Introduction to the properties of tensors

Authier, A.

doi:10.1107/97809553602060000628

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.1, pp. 3-33
https://doi.org/10.1107/97809553602060000628

Chapter 1.1. Introduction to the properties of tensors

A. Authier^a ^*

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France
Correspondence e-mail: aauthier@wanadoo.fr

This chapter introduces the notion of tensors in physics (field tensors and material property tensors), starting from the matrix of physical properties. The basic properties of vector spaces and the elementary mathematical properties of tensors are then recalled. The most important part of the chapter, Section 1.1.4 , is devoted to the symmetry properties of tensors: Neumann's principle, Curie laws and the reduction of the components of polar and axial tensors of rank up to 4 for the 32 crystallographic groups. The relations between the thermodynamic functions and the physical property tensors are developed in Section 1.1.5 .

Keywords: Curie laws; Neumann's principle; Onsager relations; Voigt matrix; adiabatic coefficients; antisymmetric tensors; axial tensors; axial vectors; bilinear forms; contracted product; contraction; contravariant; covariant; dielectric constant; dielectric polarization; dielectric susceptibility; direct inspection method; dual space; electric field; electric polarization; electrostriction; enantiomorphic groups; extensive quantities; intensive quantities; intrinsic symmetry; linear forms; magnetic induction; material tensors; matrix of physical properties; matrix method; metric tensor; outer product; permutation tensor; photoelastic effect; physical property tensors; piezoelectric constants; piezoelectric effect; piezoelectric tensor; piezoelectricity; piezo-optic effect; piezo-optic tensor; polar tensors; polar vectors; pseudoscalar; pseudovectors; pyroelectric coefficients; pyroelectric effect; pyroelectricity; reciprocal basis; reciprocal lattice; reciprocal space; reduction of tensor components; representation surface; rotation matrix; symmetric tensors; tensor derivatives; tensor product; tensors; thermal expansion; uniaxial crystals; vector product; vector spaces.

1.1.1. The matrix of physical properties

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1.1.1.1. Notion of extensive and intensive quantities

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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:

(a) thermal expansion and piezocalorific effect: mechanical reaction to a thermal impetus or the reverse;
(b) pyroelectricity and electrocalorific effect: electrical response to a thermal impetus or the reverse;
(c) piezoelectricity and electrostriction: electric response to a mechanical impetus;
(d) piezomagnetism and magnetostriction: magnetic response to a mechanical impetus;
(e) photoelasticity: birefringence produced by stress;
(f) acousto-optic effect: birefringence produced by an acoustic wave;
(g) electro-optic effect: birefringence produced by an electric field;
(h) magneto-optic effect: appearance of a rotatory polarization under the influence of a magnetic field.

The physical quantities that are involved in these relations can be divided into two categories:

(i) extensive quantities, which are proportional to the volume of matter or to the mass, that is to the number of molecules in the medium, for instance entropy, energy, quantity of electricity etc. One uses frequently specific extensive parameters, which are given per unit mass or per unit volume, such as the specific mass, the electric polarization (dipole moment per unit volume) etc.

(ii) intensive parameters, quantities whose product with an extensive quantity is homogeneous to an energy. For instance, volume is an extensive quantity; the energy stored by a gas undergoing a change of volume dV under pressure p is $[p\,\,{\rm d}V]$ . Pressure is therefore the intensive parameter associated with volume. Table 1.1.1.1 gives examples of extensive quantities and of the related intensive parameters.

Table 1.1.1.1 | top | pdf |
Extensive quantities and associated intensive parameters

The last four lines of the table refer to properties that are time dependent.

Extensive quantities	Intensive parameters
Volume	Pressure
Strain	Stress
Displacement	Force
Entropy	Temperature
Quantity of electricity	Electric potential
Electric polarization	Electric field
Electric displacement	Electric field
Magnetization	Magnetic field
Magnetic induction	Magnetic field
Reaction rate	Chemical potential
Heat flow	Temperature gradient
Diffusion of matter	Concentration gradient
Electric current	Potential gradient

1.1.1.2. Notion of tensor in physics

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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.

