Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section A / A3 / Solution

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 $b_K=a{C}_{\infty }$.