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
S(\vec{k},\omega)
\vec{k}
\vec{q}
\omega
\hbar\omega
S(\vec{k},\omega)\equiv
| 1 | |
| 2\pi |
| infty | |
| \int | |
| -infty |
F(\vec{k},t)\exp(i\omegat)dt
Here
F(\vec{k},t)
G(\vec{r},t)
F(\vec{k},t)\equiv\intG(\vec{r},t)\exp(-i\vec{k} ⋅ \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
\rho
F(\vec{k},t)=
| 1 | |
| N |
\langle\rho\vec{k
The dynamic structure is exactly what is probed in coherent inelastic neutron scattering. The differential cross section is :
| d2\sigma | |
| d\Omegad\omega |
=
| ||||
| a |
\right)1/2S(\vec{k},\omega)
where
a
The van Hove function for a spatially uniform system containing
N
G(\vec{r},t)=\left\langle
| 1 | |
| N |
\int
| N | |
| \sum | |
| i=1 |
| N | |
| \sum | |
| j=1 |
\delta[\vec{r}'+\vec{r}-\vec{r}j(t)]\delta[\vec{r}'-\vec{r}i(0)]d\vec{r}'\right\rangle
G(\vec{r},t)=\left\langle
| 1 | |
| N |
\int\rho(\vec{r}'+\vec{r},t)\rho(\vec{r}',0)d\vec{r}'\right\rangle