The radial distribution function

Consider a simple atomic liquid  or colloidal suspension . The average density $\rho (r)$ at a distance r of a given particle is defined to be $\rho g(r)$, i.e.

$$\rho (r)=\rho g(r).$$ (2.3)

Here $\rho$ is the average density in the fluid. Notice that $\rho (r)$ is a conditional density; it is the density at $\vec{r}$, given a particle is present in the origin. A qualitative picture of g(r) is given in Fig. (2.1); it is called the radial distribution function .

It is clear that g(r) should go to 1 for large r. At very short rthe radial distribution function must be zero, because two particles cannot occupy the same space. Outside the van der Waals diameter, there is a peak because the remaining N-1 particles try to diffuse into the region occupied by the one at the origin. The surplus of particles in the first peak causes a lack of particles a little bit further on, explaining the minimum of g(r) around $r=2\sigma$.

  

Figure 2.1: Radial distribution function.

Radial distribution functions may be measured by means of scattering experiments, for example by means of X-ray or neutron scattering   in the case of atomic liquids, or by means of SAXS, SANS or light scattering  in the case of colloidal suspensions. Generally the sample produces a scattering amplitude 

$$F( \vec{k})=\int d^{3}rB(\vec{r})\exp \{i\vec{k}\cdot \vec{r}\}$$ (2.4)

at the detector. Here $B(\vec{r})$ is the scattering power at position $\vec{r}$, and $\vec{k}$ is defined by

$$\vec{k}=\vec{k}_{\mathrm{out}}-\vec{k}_{\mathrm{in}}$$ (2.5)

where $\vec{k}_{\mathrm{out}}$ and $\vec{k}_{\mathrm{in}}$ are the wavevectors  of the outgoing and incoming wave respectively. The measured intensity is proportional to

$$\langle \vert F( \vec{k})\vert^{2}\rangle =\langle \int d^{3}... ... })\exp \{i\vec{k}\cdot (\vec{r}-\vec{r}^{\prime })\}\rangle .$$ (2.6)

Writing the scattering power like

$$B(\vec{r})=\sum_{j}b(\vec{r}-\vec{r}_{j})$$ (2.7)

we find after some algebra

$$\langle \vert F(\vec{k})\vert^{2}\rangle N\vert f(\vec{k})\vert^{2}\langle \frac{1}{N} \sum_{i}\sum_{j}\exp \{i\vec{k}\cdot (\vec{r}_{i}-\vec{r}_{j})\}\rangle$$ = (2.8)

$$f(\vec{k}) \int d^{3}xb(\vec{x})\exp \{i\vec{k}\cdot \vec{x}\}$$ = (2.9)

The important quantity for us is

$${\langle \frac{1}{N}\sum_{i}\sum_{j}\exp \{i\vec{k}\cdot (\vec{r} _{i}-\vec{r}_{j})\}\rangle }$$

$$1+\frac{1}{N}\langle \sum \sum_{i\neq j}\exp \{i\vec{k}\cdot (\vec{r}_{i}- \vec{r}_{j})\}\rangle$$ =

$$1+\int d^{3}r\rho g(r)\exp \{i\vec{k}\cdot \vec{r}\}$$ =

$$1+\int d^{3}r\rho \{g(r)-1\}\exp \{i\vec{k}\cdot \vec{r}\}+(2\pi )^{3}\rho \delta (\vec{k}).$$ = (2.10)

The $\delta$-function only gives a contribution at $\vec{k}=\vec{0}$, i.e. to the forward scattering. At all other k-values one measures the scattering factor 

$$S(k)=1+\rho \int d^{3}r(g(r)-1)\exp \{i \vec{k}\cdot \vec{r}\}$$ (2.11)

i.e. essentially the Fourier transform of the total correlation function h(r)=g(r)-1.

Once we have determined the radial distribution function we may calculate several quantities of interest, like

$$U=\frac{3}{2}NkT+\frac{1}{2}N\int d^{3}r\rho g(r)\phi (r)$$ (2.12)

$$P=\frac{NkT}{V}-\frac{1}{6}\rho ^{2}\int d^{3}rg(r)r\frac{d\phi }{dr}$$ (2.13)

where $\phi (r)$ is the interaction potential  between two particles in the system, and Uand P are the thermodynamic energy and pressure  respectively. Another important relation is the compressibility equation 

$$\rho kT\kappa _{T}=1+\rho \int d^{3}r(g(r)-1)=S(k=0).$$ (2.14)

For more information we refer to any book on liquid state theory, like for example J. -P. Hansen  and I. R. McDonald , Theory of simple liquids.