Guinier expansion of a polymer scattering function ($g(k)/N=1-k^2{R}_g^2/3+O(k^4)$)

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.