Tensor of dielectric anisotropy

In general, however, tensor Qij is biaxial but the biaxiaUty is small, on the order of yPo where is the length corresponding to nematic correlations. This correlation length may be found, for example, from the light scattering in the isotropic phase close to the transition to the nematic phase. Then, at each point, that is locally, the anisotropic part of dielectric susceptibility tensor is biaxial and traceless 8ei + 5e2 + 8e3 = 0 with 8e2 8E3. 

Particularly, for t = 0 we return to the nematic tensor of dielectric anisotropy with factor 2/3 included in 8ei  

To obtain the tensor of the cholesteric helical structure one should imagine that the local tensor rotates in the laboratory co-ordinate system, or, alternatively, to introduce a rotating co-ordinate system. In the latter case, one should make transformation 

Note that for our rotation matrix, which is antisymmetric, the inverse matrix is equal to the transposed one. Now using Eqs. (4.9) and (4.10) we write 

Finally, we can write the tensors of the orientational order parameter Qy in the rotating frame for locally uniaxial and biaxial cholesteric liquid crystal (ChLC) Uniaxial ChLC  

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