In the scattering of a particle (or system) of charge $\mathrm{Ze}$ by a neutral system with an electrical polarizability $\alpha$, an electric dipole moment is induced which generates an effective potential that behaves asymptotically as $-\frac{\frac{1}{2}{Z}^2{e}^2\alpha }{r^4}$. Due to this effective long-range interaction, effective-range theory in its normal form is not applicable. Thus, for the scattering of a particle with an incident orbital angular momentum of zero, for example, the expansion of $kcot\eta \mathrm{}\left(0\right)\mathrm{}$ includes terms in $k$ and in $k^2\ln k$, in addition to the usual constant and $k^2$ terms. The effective range $r_0$ as normally defined is infinite, but one can define a quantity $r_{p0\mathrm{}}$ which explicitly takes into account the long-range character of the effective potential. For $L>\mathrm{}0\mathrm{}$ it is $k^{2}cot\eta \mathrm{}\left(L\right)\mathrm{}$ which approaches a constant as $k$ approaches zero rather than $k^{2L+1}cot\eta \mathrm{}\left(L\right)\mathrm{}$ as for a short-range potential. The above results can have serious consequences in the scattering of electrons and of positrons by neutral spherically symmetric atoms. Some detailed consideration is given to the scattering by hydrogen atoms. The formulation of effective-range theory which is given is valid when Pauli exchange between the two colliding systems is possible. The method used for taking into account the effect of the Pauli principle (this method would be the same for long-range and short-range forces) is rather more convenient than in the usual presentation of effective range theory.

• Received 12 October 1961

DOI:https://doi.org/10.1103/PhysRev.125.1300