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

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}<\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)))>

,

References