2. Physical Properties as Tensors

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

$\begin{displaymath} B_{ijk...n} = a_{ijk...npqr...u} \cdot A_{pqr...u} \qquad (i, j, k,\dots n, p, q, r,\dots u = 1, 2, 3)\end{displaymath}$

(2.1)

where the tensor [a_{ijk...npqr...u}] denotes the physical property connecting the two physical quantities.

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.

**Table 1:** Tensors representing physical properties
Property or effect	Tensor notation	Tensor rank	Maximum no. of independent components	Defining equation	Physical quantities in the defining equation
Density	$\rho$		1	$\Delta m = \rho\Delta V$	$\Delta m$ mass	$\Delta V$ volume
Specific heat	c		1	$\Delta S=\displaystyle \frac{c}{T}\Delta T$	$\Delta S$ entropy	T temperature
Pyroelectricity	[p_i]	1	3	$\Delta P_i = p_i \cdot \Delta T$	[P_i] dielectric polarization	T temperature
Electrocaloric effect	[p_i]	1	3	$\Delta S = p_i \Delta E_i$	$\Delta S$ entropy	[E_i] electric field
Dielectric permittivity	$[\varepsilon_{ij}]$	2	6	$D_i=\varepsilon_{ij}E_j$	[D_i] electric displacement	[E_j] electric field
Magnetic permeability	$[\mu_{ij}]$	2	6	$B_i=\mu_{ij}H_j$	[B_i] magnetic induction	[H_j] magnetic field
Electrical conductivity	$[\sigma_{ik}]$	2	6	$j_i=\sigma_{ik}E_k$	[j_i] current density	[E_k] electric field
Electrical resistivity	$[\rho_{ik}]$	2	6	$E_i=\rho_{ik}\cdot j_k$	[E_i] electric field	[j_k] current density
Thermal conductivity	[k_ij]	2	6	$h_i=-k_{ij}(\partial T / \partial x_j)$	[h_i] heat flux	$[\partial T /\partial x_j]$ temperature gradient
Thermal expansion	$[\alpha_{ij}]$	2	6	$\varepsilon_{ij}=\alpha_{ij}\Delta T$	$[\varepsilon_{ij}]$ strain	T temperature
Seebeck effect	$[\beta_{ik}]$	2	9	$E_i=-\beta_{ik}(\partial T/\partial x_k)$	[E_i] electric field	$[\partial T/\partial x_k]$ temperature gradient
Peltier effect	$[\pi_{ik}]$	2	9	$h_i=\pi_{ik}j_k$	[h_i] heat flux	[j_k] current density
Hall effect	$\rho_{ikl}]$	3	9	$E_i=\rho_{ikl}\cdot j_k\cdot H_l$	[E_i] electric field	[j_k] current density [H_l] magnetic field
Direct piezoelectric effect	[d_ijk]	3	18	$P_i = d_{ijk}\cdot\sigma_{jk}$	[P_i] dielectric polarization	$[\sigma_{jk}]$ stress
Converse piezoelectric effect	[d_ijk]	3	18	$\varepsilon_{jk}=d_{ijk}E_i$	$[\varepsilon_{jk}]$ strain	[E_i] electric field
Piezomagnetic effect	[q_lij]	3	18	$M_l=q_{lij}\sigma_{ij}$	[M_l] magnetic polarization	$[\sigma_{ij}]$ stress
Electro-optical effect	[r_ijk]	3	18	$\Delta a_{ij}=r_{ijk}E_k$	[a_ij] dielectric impermeability	[E_k] electric field
Second harmonic generation	[d_ijk]	3	18	$P_i^{2\omega}=d_{ijk}E_jE_k$	$[P_i^{2\omega}]$ dielectric polarization at frequency $2\omega$	$[E_j],\,[E_k]$ electric field
Second-order elastic stiffnesses	[c_ijkl]	4	21	$\sigma_{ij}=c_{ijkl}\varepsilon_{kl}$	$[\sigma_{ij}]$ stress	$[\varepsilon_{kl}]$ strain
Second-order elastic compliances	[s_ijkl]	4	21	$\varepsilon_{ij}=s_{ijkl}\sigma_{kl}$	$[\varepsilon_{ij}]$ strain	$[\sigma_{kl}]$ stress
Piezooptic effect	$[\pi_{ijkl}]$	4	36	$\Delta a_{ij}= \pi_{ijkl}\sigma_{kl}$	[a_ij] dielectric impermeability	$[\sigma_{kl}]$ stress
Quadratic electrooptic effect	[R_ijkl]	4	36	$\Delta a_{ij}= R_{ijkl}E_kE_l$	[a_ij] dielectric impermeability	$[E_k],\,[E_l]$ electric field
Electrostriction	$[\gamma_{iljk}]$	4	36	$\varepsilon_{jk}=\mu_{iljk}E_iE_l$	$[\varepsilon_{jk}]$ strain	$[E_i],\,[E_l]$ electric field
Third-order elastic stresses	[c_ijklmn]	6	56	$\Phi=\frac{1}{2}c_{ijkl}\cdot \eta_{ij}\cdot\eta_{kl} + \frac{1}{6}c_{ijklmn}\cdot\eta_{ij}\cdot\eta_{kl}\cdot\eta_{mn}]$	$\Phi$ energy of deformation [c_ijkl] second-order stiffnesses	$[\eta_{ij}],\,[\eta_{kl}],\,[\eta_{mn}]$ the Lagrange finite strain components