(i) Thermal expansion. In an isotropic medium, thermal expansion is represented by a single number, a scalar, but this is not the case in an anisotropic medium: a sphere cut in an anisotropic medium becomes an ellipsoid when the temperature is varied and thermal expansion can no longer be represented by a single number. It is actually represented by a tensor of rank 2.
(ii) Dielectric constant. In an isotropic medium of a perfect dielectric we can write, in SI units, $[\eqalignno{{\bf P} & = \varepsilon _{0}\chi_{e} {\bf E} &\cr {\bf D} & = \varepsilon _{0}{\bf E} + {\bf P} = \varepsilon _{0}(1 + \chi_{e}) {\bf E} = \varepsilon {\bf E}, &\cr}]$ where P is the electric polarization (= dipole moment per unit volume), $[\varepsilon _{0}]$ the permittivity of vacuum, $[\chi_e]$ the dielectric susceptibility , D the electric displacement and $[\varepsilon]$ the dielectric constant, also called dielectric permittivity . These expressions indicate that the electric field, on the one hand, and polarization and displacement, on the other hand, are linearly related. In the general case of an anisotropic medium, this is no longer true and one must write expressions indicating that the components of the displacement are linearly related to the components of the field: $[\left\{\matrix{D^{1}=\varepsilon_1 ^{1} E^{1}+ \varepsilon_1 ^{2} E^{2}+ \varepsilon _{1}^{3}E \cr D^{2}=\varepsilon_1 ^{2} E^{1}+\varepsilon_2 ^{2} E^{2}+ \varepsilon_2 ^{3}E \cr D^{3}=\varepsilon_1 ^{3} E^{1}+ \varepsilon_2 ^{3} E^{2}+ \varepsilon _{3}^{3}E.\cr}\right. \eqno{(1.1.1.1)}]$ The dielectric constant is now characterized by a set of nine components $[\varepsilon\kern1pt^{j}_{i}]$ ; they are the components of a tensor of rank 2. It will be seen in Section 1.1.4.5.2.1 that this tensor is symmetric ( $[\varepsilon\kern1pt^{j}_{i} = \varepsilon^{i}_{j}]$ ) and that the number of independent components is equal to six.
(iii) Stressed rod (Hooke's law). If one pulls a rod of length ℓ and cross section $[{\cal A}]$ with a force F, its length is increased by a quantity $[\Delta \ell]$ given by $[\Delta \ell/\ell = (1/E)F/{\cal A},]$ where E is Young's modulus, or elastic stiffness (see Section 1.3.3.1 ). But, at the same time, the radius, r, decreases by Δr given by $[\Delta r/r = - (\nu /E)F/{\cal A}]$ , where ν is Poisson's ratio (Section 1.3.3.4.3 ). It can be seen that a scalar is not sufficient to describe the elastic deformation of a material, even if it is isotropic. The number of independent components depends on the symmetry of the medium and it will be seen that they are the components of a tensor of rank 4. It was precisely to describe the properties of elasticity by a mathematical expression that the notion of a tensor was introduced in physics by W. Voigt in the 19th century (Voigt, 1910 ) and by L. Brillouin in the first half of the 20th century (Brillouin, 1949 ).
(iv) Expansion in Taylor series of a field of vectors. Let us consider a field of vectors $[{\bf u}({\bf r})]$ where r is a position vector. The Taylor expansion of its components is given by $[{u^{i}({\bf r} + \hbox{d}{\bf r}) = u^{i}({\bf r}) + \left({\partial u^{i} \over \partial u\hskip 1pt^{j}}\right) {\rm d}x\hskip 1pt^{j} + \textstyle{1 \over 2} \displaystyle{\left({\partial^{2} u^{i} \over \partial u\hskip 1pt^{j} \partial u^{k}}\right)} {\rm d}x\hskip 1pt^{j}\,\, {\rm d}x^{k} + \ldots} \eqno(1.1.1.2)]$ using the so-called Einstein convention, which implies that there is automatically a summation each time the same index appears twice, once as a superscript and once as a subscript. This index is called a dummy index. It will be shown in Section 1.1.3.8 that the nine partial differentials $[\partial u^{i}/\partial x\hskip 1pt^{j}]$ and the 27 partial differentials $[\partial^{2}u^{i}/(\partial x\hskip 1pt^{j}\partial x^{k})]$ are the components of tensors of rank 2 and 3, respectively.

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 ).

1.1.1.3. The matrix of physical properties

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Each extensive parameter is in principle a function of all the intensive parameters. For a variation $[\hbox{d}i_{q}]$ of a particular intensive parameter, there will be a variation $[\hbox{d}e_{p}]$ of every extensive parameter. One may therefore write $[\hbox{d}e_{p} = C_{p}^{q}\,\, \hbox{d}i_{q}. \eqno(1.1.1.3)]$ The 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: $[(\hbox{d}e) = (C) (\hbox{d}i), \eqno(1.1.1.4)]$ 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 as $[\openup4pt\left. \matrix{\hfill \hbox{d}i_{p} &= R_{p}^{q} \,\,\hbox{d}e_{q}\hfill\cr (\hbox{d}i) &= (R) (\hbox{d}e).\cr}\right\} \eqno(1.1.1.5)]$ Matrices (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, $[i_{q}]$ , varies, a variation $[\hbox{d}i_{q}]$ of this parameter is the cause of which the effect is a variation, $[\hbox{d}e_{p} = C_{p}^{q}\,\, \hbox{d}i_{q}]$ (without summation), of each of the extensive parameters. The matrix coefficients $[C_{p}^{q}]$ may therefore be considered as partial differentials: $[C_{p}^{q} = \partial e_{p} / \partial i_{q}.]$

The parameters $[C_{p}^{q}]$ that relate causes $[\hbox{d}i_{q}]$ and effects $[\hbox{d}e_{p}]$ 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: $[\pmatrix{S\cr P\cr \noalign{\vskip3pt}B\cr \noalign{\vskip3pt}\delta \sigma\cr} = \pmatrix{C_{S}^{T} &C_{S}^{E} &C_{S}^{H} &C_{S}^{\Theta}\cr\noalign{\vskip3pt}C_{P}^{T} &C_{P}^{E} &C_{P}^{H} &C_{P}^{\Theta}\cr\noalign{\vskip3pt}C_{B}^{T} &C_{B}^{E} &C_{B}^{H} &C_{B}^{\Theta}\cr\noalign{\vskip3pt}C_{\sigma}^{T} &C_{\sigma}^{E} &C_{\sigma}^{H} &C_{\sigma}^{\Theta}\cr} \pmatrix{T\cr \noalign{\vskip3pt}E\cr \noalign{\vskip3pt}H\cr \Theta\cr}. \eqno(1.1.1.6)]$

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 $[C_{p}^{q}]$ relating all the components of a given extensive parameter to the components of an intensive parameter. The leading diagonal terms, $[C_{S}^{T}]$ , $[C_{P}^{E}]$ , $[C_{B}^{H}]$ , $[C_{\sigma}^{\Theta}]$ , correspond to the principal physical properties, which are elasticity, dielectric susceptibility , magnetic susceptibility and specific heat, respectively. The non-diagonal 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:

$[C_{S}^{E}]$ and $[C_{P}^{T}]$ represent the piezoelectric effect and the converse piezoelectric effect, respectively;
$[C_{S}^{H}]$ and $[C_{B}^{T}]$ the piezomagnetic effect and the converse piezomagnetic effect;
$[C_{S}^{\Theta}]$ and $[C_{\sigma}^{T}]$ thermal expansion and the piezocalorific effect;
$[C_{P}^{T}]$ and $[C_{\sigma}^{E}]$ the pyroelectric and the electrocalorific effects;
$[C_{P}^{H}]$ and $[C_{B}^{E}]$ the magnetoelectric effect and the converse magnetoelectric effect;
$[C_{\sigma}^{H}]$ and $[C_{B}^{\Theta}]$ the pyromagnetic effect and the magnetocalorific effect.

