International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 35
Section 1.1.1. The matrix of physical properties^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France 
Physical laws express in general the response of a medium to a certain influence. Most physical properties may therefore be defined by a relation coupling two or more measurable quantities. For instance, the specific heat characterizes the relation between a variation of temperature and a variation of entropy at a given temperature in a given medium, the dielectric susceptibility the relation between electric field and electric polarization, the elastic constants the relation between an applied stress and the resulting strain etc. These relations are between quantities of the same nature: thermal, electrical and mechanical, respectively. But there are also cross effects, for instance:
The physical quantities that are involved in these relations can be divided into two categories:

Each of the quantities mentioned in the preceding section is represented by a mathematical expression. Some are direction independent and are represented by scalars: specific mass, specific heat, volume, pressure, entropy, temperature, quantity of electricity, electric potential. Others are direction dependent and are represented by vectors: force, electric field, electric displacement, the gradient of a scalar quantity. Still others cannot be represented by scalars or vectors and are represented by more complicated mathematical expressions. Magnetic quantities are represented by axial vectors (or pseudovectors), which are a particular kind of tensor (see Section 1.1.4.5.3). A few examples will show the necessity of using tensors in physics and Section 1.1.3 will present elementary mathematical properties of tensors.
Remark. Of the four examples given above, the first three (thermal expansion, dielectric constant, stressed rod) are related to physical property tensors (also called material tensors), which are characteristic of the medium and whose components have the same value everywhere in the medium if the latter is homogeneous, while the fourth one (expansion in Taylor series of a field of vectors) is related to a field tensor whose components vary at every point of the medium. This is the case, for instance, for the strain and for the stress tensors (see Sections 1.3.1 and 1.3.2 ).
Each extensive parameter is in principle a function of all the intensive parameters. For a variation of a particular intensive parameter, there will be a variation of every extensive parameter. One may therefore writeThe summation is over all the intensive parameters that have varied.
One may use a matrix notation to write the equations relating the variations of each extensive parameter to the variations of all the intensive parameters: where the intensive and extensive parameters are arranged in column matrices, (di) and (de), respectively. In a similar way, one could write the relations between intensive and extensive parameters asMatrices (C) and (R) are inverse matrices. Their leading diagonal terms relate an extensive parameter and the associated intensive parameter (their product has the dimensions of energy), e.g. the elastic constants, the dielectric constant, the specific heat etc. The corresponding physical properties are called principal properties. If one only of the intensive parameters, , varies, a variation of this parameter is the cause of which the effect is a variation, (without summation), of each of the extensive parameters. The matrix coefficients may therefore be considered as partial differentials:
The parameters that relate causes and effects represent physical properties and matrix (C) is called the matrix of physical properties. Let us consider the following intensive parameters: T stress, E electric field, H magnetic field, Θ temperature and the associated extensive parameters: S strain, P electric polarization, B magnetic induction, σ entropy, respectively. Matrix equation (1.1.1.4) may then be written:
The various intensive and extensive parameters are represented by scalars, vectors or tensors of higher rank, and each has several components. The terms of matrix (C) are therefore actually submatrices containing all the coefficients relating all the components of a given extensive parameter to the components of an intensive parameter. The leading diagonal terms, , , , , correspond to the principal physical properties, which are elasticity, dielectric susceptibility, magnetic susceptibility and specific heat, respectively. The nondiagonal terms are also associated with physical properties, but they relate intensive and extensive parameters whose products do not have the dimension of energy. They may be coupled in pairs symmetrically with respect to the main diagonal:
It is important to note that equation (1.1.1.6) is of a thermodynamic nature and simply provides a general framework. It indicates the possibility for a given physical property to exist, but in no way states that a given material will exhibit it. Curie laws, which will be described in Section 1.1.4.2, show for instance that certain properties such as pyroelectricity or piezoelectricity may only appear in crystals that belong to certain point groups.
If parameter varies by , the specific energy varies by du, which is equal to We have, therefore and, using (1.1.1.5), Since the energy is a state variable with a perfect differential, one can interchange the order of the differentiations: Since p and q are dummy indices, they may be exchanged and the last term of this equation is equal to . It follows thatMatrices and are therefore symmetric. We may draw two important conclusions from this result:
Let us now consider systems that are in steady state and not in thermodynamic equilibrium. The intensive and extensive parameters are time dependent and relation (1.1.1.3) can be written where the intensive parameters are, for instance, a temperature gradient, a concentration gradient, a gradient of electric potential. The corresponding extensive parameters are the heat flow, the diffusion of matter and the current density. The diagonal terms of matrix correspond to thermal conductivity (Fourier's law), diffusion coefficients (Fick's law) and electric conductivity (Ohm's law), respectively. Nondiagonal terms correspond to cross effects such as the thermoelectric effect, thermal diffusion etc. All the properties corresponding to these examples are represented by tensors of rank 2. The case of secondrank axial tensors where the symmetrical part of the tensors changes sign on time reversal was discussed by Zheludev (1986).
The Onsager reciprocity relations (Onsager, 1931a,b) express the symmetry of matrix . They are justified by considerations of statistical thermodynamics and are not as obvious as those expressing the symmetry of matrix (). For instance, the symmetry of the tensor of rank 2 representing thermal conductivity is associated with the fact that a circulating flow is undetectable.
Transport properties are described in Chapter 1.8 of this volume.
References
Brillouin, L. (1949). Les tenseurs en mécanique et en élasticité. Paris: Masson & Cie.Onsager, L. (1931a). Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405–426.
Onsager, L. (1931b). Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265–2279.
Voigt, W. (1910). Lehrbuch der Kristallphysik. Leipzig: Teubner. 2nd ed. (1929); photorep. (1966). New York: Johnson Reprint Corp. and Leipzig: Teubner.
Zheludev, I. S. (1986). Space and time inversion in physical crystallography. Acta Cryst. A42, 122–127.