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I have what may be a dummy question. In NMR or in the study of liquid crystals for example, an order parameter $S$ is often used: $$ S=\left\langle\frac{1}{2}\left(3\cos^2\theta−1\right)\right\rangle $$ with $\theta$ the angle of the molecule with a "director" (the magnetic field in NMR, the normal to a membrane for lipids, the global direction in a nematic phase etc). $S$ corresponds to a second-order Legendre polynomial. I have often read that in an isotropic environment, $S=0$ whereas when all the molecules are well aligned with the reference vector (director), $S=1$. I understand why $S=1$ as $\theta=0$° but I can't find why $S=0$ when all the orientations are random. Can anyone help me?

Liam

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For an isotropic substance, $$S=\left\langle\frac{1}{2}\left(3\cos^2\theta−1\right)\right\rangle=\frac{1}{8\pi^2}\int_0^{\pi}\sin(\theta)\,d\theta\int_0^{2\pi}d\phi\int_0^{2\pi}d\chi\,\frac{1}{2}\left(3\cos^2\theta−1\right)=0.$$

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