131.111.165.150 at 11:03, 3 December 2013

2013-12-03T11:03:22Z

← Older revision		Revision as of 13:03, 3 December 2013
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	~~Dusty Ledbetter~~ is ~~his name whilst~~ it ~~is not his delivery title~~. The ~~position he's been occupying~~ for ~~many years is~~ a ~~dispatcher~~. ~~New Jersey~~ is ~~wherever he is generally been dwelling~~. ~~Home brewing is the only passion his wife would not~~ [~~http~~://~~thesaurus.com/browse/approve approve~~] of. He is ~~been doing~~ the ~~job on his~~ [~~http~~:~~//www~~.~~google.com/search?q~~=~~web+site&btnI~~=~~lucky web site~~] ~~for some time now~~. ~~Test it out right here: http:~~//~~www~~.~~evcportfolio~~.nl/~~aspnet_client~~/~~nike/air~~-~~max-90~~.~~html~~<br><br>~~Here~~ is ~~my web page; Nike air max 90 (~~[~~http~~://~~www~~.~~evcportfolio.nl~~/~~aspnet_client~~/~~nike/air~~-~~max~~-90.~~html just click the up coming site~~])		In [[condensed matter physics]], the '''dynamic structure factor''' is a mathematical function that contains information about inter-particle correlations and their time evolution. It is a generalization of the [[structure factor]] which considers correlations in both space ''and'' time. Experimentally, it can be accessed most directly by [[inelastic neutron scattering]].

			The dynamic structure factor is most often denoted <math>S(\vec{k},\omega)</math>, where <math>\vec{k}</math> (sometimes <math>\vec{q}</math>) is a wave vector (or wave number for isotropic materials), and <math>\omega</math> a frequency (sometimes stated as energy, <math>\hbar\omega</math>). It is defined as:<ref name = "theoryliquids">
			{{cite book
			\|first=J.P. \|last=Hansen \|first2=I.R. \|last2=McDonald
			\|title=Theory of Simple Liquids
			\|publisher=[[Academic Press]]
			\|page=
			\|year=1986
			\|isbn=
			}}</ref>

			:<math>S(\vec{k},\omega) \equiv \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\vec{k},t)\mbox{exp}(i\omega t) dt</math>

			Here <math>F(\vec{k},t)</math>, is called the '''intermediate scattering function''' and can be measured by [[neutron spin echo]] spectroscopy. The intermediate scattering function is the spatial [[Fourier transform]] of the '''van Hove function''' <math>G(\vec{r},t)</math>:<ref>{{cite journal
			\|last=van Hove \|first=L.
			\|title=Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting Particles
			\|volume=95 \|issue=1 \|page=249
			\|journal=[[Physical Review]]
			\|year=1954
			\|doi=10.1103/PhysRev.95.249
			\|bibcode = 1954PhRv...95..249V }}</ref><ref>G. Vineyard, "Scattering of Slow Neutrons by a Liquid", Phys. Rev. 110, 999-1010
			(1958).</ref>

			:<math>F(\vec{k},t) \equiv \int G(\vec{r},t)\exp (-i\vec{k}\cdot\vec{r}) d\vec{r}</math>

			Thus we see that the dynamical structure factor is the spatial ''and'' temporal Fourier transform of [[Leon van Hove\|van Hove]]'s time-dependent pair correlation function. It can be shown (see below), that the intermediate scattering function is the correlation function of the Fourier components of the density <math>\rho</math>:

			:<math>F(\vec{k},t) = \frac{1}{N}\langle \rho_{\vec{k}}(t)\rho_{-\vec{k}} \rangle</math>

			The dynamic structure is exactly what is probed in coherent inelastic neutron scattering. The [[differential cross section]] is :

			:<math>\frac{d^2 \sigma}{d\Omega d\omega} = a^2\left(\frac{E_f}{E_i}\right)^{1/2} S(\vec{k},\omega) </math>

			where <math>a</math> is the [[scattering length]].

			==The van Hove Function==
			The van Hove Function for a spatially uniform system containing <math>N</math> point particles is defined as:<ref name = "theoryliquids"/>
			:<math>G(\vec{r},t) = \langle \frac{1}{N} \int \sum_{i=1}^{N}\sum_{j=1}^N \delta[\vec{r}'+\vec{r}-\vec{r}_j]\delta[\vec{r}'-\vec{r}_i(0)] d\vec{r}' \rangle</math>
			It can be rewritten as:
			:<math>G(\vec{r},t) = \langle \frac{1}{N}\int \rho(\vec{r}'+\vec{r},t)\rho(\vec{r}',0) d\vec{r}'\rangle </math>

			In an isotropic sample (with scalar ''r''), ''G(r,t)'' is a time dependent [[radial distribution function]].


			== References ==
			{{reflist}}

			== Further reading ==
			*

			[[Category:Condensed matter physics]]
			[[Category:Neutron scattering]]

70.182.68.199: Clarified that normal derivative is w.r.t y variable and added a constant to make the two definitions consistent.

2011-07-22T16:31:43Z

Clarified that normal derivative is w.r.t y variable and added a constant to make the two definitions consistent.

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Double layer potential - Revision history

131.111.165.150 at 11:03, 3 December 2013

70.182.68.199: Clarified that normal derivative is w.r.t y variable and added a constant to make the two definitions consistent.