= Small-angle width of nematic orientation
{title2=$\theta_{\rm rms}\simeq\sqrt{2(1-S)/3}$}
For a narrow <nematic liquid crystal> orientation distribution about its unoriented <nematic director>, expand $S=\langle(3\cos^2\theta-1)/2\rangle=1-3\langle\theta^2\rangle/2+O(\langle\theta^4\rangle)$. This proves the displayed root-mean-square width. It does not determine the full distribution or a full width at half maximum. If the two small transverse tilt components are independent equal-variance Gaussians with variance $\sigma^2$, then $\langle\theta^2\rangle\simeq2\sigma^2$, so $\sigma\simeq\sqrt{(1-S)/3}$.
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