Kuhn length ($b_K=2l_p$)

The Kuhn length is the link length of an equivalent long ideal chain with the same contour length and mean squared end-to-end distance. A three-dimensional worm-like chain satisfies $⟨R^2⟩\sim 2l_{p}L$ for $L\gg l_p$, hence $b_K=2l_p$.

For fixed bond length $b$, fixed forward bond angle $\alpha$, and independently uniform torsion angles, the bond correlations are $⟨{\mathbf{b}}_i\cdot {\mathbf{b}}_{i+s}⟩=b^2{c}^s$. Thus for $N$ bondsFor fixed $|c|<1$ and a sufficiently long chain, divide by contour length $Nb$ and sum the geometric series to obtain the title formula. Positive forward persistence raises the Kuhn length, and negative correlation lowers it relative to $b$. A chemical internal angle is supplementary to the forward-vector angle. The long-chain formula cannot be used uniformly as $c\to 1$ at fixed $N$: the exact chain obeys $R\le Nb$ and hence $l_K\le Nb$, with equality in the rigid-rod limit.