= Logarithmic energy of a planar nematic disclination
{title2=$E_q=\pi K\lambda_0^2q^2\log(L/r_0)$}
With constant amplitude outside a core, the <planar nematic divergence-elasticity identity> reduces the elastic density to $K\lambda_0^2|\nabla\chi|^2/2$. A <topological charge of a two-dimensional nematic disclination> $q$ has $\chi=q\theta+\chi_0$ and $|\nabla\chi|^2=q^2/r^2$. Integrating over an annulus gives the displayed energy, plus finite core and boundary terms. The outer cutoff is the sample size, compensating-defect separation, or another screening length. An isolated defect in an infinite ordered plane has divergent energy. A neutral pair $q,-q$ separated by $R$ instead has $E_{\rm pair}=2\pi K\lambda_0^2q^2\log(R/r_0)$ plus core terms, since its far-field <gradients> cancel.
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