whereby one can find the relevant terms for the dipole-dipole correlation function needed to characterize the line shapes in the impact approximation for isolated (atomic or) molecular lines perturbed by foreign gases. Through standard techniques this formalism could be extended to a fully quantum description of the entirety of the collision problem treated here. In addition, one could treat the problem of line-mixing with neighboring transitions within the present formalism. For purposes of clarity, neither of these steps is presented here. The cubic time-dependent terms of the exponent describing the exponential decay of the correlation function have been explicitly found in terms of averages utilizing only equilibrium velocity distributions and the variously important terms of the perturber interaction with the optically active molecule. Although both “soft” and “hard” types of collisions occur for various impact parameters for almost every line, the present theory makes no fundamental distinction between these limits, and makes a seamless transition from one to the other type of behavior.

For the velocity changing

In hydrogen molecular spectra the coherence limiting effects are very small. Accordingly one can, at some pressures, see a pronounced line narrowing. One might approach the theoretical problem by using the best model form of

In this development we wish to convert the expression

The author takes pleasure in acknowledging helpful conversations with R. Ciury