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Structure factor

2011-09-15T16:17:53Z

147.96.5.66:

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 it in computer simulations one typically uses:

:<math>S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left<exp(-i(r_i-r_j))\right> </math>

==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)]
[[category: Statistical mechanics]]

Structure factor

2011-09-15T16:15:55Z

147.96.5.66:

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 it in computer simulations one typically uses:

:<math> S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left<\exp(-i(r_i-r_j))\right> </math>

==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)]
[[category: Statistical mechanics]]

Structure factor

2011-09-15T16:15:14Z

147.96.5.66:

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 it in computer simulations one typically uses:

:<math> S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left<\exp(-i(r_i-r_j))\right>

==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)]
[[category: Statistical mechanics]]

Structure factor

2011-09-15T16:14:29Z

147.96.5.66:

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 it in computer simulations one typically uses:

:<math> S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left<\exp(-i(r_i-r_j))\right>

==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)]
[[category: Statistical mechanics]]