Sum rule for impenetrable systems
The hard sphere liquid is an idealized model, but some of its properties hold for a very large class of systems, those that have an impenetrable core of size \(R_c\) (\(g(r< 2 R_c = 0\)). Let us write the Fourier relation between \(g(r) -1\) and \(S(q) -1\) (the inverse of Eq. (4) in post II):
