1. Wigner Threshold Laws for Universal Scalings of Collision Cross Sections σ­(k) as a Function of the Wavenumber k for Two-Body Collisions in Three Dimensions .

	Inelastic
lim r→∞ V(r)	Endothermic	Exothermic	Depolarization	Elastic
0	$k^{2\mathcal{l}+1}$	$k^{2\mathcal{l}-1}$	$k^{2\mathcal{l}+2\mathcal{l}\prime }$	$k^{4\mathcal{l}}$
Ze2/r	exp(−2πμZe 2/(ℏ2 k))	k–2 exp(−2πμZe 2/(ℏ2 k))	k–2 exp(−4πμZe 2/(ℏ2 k))	k –2
–Ze 2/r	1	k –2	k –2	k –2

^aScalings are shown for elastic collisions and inelastic collisions with positive threshold energy (endothermic reactions), negative threshold energy (exothermic reactions), and changes in angular momentum (depolarization events) and are given for short-range (lim _r→∞ V(r)) and long-range (attractive Coulombic lim _r→∞ V(r) = −Ze ²/r and repulsive Coulombic lim _r→∞ V(r) = −Ze ²/r) potentials. Expressions are provided in terms of the reduced mass μ, charge number product Z, and initial $\mathcal{l}$ and final angular momentum ${\mathcal{l}}^{\prime }$ quantum numbers of the collision partners, where e is the electron charge and ℏ is reduced Planck’s constant.