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It has been demonstrated in the introduction that the dielectric susceptibility of an anisotropic medium can be described with a second-rank tensor which expresses the relation between two physical quantities i.e. the relation between the vector of polarization and the vector of the electric field. Similarly the greater part of the various physical properties may be described with a tensor which establishes the relation existing between measurable physical tensor quantities. Every scalar is a zero-rank, and every vector a first-rank tensor. Generally in crystal physics a set of 3^{r} quantities with r indices transforming under transition from the old coordinates to the new ones as the products of the components of r vectors is called a polar (or true) tensor of rank r.
Accordingly if the [B_{ijk...n}] and [A_{pqr...u}] tensors represent physical quantities the general form of the relation between these quantities may be written (in first-order approximation) using the Einstein's convention as follows
(2.1) |
It follows from the tensor algebra that if [A_{pqr...u}] denotes an f-rank and [B_{ijk...n}] a g-rank tensor the [a_{ijk...npqr...u}], denoting the physical property, must be an (f + g)-rank tensor.
Let us consider some examples. In a given state the density of matter expresses the relationship between its mass and volume, they are represented by 0-rank tensors, consequently the density is represented by a 0-rank tensor (i.e. a scalar). The pyroelectric properties of crystals are described by a first-rank tensor. The pyroelectric tensor, (essentially a vector) represents the relation between a first-rank tensor (the vector of electric polarization) and a zero-rank tensor (the temperature). Besides the dielectric susceptibility, the electrical conductivity, the heat conductivity, the thermal expansion and so on may be represented by a second-rank tensor. Further examples including properties which can be expressed with higher-rank tensors are summarized in Table 1.
Crystals have further on also some anisotropic properties which cannot be
directly represented by tensors, such properties - not to be discussed in this
paper - are for instance the tensile strength, flow stress, surface energy, rate
of growth and dissolution, and so on.
Property or effect | Tensor notation | Tensor rank | Maximum no. of independent components | Defining equation | Physical quantities in the defining equation | |
Density | 1 | mass | volume | |||
Specific heat | c | 1 | entropy | T temperature | ||
Pyroelectricity | [p_{i}] | 1 | 3 | [P_{i}] dielectric polarization | T temperature | |
Electrocaloric effect | [p_{i}] | 1 | 3 | entropy | [E_{i}] electric field | |
Dielectric permittivity | 2 | 6 | [D_{i}] electric displacement | [E_{j}] electric field | ||
Magnetic permeability | 2 | 6 | [B_{i}] magnetic induction | [H_{j}] magnetic field | ||
Electrical conductivity | 2 | 6 | [j_{i}] current density | [E_{k}] electric field | ||
Electrical resistivity | 2 | 6 | [E_{i}] electric field | [j_{k}] current density | ||
Thermal conductivity | [k_{ij}] | 2 | 6 | [h_{i}] heat flux | temperature gradient | |
Thermal expansion | 2 | 6 | strain | T temperature | ||
Seebeck effect | 2 | 9 | [E_{i}] electric field | temperature gradient | ||
Peltier effect | 2 | 9 | [h_{i}] heat flux | [j_{k}] current density | ||
Hall effect | 3 | 9 | [E_{i}] electric field | [j_{k}] current density [H_{l}] magnetic field | ||
Direct piezoelectric effect | [d_{ijk}] | 3 | 18 | [P_{i}] dielectric polarization | stress | |
Converse piezoelectric effect | [d_{ijk}] | 3 | 18 | strain | [E_{i}] electric field | |
Piezomagnetic effect | [q_{lij}] | 3 | 18 | [M_{l}] magnetic polarization | stress | |
Electro-optical effect | [r_{ijk}] | 3 | 18 | [a_{ij}] dielectric impermeability | [E_{k}] electric field | |
Second harmonic generation | [d_{ijk}] | 3 | 18 | dielectric polarization at frequency | electric field | |
Second-order elastic stiffnesses | [c_{ijkl}] | 4 | 21 | stress | strain | |
Second-order elastic compliances | [s_{ijkl}] | 4 | 21 | strain | stress | |
Piezooptic effect | 4 | 36 | [a_{ij}] dielectric impermeability | stress | ||
Quadratic electrooptic effect | [R_{ijkl}] | 4 | 36 | [a_{ij}] dielectric impermeability | electric field | |
Electrostriction | 4 | 36 | strain | electric field | ||
Third-order elastic stresses | [c_{ijklmn}] | 6 | 56 | energy of deformation [c_{ijkl}] second-order stiffnesses | the Lagrange finite strain components |
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