Past exam of the quantum condensed matter course of the University of Cambridge / 2005 / Section A / A1 / a

A nematic liquid crystal has long-range orientational order without a three-dimensional positional lattice. Its nematic director $\mathbf{n}$ identifies the preferred molecular axis, with $\mathbf{n}$ and $-\mathbf{n}$ equivalent for apolar molecules. The scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$ is zero for isotropic orientations and one for perfectly parallel axes. The tensor $Q_{ij}=⟨u_i{u}_j-{\delta }_{ij}/3⟩$ describes alignment without choosing a sign for the director.

Ordering balances orientational entropy against interactions and packing. For long rods, alignment can increase available translational configurations by reducing mutual excluded volume; thermotropic materials additionally have temperature-dependent energetic preferences. Spatially varying alignment costs Frank elastic energy through splay, twist and bend. Surface anchoring and electric or magnetic fields select the director in a finite specimen.

Crossed polarizers reveal birefringence and defect textures, while scattering tests positional order independently. A particularly direct response experiment is the Fréedericksz transition: confine a planar-aligned layer between anchored plates, apply a transverse field and monitor the transmitted polarization as the director tilts above an elastic threshold. This tests orientational elasticity and dielectric anisotropy rather than presuming that an optically anisotropic fluid has crystalline positional order.