Birefringence of a Rigid Rod Polymer

The Lorentz-Lorenz equation can be used to express the components of the refractive index tensor in terms of the polarizability tensor. Recognizing that the birefringence normalized by the mean refractive index is normally very small, ( A/i / 1), it is assumed that Aa /a 1, where the mean polarizability is a = (al + 2oc2)/3 and the polarizability anisotropy is Aa = a1-a2. It is expected that the macroscopic refractive 

This expression can be combined with equation (4.12) to obtain the following expression for the refractive index tensor as a function of the orientation distribution of the rods making up the bulk sample  

Using this model, the birefringence and orientation angle, %, of the principal directions of the refractive index tensor can be determined. If light is propagating along the y axis, we find that 

If the sample is uniaxial and aligned along the z axis, then 

As before, the polarizability of the ith individual segment is a. = a2I+ (o - a2) (u.u .). The polarizability of the entire chain is the summation of this polarizability over all the segments and its average over the conformation of the macromolecule. In other words, 

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