E tensor

Cartesian tensors, i.e., tensors in a Cartesian coordinate system, will be discussed. Three Independent quantities are required to describe the position of a point in Cartesian coordinates. This set of quantities is X where X is (x, X2, X3). The index i in X has values 1,2, and 3 because of the three coordinates in three-dimensional space. The indices i and j in a j mean, therefore, that a j has nine components. Similarly, byi has 27 components, Cp has 81 components, etc. The indices are part of what is called index notation. The number of subscripts on the symboi denotes the order of the tensor. For example, a is a zero-order tensor... 

However, the components of the yj2) e, e tensor are chiral (i.e., only present in a chiral isotropic medium), whereas the components of the tensors y 2) and y(2) meeare achiral (i.e., present in any isotropic medium, chiral or achiral). Hence, only the electric dipole response of chiral isotropic materials is related to chirality. The experimental work on chiral polymers described in Section 4 showed that large magnetic contributions to the nonlinearity are due to chirality. However, such contributions will therefore not survive in chiral isotropic media. In this respect, the electric dipole contributions associated with chirality may prove more interesting for applications. 

On a scale of the order of atomic size, molecular multipole fields vary strongly with orientation and separation. As a consequence, one will generally find induced dipole components arising from field gradients of first and higher order which interact with the so-called dipole-multipole polarizability tensor components, such as the A and E tensors. 

Basic equations of the homogenization theory are applicable not only for scalar values of the constiments dielectric constants e, but also for the tensorial quantities. Indeed, evaluation of these equations does not demand any special requirements regarding the character of the electric displacement D and electric intensity E relation D = eeoE in which dielectric constant e can be tensorial. This fact enables us to calculate the effective dielectric constant Ceff for the composites with anisotropic constituents. It can be, for example, polymer composites with magnetic granular, the permittivity of which is tensorial quantity. The explicit form of the e tensor for magnetic media will be given in the next section. 

Let us consider the simplest case of an optically isotropic ferromagnet. The presence of magnetization reduces the symmetry to the single-axis one. The e tensor can be represented as a sum of symmetric and asymmetric tensors, which is given by [26]... 

In crystals the dependence of the e tensor on M is more complicated, namely,... 

E. Tensor Surface Harmonic Theory Approximation of Borane Deltahedra by Spheres... 

The electric flux density D and electric field strength E are vectors, i.e. tensors of first order and therefore may be related via a tensor of second order with nine constants for the three dimensions. Due to the potential property also observed for electrostatic fields, the tensor is symmetric and thus contains six independent entries. The electrostatic constitutive relation can be expressed with the aid of the dielectric permittivity matrix e (to be distinguished from the strains e) or its inverse p ... 

Dielectric Spectroscopy To Find the yand e Tensor Components.617... 

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