 Nematic order parameter

A nematic order parameter measures orientational alignment without distinguishing the two ends of a molecular axis. For unit molecular axes $\mathbf{u}$ in three dimensions, a common tensor convention isIt is symmetric and traceless, and is unchanged by $\mathbf{u}\mapsto -\mathbf{u}$. An isotropic distribution gives $Q=0$ even though individual molecules retain definite axes.

For a uniaxial nematic liquid crystal, this convention gives $Q=S(\mathbf{nn}-I/3)$, where $\mathbf{n}$ is the unoriented nematic director and $S=⟨(3{\cos }^2\theta -1)/2⟩$. Perfect alignment has $S=1$ and eigenvalues $2/3,-1/3,-1/3$. Other normalizations multiply $Q$ by a constant, so coefficients in a Landau-Ginzburg theory must use the same convention. The tensor representation handles head–tail symmetry and permits alignment to vanish at defect cores.