Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A4 / b

The Kuhn length $l_K$ is the segment length of an equivalent freely jointed chain chosen to match contour length $L$ and the unperturbed mean-square end-to-end distance: $⟨R^2⟩=L{l}_K$. Since $R\le L$, its maximum finite-chain value is $l_K=L$, attained by a straight rigid rod. Independent steps of length $b$ have $L=Nb$ and $⟨R^2⟩=N{b}^2$, soFor a long freely rotating chain with fixed angle $\alpha$ between successive forward bond vectors, orientational correlations decay as $c^s$ with $c=\cos \alpha$. Summing them gives the Kuhn length of a freely rotating chain:Positive persistence increases the statistical segment length, while negative bond correlations can reduce it. A chemical internal bond angle is supplementary to this forward-vector angle, so the cosine convention matters. Near the perfectly aligned limit the finite-chain bound $l_K\le L$ replaces the long-chain asymptotic formula.