Double layer potential: Difference between revisions

Latest revision as of 12:03, 3 December 2013

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 $S (\vec{k}, ω)$ , where $\vec{k}$ (sometimes $\vec{q}$ ) is a wave vector (or wave number for isotropic materials), and $ω$ a frequency (sometimes stated as energy, $ℏ ω$ ). It is defined as:^[1]

S (\vec{k}, ω) \equiv \frac{1}{2 π} \int_{- \infty}^{\infty} F (\vec{k}, t) exp (i ω t) d t

Here $F (\vec{k}, t)$ , 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 $G (\vec{r}, t)$ :^[2]^[3]

F (\vec{k}, t) \equiv \int G (\vec{r}, t) \exp (- i \vec{k} \cdot \vec{r}) d \vec{r}

Thus we see that the dynamical structure factor is the spatial and temporal Fourier transform of 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 $ρ$ :

F (\vec{k}, t) = \frac{1}{N} ⟨ ρ_{\vec{k}} (t) ρ_{- \vec{k}} ⟩

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

\frac{d^{2} σ}{d Ω d ω} = a^{2} {(\frac{E_{f}}{E_{i}})}^{1 / 2} S (\vec{k}, ω)

where $a$ is the scattering length.

The van Hove Function

The van Hove Function for a spatially uniform system containing $N$ point particles is defined as:^[1]

G (\vec{r}, t) = ⟨ \frac{1}{N} \int \sum_{i = 1}^{N} \sum_{j = 1}^{N} δ [{\vec{r}}^{'} + \vec{r} - {\vec{r}}_{j}] δ [{\vec{r}}^{'} - {\vec{r}}_{i} (0)] d {\vec{r}}^{'} ⟩

It can be rewritten as:

G (\vec{r}, t) = ⟨ \frac{1}{N} \int ρ ({\vec{r}}^{'} + \vec{r}, t) ρ ({\vec{r}}^{'}, 0) d {\vec{r}}^{'} ⟩

In an isotropic sample (with scalar r), G(r,t) is a time dependent radial distribution function.

References

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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↑ G. Vineyard, "Scattering of Slow Neutrons by a Liquid", Phys. Rev. 110, 999-1010 (1958).

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[3] G. Vineyard, "Scattering of Slow Neutrons by a Liquid", Phys. Rev. 110, 999-1010 (1958).

[1]

[2]

[3]

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+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]]

Double layer potential: Difference between revisions

Latest revision as of 12:03, 3 December 2013

The van Hove Function

References

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