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For a narrow nematic liquid crystal orientation distribution about its unoriented nematic director, expand $S=⟨(3{\cos }^2\theta -1)/2⟩=1-3⟨{\theta }^2⟩/2+O(⟨{\theta }^4⟩)$. 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 $⟨{\theta }^2⟩\simeq 2{\sigma }^2$, so $\sigma \simeq \sqrt{(1-S)/3}$.

A smectic A phase has layered positional order with the average molecular orientation along the layer normal. Molecular positions within each layer remain liquid-like. Its orientational axis is described by a nematic director.

A smectic C phase has layered positional order with the nematic director tilted away from the layer normal. The tilt supplies additional orientational information beyond the density modulation of a smectic phase.

In simple shear flow, a persistent polar order parameter with $|\xi |<1$ has no fixed angle and rotates continuously. Integrating $\dot{\theta }=g(\xi \cos 2\theta -1)/2$ through $2\pi$ gives the displayed period. An unoriented nematic director repeats after half that time. At $\xi =0$, the magnitude relaxes independently while the angular speed is $-g/2$.

A uniaxial nematic order parameter has one distinguished nematic director and equal eigenvalues in its perpendicular plane. In the displayed normalization the eigenvalues are $\lambda ,-\lambda /2,-\lambda /2$, giving $\mathrm{Tr}{Q}^2=3{\lambda }^2/2$ and $\mathrm{Tr}{Q}^3=3{\lambda }^3/4$. Positive and negative $\lambda$ describe prolate and oblate orientational anisotropy, respectively; reversing the nematic director leaves the tensor unchanged.