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.

1.1.1.4. Symmetry of the matrix of physical properties

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If parameter $[e_{p}]$ varies by $[\hbox{d}e_{p}]$ , the specific energy varies by du, which is equal to $[\hbox{d}u = i_{p}\,\,\hbox{d}e_{p}.]$ We have, therefore $[i_{p}= {\partial u \over \partial e_{p}}]$ and, using (1.1.1.5), $[R_{p}^{q} = {\partial i_{p} \over \partial e_{q}} = {\partial^{2} u \over \partial e_{p} \partial e_{q}}.]$ Since the energy is a state variable with a perfect differential, one can interchange the order of the differentiations: $[R_{p}^{q} = {\partial^{2} u \over \partial e_{q} \partial e_{p}} = {\partial i_{q} \over \partial e_{p}}.]$ Since p and q are dummy indices, they may be exchanged and the last term of this equation is equal to $[R_{q}^{p}]$ . It follows that $[R_{p}^{q} = R_{q}^{p}.]$ Matrices $[R_{p}^{q}]$ and $[C_{p}^{q}]$ are therefore symmetric. We may draw two important conclusions from this result:

(i) The submatrices associated with the principal properties are symmetric with respect to interchange of the indices related to the causes and to the effects: these properties are represented by symmetric tensors. For instance, the dielectric constant and the elastic constants are represented by symmetric tensors of rank 2 and 4 , respectively (see Section 1.1.3.4 ).
(ii) The submatrices associated with terms that are symmetric with respect to the main diagonal of matrices (C) and (R) and that represent cross effects are transpose to one another. For instance, matrix ( $[C_{S}^{E}]$ ) representing the converse piezoelectric effect is the transpose of matrix ( $[C_{P}^{T}]$ ) representing the piezoelectric effect. It will be shown in Section 1.1.3.4 that they are the components of tensors of rank 3.

1.1.1.5. Onsager relations

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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 $[J_{m} = L_{mn}X_{n},]$ where the intensive parameters $[X_{n}]$ are, for instance, a temperature gradient, a concentration gradient, a gradient of electric potential. The corresponding extensive parameters $[J_{m}]$ are the heat flow, the diffusion of matter and the current density. The diagonal terms of matrix $[L_{mn}]$ correspond to thermal conductivity (Fourier's law), diffusion coefficients (Fick's law) and electric conductivity (Ohm's law), respectively. Non-diagonal 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 second-rank 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 ) $[L_{mn} = L_{nm}]$ express the symmetry of matrix $[L_{mn}]$ . They are justified by considerations of statistical thermodynamics and are not as obvious as those expressing the symmetry of matrix ( $[C_{p}^{q}]$ ). 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.

1.1.2. Basic properties of vector spaces

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[The reader may also refer to Section 1.1.4 of Volume B of International Tables for Crystallography (2001).]

1.1.2.1. Change of basis

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Let us consider a vector space spanned by the set of n basis vectors $[{\bf e}_{1}]$ , $[{\bf e}_{2}]$ , $[{\bf e}_{3},\ldots, {\bf e}_{n}]$ . The decomposition of a vector using this basis is written $[{\bf x} = x^{i}{\bf e}_{i} \eqno(1.1.2.1)]$ using the Einstein convention. The interpretation of the position of the indices is given below. For the present, we shall use the simple rules:

(i) the index is a subscript when attached to basis vectors;
(ii) the index is a superscript when attached to the components. The components are numerical coordinates and are therefore dimensionless numbers.

Let us now consider a second basis, $[{\bf e}'_{j}]$ . The vector x is independent of the choice of basis and it can be decomposed also in the second basis: $[{\bf x} = x'^{i}{\bf e}'_{i}. \eqno(1.1.2.2)]$

If $[A\hskip1pt_{i}^{j}]$ and $[B_{j}^{i}]$ are the transformation matrices between the bases $[{\bf e}_{i}]$ and $[{\bf e}'_{j}]$ , the following relations hold between the two bases: $[\left.\matrix{{\bf e}_{i} = A\hskip1pt_{i}^{j}{\bf e}'_{j}\semi\hfill &{\bf e}'_{j} = B_{j}^{i}{\bf e}_{i}\hfill \cr x^{i} = B_{j}^{i} x'^{j}\semi &x'^{j} = A\hskip1pt_{i}^{j} x^{i}\cr}\right\} \eqno(1.1.2.3)]$ (summations over j and i, respectively). The matrices $[A\hskip1pt_{i}^{j}]$ and $[B_{j}^{i}]$ are inverse matrices: $[A\hskip1pt_{i}^{j} B_{j}^{k} = \delta_{i}^{k} \eqno(1.1.2.4)]$ (Kronecker symbol: $[\delta_{i}^{k} = 0]$ if $[i \neq k, = 1]$ if [i = k] ).

Important Remark. The behaviour of the basis vectors and of the components of the vectors in a transformation are different. The roles of the matrices $[A\hskip1pt_{i}^{j}]$ and $[B_{j}^{i}]$ are opposite in each case. The components are said to be contravariant. Everything that transforms like a basis vector is covariant and is characterized by an inferior index. Everything that transforms like a component is contravariant and is characterized by a superior index. The property describing the way a mathematical body transforms under a change of basis is called variance.

