Electric susceptibility tensor

As implied by the trace expression for the macroscopic optical polarization, the macroscopic electrical susceptibility tensor at any order can be written in temis of an ensemble average over the microscopic nonlmear polarizability tensors of the individual constituents. 

Excitation wavevector X Linear electric susceptibility tensor... 

Estimation of the Raman phonon intensities is even more complex, though a model has been proposed for this quantity that is suitable for potential based methods (Kleinman and Spitzer 1962). The electric susceptibility tensor is given by ... 

The matrix representing the electric susceptibility tensor is diagonal because one of the axes, the z-axis, points along the director. If one of the axes does not point along the director, then x is no longer diagonal. However, in order for it to have physical meaning, it must be symmetric (Xjj = Xji)... 

In order to draw out the anisotropy in the electric susceptibility tensor, x is usually written as an average part plus an anisotropic part. 

This can be substituted into Eq. (47) of Sec. 1 to obtain the anisotropic part of the electric susceptibility tensor. The isotropic part of the result obtained for isotropic fluids ... 

The polarization per imit electric field of an anisotropic material is described by the electric susceptibility tensor as-... 

Note that Equation [51] represents formally the tensorial relationship, while Equation [52] expresses this relation by the (Cartesian) components of P and E and some coefficients whereas Equation [53] states this relation by using Einstein s summation convention. Here, the coefficients are the components of the electric susceptibility tensor which is a tensor of rank 2. The tensor % is an example of what is usually called a property tensor or matter tensor. Strictly speaking, property tensors describe physical properties of the static crystal which belong to the totally symmetric irreducible representation of the relevant point group. Properties, however, that depend on vibrations of the crystal lattice are described by tensors which belong to the different irreducible representations. The corresponding tensors are then often designated as tensorial covariants. 

The apphcation of an electric field E to a liquid crystal sample produces a dipole moment per unit volume called the polarisation, denoted by P. We shall employ a description that uses SI units. For clarity of exposition we begin by assuming that n is aligned parallel to the 2 -axis in a Cartesian system of coordinates, that is, n = no = (0,0,1) as introduced in equation (2.7) such a local description is always possible. The anisotropy of the liquid crystal generally forces E and P to have different directions. They are related by the electric susceptibility tensor Xe via the equation... 

---