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Write the end-to-end random vector as $\mathbf{r}={\sum }_{j=1}^{N}b{\mathbf{u}}_j$, with independent isotropically distributed unit vectors. At zero applied force, $⟨{\mathbf{u}}_j⟩=0$, ${\mathbf{u}}_j\cdot {\mathbf{u}}_j=1$, and for $i\ne j$ independence gives $⟨{\mathbf{u}}_i\cdot {\mathbf{u}}_j⟩=0$. ThereforeHere “size” is the root mean square end-to-end distance of a polymer chain. It is not the magnitude of the mean random vector, which is zero, nor the mean distance $⟨r⟩$. In the long-chain Gaussian limit of the end-to-end distribution of a freely jointed chain, $⟨r⟩\simeq b\sqrt{8N/(3\pi )}$, illustrating why these definitions cannot be interchanged.

Each Cartesian component of one link has zero mean and variance $b^2/3$, with zero cross-component covariance. By the multivariate central limit theorem, for large $N$ on the coil scale the vector probability density isIt is normalized with respect to $d^{3}r$; the radial probability density is $4\pi r^{2}p(\mathbf{r})$. The Gaussian limit of the end-to-end distribution of a freely jointed chain is an approximation, not the exact finite-$N$ law near $r=Nb$.

At prescribed vector separation the configurational entropy is $S(\mathbf{r})=k_B\ln p(\mathbf{r})+\text{constant}$. The ideal chain has no extension-dependent internal energy, so its Helmholtz free energy isThe externally required force is ${\nabla }_{\mathbf{r}}A$, while the chain's restoring force has the opposite sign. ThusThe last equation also follows by multiplying $p(\mathbf{r})$ by the Boltzmann factor $e^{fz/(k_{B}T)}$ and completing the square. This entropic spring resists extension because fewer link configurations realize a stretched chain. At the same small applied force, increasing temperature reduces extension; at the same extension it increases the required force. Do not differentiate $-k_{B}T\ln [4\pi r^{2}p(\mathbf{r})]$ to obtain the fixed-vector Hooke law, since that radial measure includes an additional geometric multiplicity. The low-force condition is $fb/(k_{B}T)\ll 1$, equivalently $⟨z⟩\ll Nb$.

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.

An ideal chain can be treated as a random walk when effective segment orientations are independent and isotropic, segment length is fixed, and interactions such as long-range excluded volume are absent or effectively screened. A large number of segments is needed for the Gaussian continuum approximation. Write $\mathbf{R}={\sum }_{j=1}^N{\mathbf{a}}_j$. Each Cartesian component has mean zero and variance $N{a}^2/3$, so the central limit theorem gives the Gaussian limit of the end-to-end distribution of a freely jointed chainEquivalently, one step has characteristic function $\sin (ka)/(ka)$; taking its $N$th power and expanding its logarithm gives $e^{-N{a}^2{k}^2/6}$, whose inverse Fourier transform is this density. It is a density per vector volume $d^{3}R$. The scalar end-to-end-distance density is $4\pi R^{2}p(\mathbf{R},N)$. Thus the printed expression must be interpreted as a vector density; it is also a large-$N$ approximation, not the exact finite-chain distribution near $R=Na$.

For a freely rotating chain, neighboring forward bond vectors have fixed angle $\theta$, and independently uniform torsion angles give $⟨{\mathbf{a}}_i\cdot {\mathbf{a}}_j⟩=a^2{\alpha }^{|j-i|}$ with $\alpha =\cos \theta$. Squaring the sum of bonds therefore givesThere are $N-k$ pairs separated by $k$ bonds, so the characteristic ratio of a freely rotating chain isFor fixed $|\alpha |<1$, the geometric sums give $C_N=(1+\alpha )/(1-\alpha )-2\alpha (1-{\alpha }^N)/[N(1-\alpha {)}^2]$. The correction vanishes as $N\to \infty$, yieldingFor the specified polyethylene parameters, $\alpha =\cos (-70.5^{\circ })\simeq 0.333807$, so $C_{\infty }\simeq 2.00213$ and $C_{9160}\simeq 2.00197$. HenceThe negative sign of the stated angle does not affect its cosine. The forward-bond angle is supplementary to the chemical internal bond angle. This freely rotating estimate includes fixed-angle correlations but omits torsional energy preferences; the Kuhn length of a freely rotating chain expresses the same long-chain persistence as $b_K=a{C}_{\infty }$.