Structure factor

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

\[S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~{\mathrm {d}}r\]

where \(g_2(r)\) is the radial distribution function, and \(k\) is the scattering wave-vector modulus

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

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

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

At zero wavenumber, i.e. \(|\mathbf{k}|=0\),

\[S(0) = k_BT \left. \frac{\partial \rho}{\partial p}\right\vert_T\]

from which one can calculate the isothermal compressibility.

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

\[S(k) = \frac{1}{N} \sum^{N}_{n,m=1} \langle\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m)) \rangle \],

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: \[S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} \langle \exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0))) \rangle \],

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