Effective orientation angle from nematic order (${\theta }_{\mathrm{eff}}=\arccos \sqrt{(2S+1)/3}$)

For scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$, the displayed angle has ${\cos }^2{\theta }_{\mathrm{eff}}=⟨{\cos }^2\theta ⟩$. It summarizes a second moment, not the literal mean angle. For example an isotropic orientation distribution has $S=0$, giving ${\theta }_{\mathrm{eff}}=\arccos (1/\sqrt{3})$; the average acute angle to its unoriented axis is instead one radian, obtained by integrating $\theta \sin \theta$ over $[0,\pi /2]$.