# Orienational order parameter in isotropic systems

1. Apr 25, 2014

### Liam79

Hi everyone,

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

2. Apr 25, 2014

### nasu

For isotropic distribution the average of cos^2 is 1/3.
The average (these brackets) imply an integral over solid angle of 4π.