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

{{cite book

|first1=J. P. |last1=Hansen |first2=I. R. |last2=McDonald

|title=Theory of Simple Liquids

|publisher=Academic Press

|page=

|year=1986

|isbn=

}}

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

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 $G(\vec{r},t)$ :{{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 }}{{cite journal|last1=Vineyard|first1=George H.|title=Scattering of Slow Neutrons by a Liquid|journal=Physical Review|volume=110|issue=5|year=1958|pages=999–1010|issn=0031-899X|doi=10.1103/PhysRev.110.999|bibcode=1958PhRv..110..999V }}

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

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 $\rho$ :

: $F(\vec{k},t) = \frac{1}{N}\langle \rho_{\vec{k}}(t)\rho_{-\vec{k}}(0) \rangle$

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

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

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:

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

It can be rewritten as:

: $G(\vec{r},t) = \left\langle \frac{1}{N}\int \rho(\vec{r}'+\vec{r},t)\rho(\vec{r}',0) d\vec{r}'\right\rangle$

dynamic structure factor

The van Hove function

References

Further reading