title: Dynamic structure factor
text: # 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:
  
      $S({\vec {k}},\omega )\equiv {\frac {1}{2\pi }}\int _{-\infty }^{\infty }F({\vec {k}},t)\exp(i\omega t)\,dt$
  
  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 $G({\vec {r}},t)$:
  
      $F({\vec {k}},t)\equiv \int G({\vec {r}},t)\exp(-i{\vec {k}}\cdot {\vec {r}})\,d{\vec {r}}$
  
  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:
  
      $G({\vec {r}},t)=\left\langle {\frac {1}{N}}\int \rho ({\vec {r}}'+{\vec {r}},t)\rho ({\vec {r}}',0)d{\vec {r}}'\right\rangle$
  