On Collision Problems Involving Large Interactions

Many collisions, particularly those in which there is an exchange of electronic energy from one atom or molecule to another, involve interactions between the colliding bodies, which (being treated as perturbations) are too large to be handled by the ordinary Born method. A method is given for the treatment of such problems. Two cases must be distinguished, that of good and that of poor resonance, resonance being good if the transition which takes place at the collision does not involve the transfer of much energy from internal energy to relative translational energy, or vice versa. The case of good resonance is handled by Dirac's perturbation method (variation of constants), and the probability that a transition take place at a collision of given distance of closest approach found in terms of the perturbation (interaction) matrix component for that collision. In the case of poor resonance, we first assume the two atoms or molecules are held at a fixed distance from each other, and apply the perturbation due to the interaction between them constructing potential energy curves as a function of the distance, as if the whole system were a large molecule. We then allow the translational motion to take place. This introduces further perturbations, which in the case of poor resonance are always small, so that relatively few transitions will occur. Since the amount of energy which we may have transferred from internal to translational, or vice versa, and still have the case of good resonance may be determined in special cases as a function of the distance of closest approach, we may tell in any special case how close the two molecules or atoms must come to each other in order for there to be a transition at the collision. Thus a radius of action can be found. In general it is concluded that if the transition is to take place with a large radius of action (compared with kinetic theory radii), then less energy can be transferred from internal to translational, or vice versa, than has hitherto been supposed.