1.1.2.2. Metric tensor

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We shall limit ourselves to a Euclidean space for which we have defined the scalar product. The analytical expression of the scalar product of two vectors $[{\bf x} = x^{i}{\bf e}_{i}]$ and $[{\bf y} = y\hskip 2pt^{j}{\bf e}_{j}]$ is $[{\bf x} \cdot {\bf y} = x^{i}{\bf e}_{i} \cdot y\hskip 2pt^{j}{\bf e}_{j}.]$ Let us put $[{\bf e}_{i} \cdot {\bf e}_{j} = g_{ij}. \eqno(1.1.2.5)]$ The nine components $[g_{ij}]$ are called the components of the metric tensor. Its tensor nature will be shown in Section 1.1.3.6.1 . Owing to the commutativity of the scalar product, we have $[g_{ij}= {\bf e}_{i} \cdot {\bf e}_{j} = {\bf e}_{j} \cdot {\bf e}_{i} = g_{ji}.]$

The table of the components $[g_{ij}]$ is therefore symmetrical. One of the definition properties of the scalar product is that if $[{\bf x} \cdot {\bf y} = 0]$ for all x, then $[{\bf y} = {\bf 0}]$ . This is translated as $[x^{i}y\hskip 2pt^{j}g_{ij}= 0 \quad \forall x^{i}\ \Longrightarrow\ y\hskip 2pt^{j}g_{ij} = 0.]$

In order that only the trivial solution $[(y\hskip 2pt^{j} = 0)]$ exists, it is necessary that the determinant constructed from the $[g_{ij}]$ 's is different from zero: $[\Delta (g_{ij}) \neq 0.]$ This important property will be used in Section 1.1.2.4.1 .

1.1.2.3. Orthonormal frames of coordinates – rotation matrix

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An orthonormal coordinate frame is characterized by the fact that $[g_{ij} = \delta_{ij} \quad (= 0 \quad \hbox{if} \ \ i \neq j \ \hbox{ and } \ = 1 \ \hbox{ if } \ i = j). \eqno(1.1.2.6)]$ One deduces from this that the scalar product is written simply as $[{\bf x} \cdot {\bf y} = x^{i}y\hskip 2pt^{j}g_{ij} = x^{i}y^{i}.]$

Let us consider a change of basis between two orthonormal systems of coordinates: $[{\bf e}_{i} = A\hskip1pt_{i}^{j}{\bf e}'_{j}.]$ Multiplying the two sides of this relation by $[{\bf e}'_{j}]$ , it follows that $[{\bf e}_{i} \cdot {\bf e}'_{j} = A\hskip1pt_{i}^{j} {\bf e}'_{k} \cdot e'_{j} = A\hskip1pt_{i}^{j} g'_{kj} = A\hskip1pt_{i}^{j} \delta_{kj} \ \hbox{ (written correctly),}]$ which can also be written, if one notes that variance is not apparent in an orthonormal frame of coordinates and that the position of indices is therefore not important, as $[{\bf e}_{i} \cdot {\bf e}'_{j} = A\hskip1pt_{i}^{j} \ \hbox{ (written incorrectly)}.]$

The matrix coefficients, $[A\hskip1pt_{i}^{j}]$ , are the direction cosines of $[{\bf e}'_{j}]$ with respect to the $[{\bf e}_{i}]$ basis vectors. Similarly, we have $[B_{j}^{i} = {\bf e}_{i} \cdot {\bf e}'_{j}]$ so that $[A\hskip1pt_{i}^{j} = B_{j}^{i} \quad \hbox{or} \quad A = B^{T},]$ where ^T indicates transpose. It follows that $[A = B^{T} \hbox{ and } A = B^{-1}]$ so that $[\left. \matrix{A^{T} \ = A^{-1}\quad \Rightarrow\quad A^{T}A = I\cr B^{T} \ = B^{-1}\quad \Rightarrow\quad B^{T}B = I.\cr}\right\} \eqno(1.1.2.7)]$ The matrices A and B are unitary matrices or matrices of rotation and $[\Delta (A)^{2} = \Delta (B)^{2} = 1 \Rightarrow \Delta (A) = \pm 1. \eqno(1.1.2.8)]$

If $[\Delta (A) = 1]$ the senses of the axes are not changed – proper rotation.
If $[\Delta (A) = - 1]$ the senses of the axes are changed – improper rotation. (The right hand is transformed into a left hand.)

One can write for the coefficients $[A\hskip1pt_{i}^{j}]$ $[A\hskip1pt_{i}^{j}B_{j}^{k} = \delta_{i}^{k};\quad A\hskip1pt_{i}^{j}A_{j}^{k} = \delta_{i}^{k},]$ giving six relations between the nine coefficients $[A\hskip1pt_{i}^{j}]$ . There are thus three independent coefficients of the $[3 \times 3]$ matrix A.

1.1.2.4. Covariant coordinates – dual or reciprocal space

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1.1.2.4.1. Covariant coordinates

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Using the developments (1.1.2.1) and (1.1.2.5), the scalar products of a vector x and of the basis vectors $[{\bf e}_{i}]$ can be written $[x_{i} = {\bf x} \cdot {\bf e}_{i} = x\hskip 1pt^{j}{\bf e}_{j} \cdot {\bf e}_{i} = x\hskip 1pt^{j}g_{ij}. \eqno(1.1.2.9)]$ The n quantities $[x_{i}]$ are called covariant components, and we shall see the reason for this a little later. The relations (1.1.2.9) can be considered as a system of equations of which the components $[x\hskip 1pt^{j}]$ are the unknowns. One can solve it since $[\Delta (g_{ij}) \neq 0]$ (see the end of Section 1.1.2.2 ). It follows that $[x\hskip 1pt^{j} = x_{i}g^{ij} \eqno(1.1.2.10)]$ with $[g^{ij}g_{jk} = \delta_{k}^{i}. \eqno(1.1.2.11)]$

The table of the $[g^{ij}]$ 's is the inverse of the table of the $[g_{ij}]$ 's. Let us now take up the development of x with respect to the basis $[{\bf e}_{i}]$ : $[{\bf x} = x^{i}{\bf e}_{i}.]$

Let us replace $[x^{i}]$ by the expression (1.1.2.10): $[{\bf x} = x_{j}g^{ij}{\bf e}_{i}, \eqno(1.1.2.12)]$ and let us introduce the set of n vectors $[{\bf e}\hskip 1pt^{j} = g^{ij}{\bf e}_{i} \eqno(1.1.2.13)]$ which span the space $[E^{n}\,\,(j = 1, \ldots, n)]$ . This set of n vectors forms a basis since (1.1.2.12) can be written with the aid of (1.1.2.13) as $[{\bf x} = x_{j}{\bf e}\hskip 1pt^{j}. \eqno(1.1.2.14)]$

