# Structure factor

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

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

where ${\displaystyle g_{2}(r)}$ is the radial distribution function, and ${\displaystyle k}$ is the scattering wave-vector modulus

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

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

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

At zero wavenumber, i.e. ${\displaystyle |\mathbf {k} |=0}$,

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

from which one can calculate the isothermal compressibility.

To calculate ${\displaystyle S(k)}$ in molecular simulations one typically uses:

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

where ${\displaystyle N}$ is the number of particles and ${\displaystyle \mathbf {r} _{n}}$ and ${\displaystyle \mathbf {r} _{m}}$ are the coordinates of particles ${\displaystyle n}$ and ${\displaystyle m}$ respectively.

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

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

The ratio between the dynamic and the static structure factor, ${\displaystyle S(k,t)/S(k,0)}$, is known as the collective (or coherent) intermediate scattering function.