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 \( S({\vec {k}},\omega ) \), 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:[1]

\( {\displaystyle S({\vec {k}},\omega )\equiv {\frac {1}{2\pi }}\int _{-\infty }^{\infty }F({\vec {k}},t)\exp(i\omega t)\,dt} \)

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 \( G({\vec {r}},t) \) :[2][3]

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

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 ρ {\displaystyle \rho } \rho :

\( {\displaystyle F({\vec {k}},t)={\frac {1}{N}}\langle \rho _{\vec {k}}(t)\rho _{-{\vec {k}}}(0)\rangle } \)

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:[1]

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

It can be rewritten as:

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

In an isotropic sample G(r,t) depends only on the distance r and is the time dependent radial distribution function.
References

Hansen, J. P.; McDonald, I. R. (1986). Theory of Simple Liquids. Academic Press.
van Hove, L. (1954). "Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting Particles". Physical Review. 95 (1): 249. Bibcode:1954PhRv...95..249V. doi:10.1103/PhysRev.95.249.

Vineyard, George H. (1958). "Scattering of Slow Neutrons by a Liquid". Physical Review. 110 (5): 999–1010. doi:10.1103/PhysRev.110.999. ISSN 0031-899X.