Structure factor

The static structure factor, $S(k)$, for a monatomic system composed of spherical scatterers is defined by (Eq. 1 in ):

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

where $g_2(r)$ is the radial distribution function, and $k$ 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

Related reading

• F. Zernike and J. A. Prins "Die Beugung von Röntgenstrahlen in Flüssigkeiten als Effekt der Molekülanordnung", Zeitschrift für Physik 41 pp. 184-194 (1920)
• P. Debye and H. Menke "", Physik. Zeits. 31 pp. 348- (1930)
• B. E. Warren "X-Ray Diffraction", Dover Publications (1969) ISBN 0486663175 § 10.4
• Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids" (Third Edition) Chapter 4: "Distribution-function Theories" § 4.1