Orienational order parameter in isotropic systems

In summary, the conversation is about the use of an order parameter, S, in NMR and liquid crystal studies. S represents the angle of a molecule with a reference vector and corresponds to a second-order Legendre polynomial. It is often observed that in an isotropic environment, S=0, and in a well-aligned environment, S=1. The reason for S=0 in an isotropic environment is due to the average of cos^2 being 1/3 over a solid angle of 4π.
  • #1
Liam79
1
0
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
 
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  • #2
For isotropic distribution the average of cos^2 is 1/3.
The average (these brackets) imply an integral over solid angle of 4π.
 

Related to Orienational order parameter in isotropic systems

1. What is an orientational order parameter in isotropic systems?

An orientational order parameter is a measure of the degree of alignment or arrangement of molecules in an isotropic (random) system. It provides information about the anisotropy (directional dependence) of the system, specifically the degree of order or disorder in the orientation of molecules.

2. How is the orientational order parameter calculated?

The orientational order parameter is typically calculated using a mathematical equation that takes into account the orientations of individual molecules in a system. This equation can vary depending on the specific system and the type of order parameter being used, but it often involves calculating the average or variance of the orientational distribution function.

3. What is the significance of the orientational order parameter in isotropic systems?

The orientational order parameter is important because it provides insight into the structure and dynamics of molecules in a system. It can reveal information about phase transitions, molecular interactions, and other physical properties of the system. It is also useful in characterizing the behavior of liquid crystals, polymers, and other complex systems.

4. How does the orientational order parameter differ from other order parameters?

The orientational order parameter specifically measures the alignment or arrangement of molecules in a system, while other order parameters may measure different aspects such as positional order or dipole moments. Additionally, the orientational order parameter is typically used in isotropic systems, whereas other order parameters may be used in systems with more specific structures or symmetries.

5. Can the orientational order parameter be experimentally measured?

Yes, the orientational order parameter can be measured using various experimental techniques such as X-ray diffraction, NMR spectroscopy, or polarized light microscopy. These methods involve analyzing the scattering, spectroscopic, or optical properties of a system to determine the degree of orientational order. However, the specific technique used will depend on the system being studied and the type of order parameter being measured.

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