The $[x_{j}]$ 's are the components of x in the basis $[{\bf e}\hskip 1pt^{j}]$ . This basis is called the dual basis. By using (1.1.2.11) and (1.1.2.13), one can show in the same way that $[{\bf e}_{j} = g_{ij}{\bf e}\hskip 1pt^{j}. \eqno(1.1.2.15)]$

It can be shown that the basis vectors $[{\bf e}\hskip 1pt^{j}]$ transform in a change of basis like the components $[x\hskip 1pt^{j}]$ of the physical space. They are therefore contravariant. In a similar way, the components $[x_{j}]$ of a vector x with respect to the basis $[{\bf e}\hskip 1pt^{j}]$ transform in a change of basis like the basis vectors in direct space, $[{\bf e}_{j}]$ ; they are therefore covariant: $[\left. \matrix{{\bf e}\hskip 1pt^{j} = B\hskip1pt_{k}^{j} {\bf e}'^{k}\semi &{\bf e}'^{k} = A_{j}^{k} {\bf e}\hskip 1pt^{j}\cr x_{i} = A\hskip1pt_{i}^{j} x'_{j}\semi &x'_{j} = B_{j}^{i}x_{i}.\cr}\right\} \eqno(1.1.2.16)]$

1.1.2.4.2. Reciprocal space

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Let us take the scalar products of a covariant vector $[{\bf e}_i]$ and a contravariant vector $[{\bf e}\hskip 1pt^{j}]$ : $[{\bf e}_{i} \cdot {\bf e}\hskip 1pt^{j} = {\bf e}_{i} \cdot g\hskip 1pt^{jk}{\bf e}_{k} = {\bf e}_{i} \cdot {\bf e}_{k} g\hskip 1pt^{jk} = g_{ik}g\hskip 1pt^{jk} = \delta\hskip1pt_{i}^{j}]$ [using expressions (1.1.2.5), (1.1.2.11) and (1.1.2.13)].

The relation we obtain, $[{\bf e}_{i} \cdot {\bf e}\hskip1pt^{j} = \delta\hskip1pt_{i}^{j}]$ , is identical to the relations defining the reciprocal lattice in crystallography; the reciprocal basis then is identical to the dual basis $[{\bf e}^{i}]$ .

1.1.2.4.3. Properties of the metric tensor

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In a change of basis, following (1.1.2.3) and (1.1.2.5), the $[g_{ij}]$ 's transform according to $[\left. \matrix{g_{ij} \ = A_{i}^{k} A_{j}^{m} g'_{km}\cr g'_{ij} \ = B_{i}^{k} B_{j}^{m} g_{km}.\cr}\right\} \eqno(1.1.2.17)]$ Let us now consider the scalar products, $[{\bf e}^{i} \cdot {\bf e}\hskip 1pt^{j}]$ , of two contravariant basis vectors. Using (1.1.2.11) and (1.1.2.13), it can be shown that $[{\bf e}^{i} \cdot {\bf e}\hskip 1pt^{j} = g^{ij}. \eqno(1.1.2.18)]$

In a change of basis, following (1.1.2.16), the $[g^{ij}]$ 's transform according to $[\left. \matrix{g^{ij} \ = B\hskip1pt_{k}^{i} B\hskip1pt_{m}^{j} g'^{km}\cr g'^{ij} \ = A_{k}^{i} A\hskip1pt_{m}^{j} g^{km}.\cr}\right\} \eqno(1.1.2.19)]$ The volumes V ′ and V of the cells built on the basis vectors $[{\bf e}'_{i}]$ and $[{\bf e}_{i}]$ , respectively, are given by the triple scalar products of these two sets of basis vectors and are related by $[\eqalignno{V' &= ({\bf e}'_{1}, {\bf e}'_{2}, {\bf e}'_{3}) &\cr &= \Delta (B_{j}^{i}) ({\bf e}_{1}, {\bf e}_{2}, {\bf e}_{3}) &\cr & = \Delta (B_{j}^{i}) V, &(1.1.2.20)\cr}]$ where $[\Delta (B_{j}^{i})]$ is the determinant associated with the transformation matrix between the two bases. From (1.1.2.17) and (1.1.2.20), we can write $[\Delta (g'_{ij}) = \Delta (B_{i}^{k}) \Delta (B_{j}^{m}) \Delta(g_{km}).]$

If the basis $[{\bf e}_{i}]$ is orthonormal, $[\Delta (g_{km})]$ and V are equal to one, $[\Delta (B_{j})]$ is equal to the volume V ′ of the cell built on the basis vectors $[{\bf e}'_{i}]$ and $[\Delta (g'_{ij}) = V'^{2}.]$ This relation is actually general and one can remove the prime index: $[\Delta (g_{ij}) = V^{2}. \eqno(1.1.2.21)]$

In the same way, we have for the corresponding reciprocal basis $[\Delta (g^{ij}) = V^{*2},]$ where $[V^{*}]$ is the volume of the reciprocal cell. Since the tables of the $[g_{ij}]$ 's and of the $[g^{ij}]$ 's are inverse, so are their determinants, and therefore the volumes of the unit cells of the direct and reciprocal spaces are also inverse, which is a very well known result in crystallography.

1.1.3. Mathematical notion of tensor

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1.1.3.1. Definition of a tensor

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For the mathematical definition of tensors, the reader may consult, for instance, Lichnerowicz (1947), Schwartz (1975) or Sands (1995).

