= Solution
A <liquid crystal> flows like a liquid while retaining some molecular order. In a <nematic liquid crystal>, elongated molecules have long-range orientational order but no ordinary positional lattice. The <nematic director> $\boldsymbol n$ specifies the preferred axis, with $\boldsymbol n$ and $-\boldsymbol n$ equivalent for head-tail-symmetric molecules. The <nematic order parameter> $S=\langle(3\cos^2\theta-1)/2\rangle$ distinguishes an isotropic distribution, $S=0$, from perfect parallel alignment, $S=1$. A <smectic phase> additionally has a periodic layered density modulation, while molecules remain mobile within layers.
The broken rotational symmetry allows <birefringence> and distinct elastic costs for <nematic splay>, <nematic twist> and <nematic bend>. Under crossed polarizers, rotating a thin aligned sample changes the transmitted intensity because its two optical polarization components acquire different phases. Heating through a nematic-to-isotropic transition removes that orientational birefringence. An <electric field> can rotate the director if the <dielectric anisotropy of a nematic> favors alignment, explaining the optical switching principle of a liquid-crystal cell. Molecular shape, <temperature>, concentration and surface anchoring determine which ordered phase appears.
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