Correction to the Debye-Hückel Theory

The problem of a gas of particles all of the same charge, imbedded in a neutralizing medium of uniformly distributed charge of the opposite sign, is considered in terms of classical statistical mechanics. If a dimensionless parameter $\epsilon$, roughly the inverse of the number of particles contained inside a Debye sphere, is small compared to unity the Debye-Hückel theory is a good first approximation. For this case the corrections in the next order in $\epsilon$ are derived for the potential of mean force and the interaction energy. It is shown how this correction has to be modified for very small particle separation; the expansion in powers of $\epsilon$ is not strictly a Taylor expansion and factors such as $\ln \epsilon$ appear in the higher terms. Methods are given for numerical calculation of some auxiliary functions even when the parameter $\epsilon$ is not small.