Radial Distribution Function of a Gas of Hard Spheres and the Superposition Approximation

The term g₂(r) proportional to the square of the density in the expansion of the radial distribution function g(r) of an imperfect gas in powers of the density is calculated exactly in the case of a gas consisting of hard spheres. The result is checked by means of Boltzmann's value of the 4th virial coefficient of such a gas. The integral equation for g(r), obtained on applying the superposition approximation introduced by Kirkwood and by Born and Green, can also be solved by an expansion in powers of the density. For the case of hard spheres the approximate g₂^′(r) found in this way is compared with the exact g₂(r). As a further application of our result on g₂(r) a certain integral is discussed, which is of interest in the treatment of interference effects in neutron scattering problems.