Physical range of a uniaxial nematic order parameter ($-\frac{1}{2}\le Q\le 1$)

For $S=⟨3\mathbf{s}{\mathbf{s}}^{\mathsf{T}}-I⟩=Q(3\mathbf{n}{\mathbf{n}}^{\mathsf{T}}-I)$ and unit molecular orientations $\mathbf{s}$, contraction with $\mathbf{n}$ gives $2Q=3⟨(\mathbf{s}\cdot \mathbf{n}{)}^2⟩-1$. Since the squared projection is between zero and one, the displayed bounds follow. The eigenvalues in this normalization are $2Q,-Q,-Q$. A differently normalized nematic order parameter has correspondingly rescaled bounds; its numerical coefficients cannot be transferred unchanged.