Impermeability tensor

If all three principal values are positive, the quadric surface is an ellipsoid with semiaxes a, = TfiVz, but if one or two of the principal values are negative the quadric surface is a hyperboloid. For example, the (relative) impermeability tensor 3 is defined by nfn, where k is the permittivity and n0 is the permittivity of free space. As for any symmetric 7(2) the components of 3 define the representation quadric I3ijxixj= 1, which here is called the indicatrix or optical index ellipsoid. Referred to principal axes the indicatrix is... 

Generally the linear approximation suffices, but, because the refractive index can be measured with considerable precision, the change in the impermeability tensor due to stress and electric field should be written as... 

Magnetic permittivity of free space, (1.1). impermeability tensor, (2.64). 

It is customary to develop the optical theory of crystals in terms of the relative impermeability tensor By = (er 1), which is a second-rank tensor. It can be shown that By is symmetrical, a property that requires By = By. It can also be shown that because By is a symmetrical second-rank tensor it can be represented by... 

In contrast to the SHG coefficient, the electro-optic coefficient r,y/t is symmetric in its first two indices for a lossless, not optically active, material. This follows from Eq. (11), which relates the electro-optical coefficient to the impermeability tensor Tiy (the matrix inverse of the dielectric permeability tensor y), a real and symmetric tensor under the stated conditions [3],... 

Next we will introduce the optical dielectric impermeability tensor of a crystal. The coefficients (17, ) of this tensor depend on the distribution of bond charges in the material [15,71]. The 17, are found by taking the reciprocal of the relative permittivity or dielectric constant [71]. The 17, have been defined in terms of the refractive index of the crystal as [71]... 

The application of an external electric field deforms the optical indicatrix (Fresnel ellipsoid) of crystals lacking a center of symmetry in such a way that its birefringence is changed. The dependence of the birefringence on the electric field E is linear, and can be analytically described by a change of the impermeability tensor a = (e ) by the electric field E [see Eq. (8.4a)] and the polarization P, respectively ... 

We begin with a lamella on which no compnssional or shear stresses arc acting i.e., no mechanical work is exchanged between the lamella and its enviromnent. Hence, Hseparated from its environment by a fictitious, impermeable, diathermal membrane so that the number of molecules that constitute the lamella remains fixed that is, 8N = 8N = 0. In other words, only heat is exchanged between the lamella and its surrotmdings. Under these conditions, we have from Eqs. (1.44), (1.45b), and (1.49a)... 

The quantities Pjk are also dielectric material coefficients. Their dimension is obviously reciprocal to the dimension of the permittivity. They are also the coordinates of a symmetric second rank tensor for which several names have been in use. Following Thurston (1974) we call it impermittivity. (Alternatively, the names dielectric impermeability (Cady 1964, p. 163), dielectric permeability (Grindlay 1970, p. 56), vetivity , derived from the latin word vetare as the opposite to the latin permittere (Voigt 1910, p. 441) have been proposed.) The effect described by Eq. (4.31) is the inverse effect to Eq. (4.30). It is obvious that E and D have exchanged their roles as independent and dependent variables and, therefore, Sij and Pij are material coefficients derived from different thermodynamic potentials. 

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