Dynamic structure factor

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 $S(\overrightarrow{k},\omega )$, where $\overrightarrow{k}$ (sometimes $\overrightarrow{q}$) is a wave vector (or wave number for isotropic materials), and $\omega$ a frequency (sometimes stated as energy, $\hslash \omega$). It is defined as:

$S(\overrightarrow{k},\omega )\equiv \frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}F(\overrightarrow{k},t)\exp (i\omega t)dt$

Here $F(\overrightarrow{k},t)$, 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 $G(\overrightarrow{r},t)$:

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

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 $\rho$:

$F(\overrightarrow{k},t)=\frac{1}{N}⟨{\rho }_{\overrightarrow{k}}(t){\rho }_{-\overrightarrow{k}}(0)⟩$

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

$\frac{d^2\sigma }{d\Omega d\omega }=a^2{\left(\frac{E_f}{E_i}\right)}^{1/2}S(\overrightarrow{k},\omega )$

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:

$G(\overrightarrow{r},t)=⟨\frac{1}{N}\int \sum _{i=1}^N\sum _{j=1}^N\delta [{\overrightarrow{r}}^{\prime }+\overrightarrow{r}-{\overrightarrow{r}}_j(t)]\delta [{\overrightarrow{r}}^{\prime }-{\overrightarrow{r}}_i(0)]d{\overrightarrow{r}}^{\prime }⟩$

It can be rewritten as:

$G(\overrightarrow{r},t)=⟨\frac{1}{N}\int \rho ({\overrightarrow{r}}^{\prime }+\overrightarrow{r},t)\rho ({\overrightarrow{r}}^{\prime },0)d{\overrightarrow{r}}^{\prime }⟩$

Further reading

• Lovesey, Stephen W. (1986). Theory of Neutron Scattering from Condensed Matter - Volume I: Nuclear Scattering. Oxford University Press. .