1.1.3.1.1. Linear forms

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A linear form in the space $[E_{n}]$ is written $[T({\bf x}) = t_{i}x^{i},]$ where $[T({\bf x})]$ is independent of the chosen basis and the $[t_{i}]$ 's are the coordinates of T in the dual basis . Let us consider now a bilinear form in the product space $[E_{n} \otimes F_{p}]$ of two vector spaces with n and p dimensions, respectively: $[T({\bf x},{\bf y}) = t_{ij}x^{i}y\hskip1pt^{j}.]$

The np quantities $[t_{ij}]$ 's are, by definition, the components of a tensor of rank 2 and the form $[T({\bf x},{\bf y})]$ is invariant if one changes the basis in the space $[E_{n} \otimes F_{p}]$ . The tensor $[t_{ij}]$ is said to be twice covariant. It is also possible to construct a bilinear form by replacing the spaces $[E_{n}]$ and $[F_{p}]$ by their respective conjugates $[E^{n}]$ and $[F^{p}]$ . Thus, one writes $[T({\bf x},{\bf y}) = t_{ij}x^{i}y\hskip1pt^{j} = t\hskip1.5pt_{i}^{j} x^{i}y_{j} = t\hskip1pt^{i}_{j}x_{i}y\hskip1pt^{j} = t^{ij}x_{i}y_{j},]$ where $[t^{ij}]$ is the doubly contravariant form of the tensor, whereas $[t\hskip1pt_{i}^{j}]$ and $[t^{i}_{j}]$ are mixed, once covariant and once contravariant .

We can generalize by defining in the same way tensors of rank 3 or higher by using trilinear or multilinear forms. A vector is a tensor of rank 1, and a scalar is a tensor of rank 0.

1.1.3.1.2. Tensor product

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Let us consider two vector spaces, $[E_{n}]$ with n dimensions and $[F_{p}]$ with p dimensions, and let there be two linear forms, $[T({\bf x})]$ in $[E_{n}]$ and $[S({\bf y})]$ in $[F_{p}]$ . We shall associate with these forms a bilinear form called a tensor product which belongs to the product space with np dimensions, $[E_{n} \otimes F_{p}]$ : $[P({\bf x},{\bf y}) = T({\bf x}) \otimes S({\bf y}).]$

This correspondence possesses the following properties:

(i) it is distributive from the right and from the left;
(ii) it is associative for multiplication by a scalar;
(iii) the tensor products of the vectors with a basis $[E_{n}]$ and those with a basis $[F_{p}]$ constitute a basis of the product space.

The analytical expression of the tensor product is then $[\left.\matrix{T({\bf x}) = t_{i} x\hskip1pt^{j} \cr S({\bf y}) = s_{j} y^{i}\cr}\right\} P({\bf x},{\bf y}) = p_{ij}x^{i}y\hskip1pt^{j} = t_{i}x^{i}s_{j}y\hskip1pt^{j}= t_{i}s_{j}x^{i}y\hskip1pt^{j}.]$ One deduces from this that $[p_{ij} = t_{i}s_{j}.]$

It is a tensor of rank 2. One can equally well envisage the tensor product of more than two spaces, for example, $[E_{n} \otimes F_{p} \otimes G_{q}]$ in npq dimensions. We shall limit ourselves in this study to the case of affine tensors, which are defined in a space constructed from the product of the space $[E_{n}]$ with itself or with its conjugate $[E^{n}]$ . Thus, a tensor product of rank 3 will have $[n^{3}]$ components. The tensor product can be generalized as the product of multilinear forms. One can write, for example, $[\left.\matrix{P({\bf x}, {\bf y}, {\bf z}) = T({\bf x},{\bf y}) \otimes S({\bf z})\hfill\cr p\hskip1pt_{ik}^{j}x^{i}y_{j}z^{k} = t\hskip1pt_{i}^{j}x^{i}y_{j}s_{k}z^{k}.\hfill\cr}\right\} \eqno(1.1.3.1)]$

1.1.3.2. Behaviour under a change of basis

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A multilinear form is, by definition, invariant under a change of basis. Let us consider, for example, the trilinear form (1.1.3.1). If we change the system of coordinates, the components of vectors x, y, z become $[x^{i} = B_{\alpha}^{i} x'^{\alpha };\quad y_{j}= A_{j}^{\beta} y'_{\beta};\quad z^{k}= B_{\gamma}^{k}z'^{\gamma}.]$

Let us put these expressions into the trilinear form (1.1.3.1): $[P({\bf x}, {\bf y}, {\bf z}) = p\hskip1pt_{ik}^{j} B_{\alpha }^{i} x'^{\alpha}A_{j}^{\beta} y'_{\beta}B_{\gamma}^{k} z'^{\gamma}.]$

Now we can equally well make the components of the tensor appear in the new basis: $[P({\bf x}, {\bf y}, {\bf z}) = p'^{\beta}_{\alpha \gamma} x'^{\alpha}y'_{\beta }z'^{\gamma}.]$

As the decomposition is unique, one obtains $[p'^{\beta}_{\alpha \gamma} = p\hskip1pt_{ik}^{j} B_{\alpha}^{i} A_{j}^{\beta} B^{k}_{\gamma}. \eqno{(1.1.3.2)}]$

One thus deduces the rule for transforming the components of a tensor q times covariant and r times contravariant : they transform like the product of q covariant components and r contravariant components.

This transformation rule can be taken inversely as the definition of the components of a tensor of rank [n = q + r] .

Example. The operator O representing a symmetry operation has the character of a tensor. In fact, under a change of basis, O transforms into O′: $[O' = A O A^{-1}]$ so that $[O'^{i}_{j} = A_{k}^{i} O_{l}^{k} (A^{-1})_{j}^{l}.]$ Now the matrices A and B are inverses of one another: $[O'^{i}_{j} = A_{k}^{i} O_{l}^{k} B_{j}^{l}.]$ The symmetry operator is a tensor of rank 2, once covariant and once contravariant .

1.1.3.3. Operations on tensors

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1.1.3.3.1. Addition

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It is necessary that the tensors are of the same nature (same rank and same variance).

1.1.3.3.2. Multiplication by a scalar

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This is a particular case of the tensor product.

1.1.3.3.3. Contracted product, contraction

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Here we are concerned with an operation that only exists in the case of tensors and that is very important because of its applications in physics. In practice, it is almost always the case that tensors enter into physics through the intermediary of a contracted product.

