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. $|\mathbf {k} |=0$,

$S(0)=k_{B}T\left.{\frac {\partial \rho }{\partial p}}\right\vert _{T}$

from which one can calculate the isothermal compressibility.

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

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

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:

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

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