Exchange collision kernel.

Abstract

In the analysis of laser-spectroscopic line shapes, velocity-changing collisions are accounted for by adding to the density-matrix equations-of-motion terms involving the direct collision kernel. The direct collision kernel is the probability density per unit time that an atom has a specific velocity after a collision, as a function of its velocity before the collision. If the collision partner's velocity distribution is not thermal, however, the direct collision kernel alone does not completely characterize the effect of collisions. In such cases it is necessary to include terms involving the exchange collision kernel. The exchange kernel is the conditional probability density per unit time that the atom has a specific velocity after the collision as a function of its collision partner's velocity before the collision. Both the direct and the exchange collision kernels are obtained from the linearized Boltzmann equation. We derive an expression for the exchange kernel in terms of the colliding-pair scattering cross section, and calculate the exchange kernel for hard-sphere collisions. Moreover, we propose a phenomenological exchange kernel similar to the Keilson-Storer direct kernel [J. Keilson and K. E. Storer, J. Appl. Math. 10, 243 (1952)] and compare it with the hard-sphere kernel. In an earlier paper [P. R. Berman, J. E. M. Haverkort, and J. P. Woerdman, Phys. Rev. A 34, 4647 (1986)], a connection was made between the collision kernels of laser spectroscopy and the collision integrals of classical transport theory. We expand on this earlier work by including the exchange collision kernel, and indicate how results from classical transport theory might be used in setting the parameters appearing in the phenomenological exchange kernel. We interpret an excitation-transfer experiment in terms of the exchange kernel.