Structure factor

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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

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

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

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