On the Behavior of Cross Sections Near Thresholds

The energy dependence of the cross section for the formation of a product, near the threshold energy for that formation, is considered. It is shown that the cross section is, apart from a constant, in the neighborhood of the threshold the same function of energy, no matter what the reaction mechanism is, as long as the long-range interaction of the product particles is the same. The same must hold, because of the principle of detailed balance, for the back reaction, i.e., the reaction between particles with very low relative velocities. In this case, the cross section, as function of the energy, depends only on the long-range interaction of the reacting particles. The energy dependence of the cross section is determined for three types of interactions, viz. no interaction, Coulomb repulsion and Coulomb attraction. The rule for a $\frac{1}{r^2}$ interaction can be obtained from the first case. Reasons are adduced to show that two interactions, the difference of which goes to zero at least as fast as $r^{-2-\epsilon }$ with $\epsilon >0\mathrm{}$, give the same energy dependence of the cross section. Hence, long-range interaction in the above connection should mean an interaction which, at large distances of the particles, does not go to zero faster than $r^{-2\mathrm{}}$. The effect of small perturbations in the long-range interaction is discussed in general.