Structure factor

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

$S(k)=1+{\frac {4\pi \rho }{k}}\int _{0}^{\infty }(g_{2}(r)-1)r\sin(kr)~dr$

where $k$ is the scattering wave-vector modulus

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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. $|\mathbf {k} |=0$,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S(0)=k_{B}T\left.{\frac {\partial \rho }{\partial p}}\right\vert _{T}}

from which one can calculate the isothermal compressibility.

To calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k)} in molecular simulations one typically uses:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m))> } ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} is the number of particles and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_n} and $\mathbf {r} _{m}$ are the coordinates of particles Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} respectively.

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> } ,

The ratio between the dynamic and the static structure factor, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)