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

:$S(\vec,\omega) \equiv \frac\int_^ F(\vec,t)\exp(i\omega t)\,dt$

Here $F(\vec,t)$, 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 $G(\vec,t)$:

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

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 $\rho$:

:$F(\vec,t) = \frac\langle \rho_(t)\rho_(0) \rangle$

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

:$\frac = a^2\left(\frac\right)^ S(\vec,\omega)$

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,t) = \left\langle \frac \int \sum_^\sum_^N \delta vec'+\vec-\vec_j(t)delta vec'-\vec_i(0)d\vec' \right\rangle
It can be rewritten as:
:$G(\vec,t) = \left\langle \frac\int \rho(\vec'+\vec,t)\rho(\vec',0) d\vec'\right\rangle$

References

Further reading

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

Condensed matter physics
Neutron scattering

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