Dynamic structure factor

In condensed matter physics, the dynamic 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 which considers correlations in both space and time. Experimentally, it can be accessed most directly by inelastic neutron 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]

S({\vec {k}},\omega )\equiv {\frac {1}{2\pi }}\int _{{-\infty }}^{{\infty }}F({\vec {k}},t){\mbox{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]

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}}}}\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]

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

In an isotropic sample (with scalar r), G(r,t) is a time dependent radial distribution function.

References

1 2 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.
↑ G. Vineyard, "Scattering of Slow Neutrons by a Liquid", Phys. Rev. 110, 999-1010 (1958).

Dynamic structure factor

The van Hove Function

References

Further reading