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Order Parameter

The order parameter, typically denoted as \vec{n}, quantifies the degree of orientational order in a liquid crystal phase, representing the average direction of molecular alignment.
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The statement of the theorem

Define the nematic director n⃗\vec{n} as the unit vector quantifying the average local orientation of the molecular long axes: n⃗=⟨u/∣u∣⟩/⟨1/∣u∣⟩\vec{n} = \langle \mathbf{u} / |\mathbf{u}| \rangle / \langle 1 / |\mathbf{u}| \rangle, where u\mathbf{u} is the unit vector along a molecule's axis. The degree of orientational order is quantified by the second-rank tensor QijQ_{ij}, the order parameter tensor: \nQij=⟨uiuj−13δijI⟩Q_{ij} = \langle u_i u_j - \frac{1}{3} \delta_{ij} \mathbf{I} \rangle \nFor uniaxial systems, this simplifies to Qij=S(ninj−13δijI)Q_{ij} = S (n_i n_j - \frac{1}{3} \delta_{ij} \mathbf{I}), where SS is the scalar order parameter (0≤S≤10 \le S \le 1).