Dynamic structure factor ($S(\omega ,\mathbf{k})$)

The dynamic structure factor is the space-time Fourier transform of an equilibrium two-point correlation function. It resolves fluctuations by angular frequency and wavevector and is related to the dissipative part of a retarded Green function by the fluctuation-dissipation theorem.

The static structure factor is the spatial Fourier transform of an equal-time two-point correlation function, with normalization and any disconnected contribution specified. It describes how spatial fluctuations contribute at each transferred wavevector.

If the dynamic structure factor is defined using the same correlation and the Fourier factor $e^{i\omega t}$, thenThe integral sets the time separation to zero. Evaluating the dynamic spectrum only at zero frequency instead selects a different quantity and generally does not produce the equal-time structure factor. A scattering experiment integrated over energy transfer can measure the static quantity after the relevant kinematic and detector corrections. A narrow peak near an ordering wavevector indicates long spatial correlations; it is not by itself a direct measurement of their relaxation time.

For $N$ equal scattering sites on one polymer, define $g(\mathbf{k})=N^{-1}{\sum }_{m,n}⟨e^{i\mathbf{k}\cdot ({\mathbf{R}}_n-{\mathbf{R}}_m)}⟩$. It has $g(0)=N$; its normalized form factor is $g/N$. Under a rotationally invariant joint distribution, each separation vector has a uniform direction at fixed magnitude $r$. The angular average is $\frac{1}{2}{\int }_{-1}^1{e}^{ikr\mu }d\mu =\sin (kr)/(kr)$, with value one at $r=0$. Thus $g$ depends only on $k=|\mathbf{k}|$, and $g(k)=N^{-1}{\sum }_{m,n}⟨\sin (k|{\mathbf{R}}_n-{\mathbf{R}}_m|)/(k|{\mathbf{R}}_n-{\mathbf{R}}_m|)⟩$. The diagonal self terms are essential in a finite chain and give the large-$k$ limit one for a discrete Gaussian chain.

Expand the isotropic angular factor as $\sin (kr)/(kr)=1-k^2{r}^2/6+O(k^4{r}^4)$. For a polymer scattering function and finite fourth pair moments, the constant double sum gives $g(0)=N$, and the quadratic sum is determined by the radius of gyration:In the range $k{R}_g\ll 1$, this is also approximated by $e^{-k^2{R}_g^2/3}$. The coefficient follows from isotropy and the definition of $R_g$ and does not require a Gaussian chain.