Inelastic Collisions between Atoms. I. General Theoretical Considerations
Proceedings of the Royal Society A
The probability of slow inelastic collisions between two atomic systems is investigated theoretically using the perturbed stationary state method (p.s.s. method) in which the kinetic energy of the relative motion is treated as the perturbation responsible for the transitions. The p.s.s. method has been extended to include cases in which resonance degeneracy occurs and when the transition concerned involves a change' of the axial com ponent of angular momentum. The scope and limitations of the
... imitations of the method are examined in detail and compared with those of Bom's approximation. It is shown that two modifications of the latter approximation are of comparable importance in most cases, one involving a change in the effective transition potential, the other the distortion of the plane waves representing the relative motion of the colliding systems. The first of these is the only one included in previous applications. It is shown that, under semi-classical conditions, the formulae of the p.s.s. method reduce to Mott's impact parameter formulae provided the important distortion effect is neglected. New semi-elassical formulae, including this effect as well as the effect of resonance degeneracy, are derived in a form suitable for numerical application. * Now at Queen's University, Belfast. Vol. 216. A. (24 February 1953) [ 437 ] 29 produce its own space-charge neutralization by the ionization of the residual gas and are thus of great technical importance. Again, to take some examples from atmospheric physics, the illumination and ionization accompanying a meteor trail are caused by the passage through the upper air of evaporated atoms having energies only of the order of 100 eV; charge exchange between slow positive ions and neutral molecules may cause changes in the ionic composition of the D, E and F layers; and collisional deactivation may influence the spectra of aurorae and of the airglow. Any attempt to disentangle in detail the various factors which determine the cross-sections for nearly adiabatic collisions must involve extensive further experi mental and theoretical research. One major difficulty about the experimental approach is that when the relative velocity in the collision is very low the chance of energy transfer is so minute that it is not easy to reduce background effects to an unimportant magnitude. This enhances the need for detailed calculations based on the p.s.s. method. It is best to begin with a case which can be treated with the minimum need to introduce approximations. A study of the excitation of hydrogen atoms by proton impact was begun. It soon became clear, however, that there were many more general aspects of the p.s.s. method which needed clarification. Thus it must be extended to allow for the existence of resonance degeneracy and the accompanying possibility of charge exchange without energy transfer. Furthermore, the relation between the p.s.s. method and the impact parameter method had not been established. An examination of this aspect revealed that Mott's formula needs modification to take account of the fact that the relative velocity of the colliding systems is not unchanged throughout the impact even when no energy transfer takes place. In other words, it does not allow for distortion, by the mutual interactions, of the waves representing the relative motion of the colliding systems. This effect is likely to be at least as important as any other ignored in Bom's approximation. A new formula has therefore been derived, both from quantal theory and from semiclassical theory, which represents the appropriate generalization of Mott's formula, reduces to it when distortion is neglected, and is in a suitable form for numerical applications. Consideration has also to be given to the choice of co-ordinates to represent the relative motion of the electrons and the colliding atoms, and to the modifications which must be introduced to deal with transitions involving states for which the component of the electronic orbital angular momentum about the internuclear axis is not zero. We present in this paper a discussion of these various aspects of the theory. Any detailed applications are reserved for a later paper. . T h e p e r t u r b e d s t a t i o n a r y s t a t e m e t h o d No resonance degeneracy We shall first consider specifically the collision between two dissimilar heavy particles A and B of masses MA and MB in which only a single electron P of mass m is involved. 438 A moving centre of co-ordinates located on the intemuclear axis will be considered in the later paper. Such a choice introduces only minor alterations to the formula developed here-indeed, essentially only formulae (49) and (50) are affected.