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I want to calculate the coherent intermediate scattering function, $S_{\text{coh}} (\mathbf{Q},t)$, from a molecular dynamics trajectory, based on its definition. The definition of the function is:

$$ S_{\text{coh}} (\mathbf{Q},t) = \frac{1}{N} \langle \sum_{j=1}^{N} \sum_{k=1}^{N} exp\left( i \mathbf{Q} \cdot \left[ \mathbf{r}_k (t) - \mathbf{r}_j (0) \right] \right) \rangle \nonumber = \\ = \frac{1}{N} \langle \sum_{j=1}^{N} \sum_{k=1}^{N}cos\left( \mathbf{Q} \cdot \left[ \mathbf{r}_k (t) - \mathbf{r}_j (0) \right] \right) + i \sum_{j=1}^{N} \sum_{k=1}^{N}sin\left( \mathbf{Q} \cdot \left[ \mathbf{r}_k (t) - \mathbf{r}_j (0) \right] \right)\rangle \nonumber \\ $$

where $N$ is the number of particles in the system, $\mathbf{Q}$, is a wave-vector, and $\mathbf{r}_k (t)$ is the position of the $k^{th}$ particle at time $t$.

When people plot $S_{\text{coh}} (\mathbf{Q},t)$ versus time, what is actually being plotted? Do they plot the complex modulus or just the real part?

$$ |S_{\text{coh}} (\mathbf{Q},t)| = \sqrt{Re^2 + Im^2} $$

$Re$ and $Im$ are the real and imaginary parts of $S_{\text{coh}} (\mathbf{Q},t)$, respectively.

PS: The imagenary part of the coherent static structure factor $S_{\text{coh}} (\mathbf{Q},0)$ is zero by definition, but I have yet to find a reason that this should also apply for $S_{\text{coh}} (\mathbf{Q},t)$.

Thank you...

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  • $\begingroup$ While this appears to be a good question in principle, it might be too long to be very useful in the format of this site as it stands. Could you maybe break the subquestions up into separate-standalone questions (you can still refer to the other ones with a link, of course)? $\endgroup$ – Danu May 27 '14 at 17:16
  • $\begingroup$ The radial distribution function $g(r)$ is trivial to compute from MD, and the structure factor $S(q)$ is related to it through a Fourier transform. Why not use this (i.e. am I missing something)? $\endgroup$ – alarge May 28 '14 at 10:23
  • $\begingroup$ @amlrg: The structure factor, S(Q), is independent of time. The question concerns the scattering function, S(Q,t), which is a dynamic property. Thanks for trying to answer. $\endgroup$ – bitted May 28 '14 at 11:52
  • $\begingroup$ I don't work with MD, but in light-scattering the ISF is equal to the electric field auto-correlation function from a system of fluctuating Brownian particles. For an ergodic system, the idea is that the particle positions will shift roughly the same over a certain delay time. So you average over all 'initial' positions and from each 'initial' position all positions specified by a certain delay time. The ISF is actually a dimensionless quantity which generally looks like an exponentially decaying function, where the decay rate relates to the speed that things change in your system. $\endgroup$ – Steve Hatcher Jul 6 '14 at 6:33
  • $\begingroup$ So to somewhat answer your question, what should be plotted is the Value of the ISF vs. the current 'delay time'. $\endgroup$ – Steve Hatcher Jul 6 '14 at 6:37
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The sum of sine terms is mathematically zero if you note that taking the ensemble average <...> requires random sampling of the directions of vector Q. Ususally true for liquids, this isotropic sampling over different solid angles causes each sine term to vanish.

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