Structure factor

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

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(0) = k_BT \left. \frac{\partial \rho}{\partial p}\right\vert_T}

References

1. A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, 6 pp. 8415-8427 (1994)