Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section A / A1 / a

For a narrowly aligned nematic liquid crystal, the scalar nematic order parameter is $S=⟨(3{\cos }^2\theta -1)/2⟩\simeq 1-\frac{3}{2}⟨{\theta }^2⟩$. Hence the small-angle width of nematic orientation isThis is a root-mean-square angular width about the unoriented nematic director. The single value of $S$ does not determine a complete distribution or its full width at half maximum. For an isotropic two-component Gaussian tilt, each transverse component would have standard deviation ${\theta }_{\mathrm{rms}}/\sqrt{2}$.