# Dynamic structure factor

Dynamic structure factor is the spatial and temporal Fourier transform of Van Hoves time dependent pair correlation function. It is written as

$$S(k,\omega)= \frac{1}{2\pi}\int F(k,t)\exp(i\omega t)dt$$

$F(k,t)$ is intermediate scattering function. My question is how did we use spatial and temporal Fourier Transform of Van Hoves to get dynamic structure factor? and what does spatial and temporal Fourier transform mean?

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A spatial Fourier transform means a Fourier transform in the spatial variable ($x\rightarrow k$), while a temporal Fourier transform is the same transformation, but in terms of the time variable ($t\rightarrow \omega$). The equation you have written is the (asymmetric) temporal Fourier transform of $F(k,t)$. The spatial transform looks like some variation of $$F(k,t) = \frac{1}{2\pi} \int G(x,t)e^{i k x} dx$$ where $G(x,t)$ is the (one-dimensional) Van Hove function.