Structure factor

2011-11-16T12:33:00Z

132.168.28.141:

The '''structure factor''', <math>S(k)</math>, for a monatomic system is defined by:

:<math>S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~dr</math>

where <math>k</math> is the scattering wave-vector modulus

:<math>k= |\mathbf{k}|= \frac{4 \pi }{\lambda} \sin \left( \frac{\theta}{2}\right)</math>

The structure factor is basically a [[Fourier analysis | Fourier transform]] of the [[pair distribution function]] <math>{\rm g}(r)</math>,

:<math>S(|\mathbf{k}|)= 1 + \rho \int \exp (i\mathbf{k}\cdot \mathbf{r}) \mathrm{g}(r) ~\mathrm{d}\mathbf{r}</math>

At zero wavenumber, ''i.e.'' <math>|\mathbf{k}|=0</math>,

:<math>S(0) = k_BT \left. \frac{\partial \rho}{\partial p}\right\vert_T</math>

from which one can calculate the [[Compressibility | isothermal compressibility]].

To calculate <math>S(k)</math> in [[Computer simulation techniques |molecular simulations]] one typically uses:

:<math>S(k) = \frac{1}{N} \sum^{N}_{n,m=1} \langle\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m)) \rangle </math>,

where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and
<math>\mathbf{r}_m</math> are the coordinates of particles
<math>n</math> and <math>m</math> respectively.

The dynamic, time dependent structure factor is defined as follows:
:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} \langle \exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0))) \rangle </math>,

The ratio between the dynamic and the static structure factor, <math>S(k,t)/S(k,0)</math>, is known
as the collective (or coherent) intermediate scattering function.
==References==
<references/>
;Related reading
*[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", J. Phys.: Condens. Matter, '''6''' pp. 8415-8427 (1994)]
[[category: Statistical mechanics]]

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Structure factor