## 25 May 2015

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

This is the third part in a series. In part I and part II we defined the basic concepts used in the theory of liquids, in particular the radial distribution function $$g(r)$$ and the structure factor $$S(q)$$.

The simplest system one can imagine is the ideal gas. There is no interaction between particles: $$u(r) = 0$$, leading to $$g(r) = 1$$ (the particle at the origin does not affect the position of its neighbors) and $$S(q) = 1$$. The ideal gas is a trivial case, but it can be seen as the reference state for other systems. In particular, one could say that the functions $$g(r) - 1$$ and $$S(q) - 1$$ that appear in Equation (4) of part II quantify the difference with respect to the ideal gas (due to the interaction potential $$u(r) \neq 0$$.)

#### Hard spheres

We will now discuss the case of a hard sphere liquid, where $$u(r) = \infty$$ for $$r < 2R$$ (with $$R$$ the hard sphere radius) and $$0$$ elsewhere.

The figure above shows the structure factor of a (three dimesional) hard sphere liquid for different volume concentrations $$\phi = \rho V_{\text{part}}$$, where $$V_{\text{part}} = (4\pi/3) R^3$$ is the particle volume. There is a single length scale in the problem, the radius $$R$$ (or the diameter $$D = 2R$$). We can thus use the normalized scattering vector $$Q = qD$$: the structure vector of systems with different $$D$$ (but the same $$\phi$$) is the same when plotted against $$Q$$.

#### Dilution law

One could object that another length scale can be obtained from the number density: $$L \sim \rho^{-1/3} = D (\pi/6)^{1/3} \phi^{-1/3}$$. In a crystal, $$L$$ would define the position of the Bragg peaks (up to a numerical constant). However, until very high values of $$\phi$$, the first maximum of $$S(q)$$, $$q_{\text{max}}$$, is constant and arises close to $$Q = 6$$ (see the right panel in the figure). The relevant distance is $$2\pi /q_{\text{max}} \simeq D$$, and not $$L$$! The hard sphere liquid does not obey the dilution law one would expect based on the value of $$L$$ (solid dots in the figure). The peak position only shifts at high concentrations $$\phi$$, close to the random close packing (RCP) limit, which marks the end of the liquid range.

To understand this apparently paradoxical behaviour, remember that $$S(q) - 1$$ is the Fourier transform of $$g(r) - 1$$ (Eq. (4) in part II). The strongest feature of $$g(r)$$ is the discontinuity in $$r=D$$ which, on Fourier transforming, causes an oscillation in $$S(q)$$ and gives rise to its peaks. The amplitude of the discontinuity increases with $$\phi$$, but its position does not change and neither does that of $$q_{\text{max}}$$.