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Orienational order parameter in isotropic systems

  1. Apr 25, 2014 #1
    Hi everyone,

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

    Liam
     
  2. jcsd
  3. Apr 25, 2014 #2
    For isotropic distribution the average of cos^2 is 1/3.
    The average (these brackets) imply an integral over solid angle of 4π.
     
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