Low Velocity Inelastic Collisions

The probability that a low velocity collision be inelastic is shown to depend in a relatively simple manner upon three factors: the magnitude of the change in total internal energy, $\Delta E$; the matrix element, with respect to the initial and final states, of the interaction energy at the closest distance of approach, $W_{12}$; and the "time of collision," $\tau$, a function of the relative velocity and of the sharpness of the collision. An approximation to this probability is $P={\left[\frac{2\pi W_{12}}{h}\tau (1+{\epsilon }_1{\beta }^{{\epsilon }_2})e^{-{\epsilon }_3\beta }\right]}^2$ where $\beta =\frac{2\pi \tau \Delta E}{h}$ and ${\epsilon }_1$, ${\epsilon }_2$, ${\epsilon }_3$ are nearly unity, their precise values depending upon the characteristics of the particular collision.