The Van Hove function describes the time dependent pair correlation between atoms,
\[G(r, t) = \frac{1}{4\pi \rho_0Nr^2}\sum_{i,j}\delta(r - |\vec{r}_i(0) - \vec{r}_j(t)|)\]\(i\) and \(j\) runs over all the \(N\) atoms in the system. \(\vec{r}_i\) refers to the position of the atom \(i\) at time \(t\). The following formula describes the fundamental construction of the Van Hove function from the dynamic structure factor \(S(Q, E)\),
\[G(r, t) = \frac{1}{2\pi\rho_0r}\int S(Q, E)sin(Qr)e^{i\omega t}Q\,dQ\,dE\]For more details and references about the formulation, refer to Ref. [1, 2]. From the above formulation, there are two main immediate derivations to reiterate,
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If we integrate out the energy (as mathematically shown in Eqn. 6 in Ref. [1]), as for a normal pair distribution function (PDF) measurement without discriminating the incoming and outgoing energy, that is just to set \(t = 0\), in the formula above – the exponential term in the Fourier transform of the energy part accordingly becomes \(1\), and we can pull out the energy relevant terms to have Eqn. 6. in Ref. [1]. Such a mathematical result indicates that in a normal PDF measurement, we are actually measuring the equal-time correlation (i.e., simultaneously catch atoms with respect to the pair distance – since we can only measure pair distances without knowing sample position directly due to the phase problem). In contrast, the non-equal-time correlation (i.e., \(t \neq 0\)) refers to something like catching atom A at \(t = 0\) but catching atom B at \(t=t_1\), again, with respect to the pair distance, in which case we then get the evolution of the real-space correlation with time.
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If we can somehow measure only the elastic scattering, that means the \(S(Q, E)\) function is a Dirac \(\delta\) function in the energy space and we know that the Fourier transform of a Dirac \(\delta\) function is a constant in the coupled space. Therefore, such a measurement will yield a static picture of the atomic configuration, i.e., correlation is constant in time space, or, atoms ‘not moving’ in the measured picture.