= Solution
A <liquid crystal> has some molecular order while retaining liquid-like motion. The distinction between orientational and positional order separates its main phases.
A <nematic liquid crystal> has long-range orientational order about a <nematic director> $\mathbf n$, with $\mathbf n$ and $-\mathbf n$ equivalent for apolar molecules, but no periodic density lattice. Its <nematic order parameter> is $S=\langle(3\cos^2\theta-1)/2\rangle$, where $\theta$ measures molecular alignment with the director. A <cholesteric liquid crystal> is a chiral nematic whose director rotates in space, producing a helical pitch while remaining locally nematic.
A <smectic phase> adds a periodic density modulation in one direction, forming layers with liquid-like motion within each layer. In a <smectic A phase>, the director is perpendicular to the layers; in a <smectic C phase>, it is tilted relative to the layer normal. A <columnar liquid crystal> instead has columns with a two-dimensional positional lattice and comparatively fluid motion along them.
These orders have different experimental signatures. <Birefringence> under crossed <polarizers> reveals orientational anisotropy, while <X-ray diffraction> can detect a smectic layer spacing $d$ through a peak near $q=2\pi/d$. Optical anisotropy alone does not establish a layer lattice.
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