(i) Contraction. Let us consider a tensor of rank 2 that is once covariant and once contravariant . Let us write its transformation in a change of coordinate system: $[t'^{j}_{i} = A\hskip1pt_{p}^{j} B_{i}^{q}t\kern1pt_{q}^{p}.]$

Now consider the quantity $[t'^{i}_{i}]$ derived by applying the Einstein convention $[(t'^{i}_{i} = t'^{1}_{1} + t'^{2}_{2} + t'^{3}_{3})]$ . It follows that $[\eqalignno{t'^{i}_{i} &= A^{i}_{p}B\hskip1pt^{q}_{i}t\hskip1pt^{p}_{q} = \delta^{q}_{p}t\hskip1pt^{q}_{q}\cr t'^{i}_{i} &= t\hskip1pt^{p}_{p}.\cr}]$

This is an invariant quantity and so is a scalar. This operation can be carried out on any tensor of rank higher than or equal to two, provided that it is expressed in a form such that its components are (at least) once covariant and once contravariant .

The contraction consists therefore of equalizing a covariant index and a contravariant index, and then in summing over this index. Let us take, for example, the tensor $[t'^{jk}_{i}]$ . Its contracted form is $[t'^{ik}_{i}]$ , which, with a change of basis, becomes $[t'^{ik}_{i} = A^{k}_{p}t\kern1pt^{qp}_{q}.]$

The components $[t\hskip1pt^{ik}_{i}]$ are those of a vector, resulting from the contraction of the tensor $[t\hskip1pt^{jk}_{i}]$ . The rank of the tensor has changed from 3 to 1. In a general manner, the contraction reduces the rank of the tensor from n to .

Example. Let us take again the operator of symmetry O. The trace of the associated matrix is equal to $[O^{1}_{1} + O^{2}_{2} + O^{3}_{3} = O^{i}_{i}.]$ It is the resultant of the contraction of the tensor O. It is a tensor of rank 0, which is a scalar and is invariant under a change of basis.
(ii) Contracted product. Consider the product of two tensors of which one is contravariant at least once and the other covariant at least once: $[p\hskip1pt^{jk}_{i} = t\hskip1pt^{j}_{i}z^{k}.]$ If we contract the indices i and k, it follows that $[p\hskip1pt^{ji}_{i} = t\hskip1pt^{j}_{i}z^{i}.]$

The contracted product is then a tensor of rank 1 and not 3. It is an operation that is very frequent in practice.
(iii) Scalar product. Next consider the tensor product of two vectors: $[t\hskip1pt^{j}_{i} = x_{i}y\hskip1pt^{j}.]$ After contraction, we get the scalar product: $[t\hskip1pt^{i}_{i} = x_{i}y^{i}.]$

1.1.3.4. Tensor nature of physical quantities

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Let us first consider the dielectric constant . In the introduction, we remarked that for an isotropic medium $[{\bf D} = \varepsilon {\bf E}.]$

If the medium is anisotropic, we have, for one of the components, $[D^{1} = \varepsilon^{1}_{1} E^{1} + \varepsilon^{1}_{2} E^{2}+ \varepsilon^{1}_{3} E^{3}.]$ This relation and the equivalent ones for the other components can also be written $[D^{i} = \varepsilon^i_{j}E\hskip1pt^{j} \eqno(1.1.3.3)]$ using the Einstein convention.

The scalar product of D by an arbitrary vector x is $[D^{i}x_{i} = \varepsilon^{i}_{j} E\hskip1pt^{j} x_{i}.]$

The right-hand member of this relation is a bilinear form that is invariant under a change of basis. The set of nine quantities $[\varepsilon^{i}_{j}]$ constitutes therefore the set of components of a tensor of rank 2. Expression (1.1.3.3) is the contracted product of $[\varepsilon^{i}_{j}]$ by $[E\hskip1pt^{j}]$ .

A similar demonstration may be used to show the tensor nature of the various physical properties described in Section 1.1.1 , whatever the rank of the tensor. Let us for instance consider the piezoelectric effect (see Section 1.1.4.4.3 ). The components of the electric polarization, $[P^{i}]$ , which appear in a medium submitted to a stress represented by the second-rank tensor $[T_{jk}]$ are $[P\hskip1pt^{i} = d\hskip1pt^{ijk}T_{jk},]$ where the tensor nature of $[T_{jk}]$ will be shown in Section 1.3.2 . If we take the contracted product of both sides of this equation by any vector of covariant components $[x_{i}]$ , we obtain a linear form on the left-hand side, and a trilinear form on the right-hand side, which shows that the coefficients $[d\hskip1pt^{ijk}]$ are the components of a third-rank tensor. Let us now consider the piezo-optic (or photoelastic ) effect (see Sections 1.1.4.10.5 and 1.6.7 ). The components of the variation $[\Delta \eta^{ij}]$ of the dielectric impermeability due to an applied stress are $[\Delta \eta^{ij} = \pi^{ijkl}T_{jl}.]$

In a similar fashion, consider the contracted product of both sides of this relation by two vectors of covariant components $[x_{i}]$ and $[y_{j}]$ , respectively. We obtain a bilinear form on the left-hand side, and a quadrilinear form on the right-hand side, showing that the coefficients $[\pi^{ijkl}]$ are the components of a fourth-rank tensor.

1.1.3.5. Representation surface of a tensor

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1.1.3.5.1. Definition

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Let us consider a tensor $[t_{ijkl\ldots}]$ represented in an orthonormal frame where variance is not important. The value of component $[t'_{1111\ldots}]$ in an arbitrary direction is given by $[t'_{1111\ldots} = t_{ijkl\ldots}B^{i}_{1}B\hskip1pt^{j}_{1}B^{k}_{1}B^{l}_{1}\ldots,]$ where the $[B^{i}_{1}]$ , $[B\hskip1pt^{j}_{1}, \ldots]$ are the direction cosines of that direction with respect to the axes of the orthonormal frame.

