Difference between revisions of "Structure factor"

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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>,
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:<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, 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

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 {\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,

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

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}_{n,m=1} \langle\exp(-i\mathbf{k}(\mathbf{r}_n-\mathbf{r}_m)) \rangle ,

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}_{n,m=1}  \langle \exp(-i\mathbf{k}(\mathbf{r}_n(t)-\mathbf{r}_m(0))) \rangle ,

The ratio between the dynamic and the static structure factor, S(k,t)/S(k,0), is known as the collective (or coherent) intermediate scattering function.

References

Related reading