Threshold Laws

Abstract

Threshold laws for two-body reactions have been investigated long ago by Wigner ¹. He showed the energy dependence of an S-matrix element near threshold to be given by ²

$$ \mathop{{\lim S \alpha }}\limits_{{k \to + 0}} \left\{ {\begin{array}{*{20}{c}} {{{e}^{{ - \pi c/k}}}} \\ {cons\tan t} \\ {_{k}1 + 1/2} \\ \end{array} } \right.\begin{array}{*{20}{c}} {for c > 0} \\ {for c < 0} \\ {for c = 0} \\ \end{array} $$

(1)

For a pair of charged particles with reduced mass m the constant c is given by

$$ c = {{Z}_{1}}{{Z}_{2}}{{e}^{2}}m\bullet $$

(2)

Eq. (1) summarises the well known result that for equally charged particles (c>0) the S-matrix decreases exponentially near threshold but it remains finite for oppositely charged particles (c<0).