Physics:Dynamic structure factor

Short description: Function in condensed matter physics

In condensed matter physics, the dynamic structure factor (or dynamical 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 that considers correlations in both space and time. Experimentally, it can be accessed most directly by inelastic neutron scattering or X-ray Raman scattering. The dynamic structure factor is most often denoted [math]\displaystyle{ S(\vec{k},\omega) }[/math], where [math]\displaystyle{ \vec{k} }[/math] (sometimes [math]\displaystyle{ \vec{q} }[/math]) is a wave vector (or wave number for isotropic materials), and [math]\displaystyle{ \omega }[/math] a frequency (sometimes stated as energy, [math]\displaystyle{ \hbar\omega }[/math]). It is defined as:

[math]\displaystyle{ S(\vec{k},\omega) \equiv \frac{1}{2\pi}\int_{-\infty}^{\infty} F(\vec{k},t)\exp(i\omega t)\,dt }[/math]

Here [math]\displaystyle{ 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]\displaystyle{ G(\vec{r},t) }[/math]:

[math]\displaystyle{ 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 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]\displaystyle{ \rho }[/math]:

[math]\displaystyle{ F(\vec{k},t) = \frac{1}{N}\langle \rho_{\vec{k}}(t)\rho_{-\vec{k}}(0) \rangle }[/math]

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

[math]\displaystyle{ \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]\displaystyle{ a }[/math] is the scattering length.

The van Hove function

The van Hove function for a spatially uniform system containing [math]\displaystyle{ N }[/math] point particles is defined as:

[math]\displaystyle{ G(\vec{r},t) = \left\langle \frac{1}{N} \int \sum_{i=1}^{N}\sum_{j=1}^N \delta[\vec{r}'+\vec{r}-\vec{r}_j(t)]\delta[\vec{r}'-\vec{r}_i(0)] d\vec{r}' \right\rangle }[/math]

It can be rewritten as:

[math]\displaystyle{ G(\vec{r},t) = \left\langle \frac{1}{N}\int \rho(\vec{r}'+\vec{r},t)\rho(\vec{r}',0) d\vec{r}'\right\rangle }[/math]

Further reading

• Ashcroft, Neil W.; Mermin, N. David (1976). Solid State Physics (Appendix N). Holt, Rinehart and Winston. ISBN 978-0-03-083993-1. https://archive.org/details/solidstatephysic00ashc. 
• Lovesey, Stephen W. (1986). Theory of Neutron Scattering from Condensed Matter - Volume I: Nuclear Scattering. Oxford University Press. ISBN:9780198520283.