Structure factor: Difference between revisions

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


To calculate <math>S(k)</math> in molecular simulations one typically uses:
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} <\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m))> </math>,
:<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
where <math>N</math> is the number of particles and <math>\mathbf{r}_n</math> and
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The dynamic, time dependent structure factor is defined as follows:
The dynamic, time dependent structure factor is defined as follows:
:<math>S(k,t) = \frac{1}{N} \sum^{N}_{n,m=1} <\exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0)))> </math>,
:<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  
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.   
as the collective (or coherent) intermediate scattering function.   
==References==
==References==
#[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)]
<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]]
[[category: Statistical mechanics]]

Revision as of 08:56, 16 September 2011

The structure factor, , for a monatomic system is defined by:


where is the scattering wave-vector modulus

The structure factor is basically a Fourier transform of the pair distribution function ,

At zero wavenumber, i.e. ,

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

Related reading