The structure factor,
, for a monatomic system is defined by:
![{\displaystyle S(k)=1+{\frac {4\pi \rho }{k}}\int _{0}^{\infty }(g_{2}(r)-1)r\sin(kr)~dr}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d3e7539fab1d6ec071e5d68606c668c235dcad)
where
is the scattering wave-vector modulus
![{\displaystyle k=|\mathbf {k} |={\frac {4\pi }{\lambda \sin \left({\frac {\theta }{2}}\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddb040479b10ca2d1a3f82e7ef95ea539c1c1a0)
The structure factor is basically a Fourier transform of the pair distribution function
,
![{\displaystyle S(|\mathbf {k} |)=1+\rho \int \exp(i\mathbf {k} \cdot \mathbf {r} )\mathrm {g} (r)~\mathrm {d} \mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e81d4c201974cd5fd727a4963a7a756344f67e62)
At zero wavenumber, i.e.
,
![{\displaystyle S(0)=k_{B}T\left.{\frac {\partial \rho }{\partial p}}\right\vert _{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56505a4e8e913dcf85d38557e14ed26fb2d35092)
from which one can calculate the isothermal compressibility.
To calculate
in molecular simulations one typically uses:
,
where
is the number of particles and
and
are the coordinates of particles
and
respectively.
The dynamic, time dependent structure factor is defined as follows:
,
The ratio between the dynamic and the static structure factor,
, is known
as the collective (or coherent) intermediate scattering function.
References
- A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, 6 pp. 8415-8427 (1994)