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):

$$\begin{equation} \label{eq:grfinal} g(r) -1 = \frac{1}{\rho} \frac{1}{(2\pi)^3} \int {\text{d}}^3 \mathbf{q}\, [S(q) - 1] \mathrm{e}^{-i \mathbf{q} \mathbf{r}} \end{equation}$$
Setting $g(0) =0$ yields the sum rule:
$$\begin{equation} \label{eq:sumrule} \frac{1}{(2\pi)^3} \int {\text{d}}^3 \mathbf{q}\, [S(q) - 1] = \frac{1}{2\pi^2} \int_{0}^{\infty} \text{d} q\, q^2 [S(q) - 1]=  - \rho \end{equation}$$

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.

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