Semi-classical treatment of atomic collisions

Abstract

The semi-classical treatment of atom-atom collisions involving electronic transitions is discussed. As is well know n difficulties occur if the classical trajectories associated with the various states of importance in a collision process differ significantly. A method designed to overcome these is described. It will be referred to as the forced-common-turning-point method. The four coupled first-order differential equations which describe the new version of the semi-classical two-state treatment for an atom-atom collision may be reduced to a pair of generalized impact parameter equations. The first Born approximation to the cross-section obtained from the straightforward semiclassical treatment differs from the corresponding cross-section obtained from the full quantal treatment mainly in that it contains an anomalous multiplying factor equal to the ratio of the initial to the final velocity of relative motion. This anomaly does not arise with the forced-common-turning-point method. A model collision process which provides a very searching test is considered. Only two states are included. The initial interaction is zero, the final interaction is Coulombic and the transition matrix element is exponential. Curve-crossing may occur. The distorted wave approximation to the excitation cross-section may be found exactly and may also be computed using the forced-common-turning-point method. There is remarkable accord between the results. Thus in a case w here the reduced m ass of the colliding systems is 2 on the chemical scale, w here the excitation energy is 3.4 eV and where the incident kinetic energy of relative motion is only 0.85 eV above this the excitation cross-sections obtained differ by as little as 0.01 %; and, moreover, the patterns of the contributions to the cross-sections from the separate partial waves are similar.