Editing Structure factor

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 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]]
@@ Line 1: / Line 1: @@
-The '''static structure factor''', <math>S(k)</math>, for a monatomic system composed of spherical scatterers is defined by (Eq. 1 in <ref>[http://dx.doi.org/10.1088/0953-8984/6/41/006 A. Filipponi, "The radial distribution function probed by X-ray absorption spectroscopy", Journal of Physics: Condensed Matter  '''6''' pp.  8415-8427 (1994)]</ref>):
+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) ~{\mathrm {d}}r</math>
-where <math>g_2(r)</math> is the [[radial distribution function]], and <math>k</math> is the scattering wave-vector modulus
+:<math>S(k) = 1 + \frac{4 \pi \rho}{k} \int_0^{\infty} ( g_2(r) -1 ) r \sin (kr) ~dr</math>
-:<math>k= |\mathbf{k}|= \frac{4 \pi }{\lambda} \sin \left( \frac{\theta}{2}\right)</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>,
@@ Line 17: / Line 18: @@
 from which one can calculate the [[Compressibility | isothermal compressibility]].
-To calculate <math>S(k)</math> in [[Computer simulation techniques |molecular simulations]] one typically uses:
+To calculate <math>S(k)</math> in computer 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>,
+:<math>S(k) = \frac{1}{N} \sum^{N}_{i,j=1} \left<exp(-i(r_i-r_j))\right> </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.
-==Binary mixtures==
-<ref>[http://dx.doi.org/10.1080/14786436508211931 T. E. Faber and J. M. Ziman "A theory of the electrical properties of liquid metals III. the resistivity of binary alloys", Philosophical Magazine '''11''' pp. 153-173 (1965)]</ref><ref>[http://dx.doi.org/10.1103/PhysRev.156.685 N. W. Ashcroft and David C. Langreth "Structure of Binary Liquid Mixtures. I", Physical Review '''156''' pp. 685–692 (1967)]</ref><ref>[http://dx.doi.org/10.1103/PhysRevB.2.3004 A. B. Bhatia and D. E. Thornton "Structural Aspects of the Electrical Resistivity of Binary Alloys", Physical Review B '''2''' pp. 3004-3012 (1970)]</ref>
 ==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)]
-;Related reading
-*[http://dx.doi.org/10.1007/BF01391926 F. Zernike and J. A. Prins "Die Beugung von Röntgenstrahlen in Flüssigkeiten als Effekt der Molekülanordnung", Zeitschrift für Physik '''41''' pp. 184-194 (1920)]
-*P. Debye and H. Menke "", Physik. Zeits. '''31''' pp. 348- (1930)
-*B. E. Warren "X-Ray Diffraction", Dover Publications (1969) ISBN 0486663175 &sect; 10.4
-*Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids" (Third Edition) [http://dx.doi.org/10.1016/B978-012370535-8/50006-9  Chapter 4: "Distribution-function Theories"] &sect; 4.1
 [[category: Statistical mechanics]]