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Independence of the segment vectors makes the end-to-end distribution an $N$-fold convolution. Use the characteristic function convention $\hat{\psi }(\mathbf{k})=\int e^{i\mathbf{k}\cdot \mathbf{r}}\psi (\mathbf{r})d^{3}r$. For a three-dimensional isotropic freely jointed chain, the radial delta function fixes $|\mathbf{r}|=b$, and angular integration givesThe characteristic function of a sum of independent variables therefore gives $\hat{\Phi }(\mathbf{k},N)=[\sin (kb)/(kb){]}^N$. In the central large-$N$ scaling, $k=O((b\sqrt{N}{)}^{-1})$, soInverting this Gaussian Fourier transform yieldsThis Gaussian limit of the end-to-end distribution of a freely jointed chain is normalized with respect to $d^{3}R$, has $⟨\mathbf{R}⟩=0$ and $⟨R^2⟩=N{b}^2$, and is valid on the coil scale $R=O(b\sqrt{N})$. If a density of the scalar distance $R$ is wanted instead, it is $4\pi R^2\Phi$. The Gaussian approximation does not describe the nearly fully stretched tail or the exact finite-chain cutoff at $R=Nb$.

For the scattering calculation, radial single-segment distributions together with the given independent-segment assumption imply rotational invariance of the whole chain. Therefore each pair separation $\mathbf{S}={\mathbf{R}}_n-{\mathbf{R}}_m$ has a uniform direction conditional on its magnitude $S$. The same angular average givesAveraging its magnitude and then summing the pairs proves the required expression for the polymer scattering function:The ratio has value one at zero separation, including every diagonal term $m=n$.

The printed gyration formula leaves $m$ unsummed. The intended radius of gyration must use a double pair sum, equivalently the mean square distance from the chain's center of mass. To establish that equivalence, put ${\mathbf{R}}_{\mathrm{cm}}=N^{-1}{\sum }_n{\mathbf{R}}_n$ and expand both sides ofThus the consistent definition is $R_g^2=(2N^2{)}^{-1}{\sum }_{m,n}⟨|{\mathbf{R}}_m-{\mathbf{R}}_n{|}^2⟩$. Expanding $\sin z/z=1-z^2/6+O(z^4)$ givesTherefore the general small-$k$ result isThis Guinier expansion of a polymer scattering function requires isotropy and finite pair moments, but not Gaussian distributions.

For a Gaussian chain, each pair difference across $\ell =|n-m|$ links has variance $⟨|{\mathbf{R}}_n-{\mathbf{R}}_m{|}^2⟩=b^2\ell$ and characteristic function $e^{-k^2{b}^2\ell /6}$. Using exactly the $N$ scattering sites in the printed double sum, the finite-chain expression isThe closed expression is understood by continuity at $z=1$, where it equals $N$. The factor $N-\ell$ counts pairs with $n=m+\ell$, with the prefactor two supplying their reversed order. In this finite-site convention,This also holds for a freely jointed chain because its link covariances give the same mean square pair distances.

To obtain the complete long-chain scaling function, set $x=N{k}^2{b}^2/6\simeq k^2{R}_g^2$ and replace contour sums by integrals, keeping $x$ fixed as $N\to \infty$. With $a=k^2{b}^2/6$,The requested Debye scattering function for a Gaussian chain is consequentlyexactly for the continuous Gaussian contour convention $R_g^2=N{b}^2/6$, and asymptotically for the long discrete chain. Its small-$x$ expansion is $\mathcal{D}(x)=1-x/3+x^2/12+\cdots$, agreeing with the general gyration result, and $\mathcal{D}(x)\sim 2/x$ at large $x$. In the continuum coil range $1\ll x\ll N$ this gives $g\sim 12/(k^2{b}^2)$. At still larger $kb$, the discrete expression instead tends to the self-scattering value one.

There is a harmless finite-size convention in the source: $N$ links have $N+1$ endpoints, whereas its scattering sum counts $N$ sites. The formulas above retain its scattering normalization; if all endpoints are counted, replace the site count by $N+1$ in the finite formulas. The large-$N$ Gaussian distribution and Debye scaling function are unchanged to leading order.