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

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

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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 static and the dynamic structure factor, <math>S(k,t)/<math>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)