Structure factor

The structure factor, $S(k)$, for a monatomic system is defined by:

$S(k)=1+\frac{4\pi \rho }{k}\underset{0}{\overset{\mathrm{\infty }}{\int }}(g_2(r)-1)r\sin (kr)dr$

where $k$ is the scattering wave-vector modulus

$k=\left|\mathbf{k}\right|=\frac{4\pi }{\lambda \sin (\frac{\theta }{2})}$

The structure factor is basically a Fourier transform of the pair distribution function $\mathrm{g}(r)$,

$S(|\mathbf{k}|)=1+\rho \int \exp (i\mathbf{k}\cdot \mathbf{r})\mathrm{g}(r)\mathrm{d}\mathbf{r}$

At zero wavenumber, i.e. $\left|\mathbf{k}\right|=0$,

$S(0)=k_{B}T{\frac{\partial \rho }{\partial p}|}_T$

from which one can calculate the isothermal compressibility.

To calculate $S(k)$ in molecular simulations one typically uses:

$S(k)=\frac{1}{N}\sum _{n,m=1}^N<\exp (-i\mathbf{k}({\mathbf{r}}_n-{\mathbf{r}}_m))>$,

where $N$ is the number of particles and ${\mathbf{r}}_n$ and ${\mathbf{r}}_m$ are the coordinates of particles $n$ and $m$ respectively.

The dynamic, time dependent structure factor is defined as follows:

$S(k,t)=\frac{1}{N}\sum _{n,m=1}^N<\exp (-i\mathbf{k}({\mathbf{r}}_n(t)-{\mathbf{r}}_m(0)))>$,

The ratio between the dynamic and the static structure factor, $S(k,t)/S(k,0)$, is known as the collective (or coherent) intermediate scattering function.

References

1. A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, 6 pp. 8415-8427 (1994)