## 25 May 2015

### The structure factor of a liquid - part IV

#### 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):

\label{eq:grfinal}
g(r) -1 = \frac{1}{\rho} \frac{1}{(2\pi)^3} \int {\text{d}}^3 \mathbf{q}\, [S(q) - 1] \mathrm{e}^{-i \mathbf{q} \mathbf{r}}

Setting $$g(0) =0$$ yields the sum rule:

\label{eq:sumrule}
\frac{1}{(2\pi)^3} \int {\text{d}}^3 \mathbf{q}\, [S(q) - 1] = \frac{1}{2\pi^2} \int_{0}^{\infty} \text{d} q\, q^2 [S(q) - 1]=  - \rho

Relation \eqref{eq:sumrule} holds for any interaction potential $$u(r)$$ as long as it has an impenetrable core: $$u(r) = \infty, \, r \leq 2 R_c$$, which is the case of almost all colloidal particles. The exceptions are so-called "soft-core" particles such as polymer coils1.

1. See e.g. A. A. Louis & al., Phys. Rev. Lett. 85, p. 2522 (2000).