The representation surface of the tensor is the polar plot of $[t'_{1111\ldots}]$ .

1.1.3.5.2. Representation surfaces of second-rank tensors

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The representation surfaces of second-rank tensors are quadrics. The directions of their principal axes are obtained as follows. Let $[t_{ij}]$ be a second-rank tensor and let $[{\bf OM} = {\bf r}]$ be a vector with coordinates $[x_{i}]$ . The doubly contracted product , $[t_{ij}x^{i}x\hskip1pt^{j}]$ , is a scalar. The locus of points M such that $[t_{ij}x^{i}x\hskip1pt^{j} = 1]$ is a quadric. Its principal axes are along the directions of the eigenvectors of the matrix with elements $[t_{ij}]$ . They are solutions of the set of equations $[t_{ij}x^{i} = \lambda x\hskip1pt^{j},]$ where the associated quantities λ are the eigenvalues.

Let us take as axes the principal axes. The equation of the quadric reduces to $[t_{11}(x^{1})^{2} + t_{22}(x^{2})^{2} + t_{33}(x^{3})^{2} = 1.]$

If the eigenvalues are all of the same sign, the quadric is an ellipsoid; if two are positive and one is negative, the quadric is a hyperboloid with one sheet; if one is positive and two are negative, the quadric is a hyperboloid with two sheets (see Section 1.3.1 ).

Associated quadrics are very useful for the geometric representation of physical properties characterized by a tensor of rank 2, as shown by the following examples:

(i) Index of refraction of a medium. It is related to the dielectric constant by $[n = \varepsilon^{1/2}]$ and, like it, it is a tensor of rank 2. Its associated quadric is an ellipsoid, the optical indicatrix, which represents its variations with the direction in space (see Section 1.6.3.2 ).
(ii) Thermal expansion. If one cuts a sphere in a medium whose thermal expansion is anisotropic, and if one changes the temperature, the sphere becomes an ellipsoid. Thermal expansion is therefore represented by a tensor of rank 2 (see Chapter 1.4 ).
(iii) Thermal conductivity. Let us place a drop of wax on a plate of gypsum, and then apply a hot point at the centre. There appears a halo where the wax has melted: it is elliptical, indicating anisotropic conduction. Thermal conductivity is represented by a tensor of rank 2 and the elliptical halo of molten wax corresponds to the intersection of the associated ellipsoid with the plane of the plate of gypsum.

1.1.3.5.3. Representation surfaces of higher-rank tensors

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Examples of representation surfaces of higher-rank tensors are given in Sections 1.3.3.4.4 and 1.9.4.2 .

1.1.3.6. Change of variance of the components of a tensor

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1.1.3.6.1. Tensor nature of the metric tensor

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Equation (1.1.2.17) describing the behaviour of the quantities $[g_{ij} = {\bf e}_{i} \cdot {\bf e}_{j}]$ under a change of basis shows that they are the components of a tensor of rank 2, the metric tensor. In the same way, equation (1.1.2.19) shows that the $[g^{ij}]$ 's transform under a change of basis like the product of two contravariant coordinates. The coefficients $[g^{ij}]$ and $[g_{ij}]$ are the components of a unique tensor, in one case doubly contravariant , in the other case doubly covariant . In a general way, the Euclidean tensors (constructed in a space where one has defined the scalar product) are geometrical entities that can have covariant , contravariant or mixed components.

1.1.3.6.2. How to change the variance of the components of a tensor

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Let us take a tensor product $[t\hskip1pt^{ij}= x^{i}y\hskip1pt^{j}.]$ We know that $[x^{i}= g^{ik}x_{k} \ \ {\rm and} \ \ y\hskip1pt^{j}= g\hskip1pt^{jl }y_{l}.]$ It follows that $[t\hskip1pt^{ij}= g^{ik}g\hskip1pt^{jl }x_{k}y_{l}.]$ $[x_{k}y_{l}]$ is a tensor product of two vectors expressed in the dual space : $[x_{k}y_{l }= t_{kl}.]$

One can thus pass from the doubly covariant form to the doubly contravariant form of the tensor by means of the relation $[t\hskip1pt^{ij}= g^{ik}g\hskip1pt^{jl }t_{kl}.]$

This result is general: to change the variance of a tensor (in practice, to raise or lower an index), it is necessary to make the contracted product of this tensor using $[g^{ij}]$ or $[g_{ij}]$ , according to the case. For instance, $[t^{l}_{k}= g\hskip1pt^{jl }t_{l k}; \quad t\hskip1pt_{k}^{ij} = g_{kl}t\hskip1pt^{ijl}.]$

Remark. $[g^i_{j}= g^{ik}g_{kj}= \delta ^i_{j}.]$ This is a property of the metric tensor.

1.1.3.6.3. Examples of the use in physics of different representations of the same quantity

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Let us consider, for example, the force, F, which is a tensor quantity (tensor of rank 1). One can define it:

(i) by the fundamental law of dynamics: $[{\bf F} = m {\boldGamma},\quad \hbox{with }F^{i}= m \,\,\hbox{d}^{2}x^{i}/ {\rm d}t^{2},]$ where m is the mass and $[\boldGamma]$ is the acceleration. The force appears here in a contravariant form.
(ii) as the derivative of the energy, W: $[F_{i}= \partial W/\partial x^{i}= \partial _{i}W.]$

The force appears here in covariant form. In effect, we shall see in Section 1.1.3.8.1 that to form a derivative with respect to a variable contravariant augments the covariance by unity. The general expression of the law of dynamics is therefore written with the energy as follows: $[m \,\,{\rm d}^{2}x^{i}/ {\rm d}t^2 = g^{ij}\partial _{j}W.]$

1.1.3.7. Outer product

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1.1.3.7.1. Definition

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The tensor defined by $[{\bf x} \bigwedge {\bf y} = {\bf x} \otimes {\bf y} - {\bf y} \otimes {\bf x}]$ is called the outer product of vectors x and y. (Note: The symbol is different from the symbol $[\wedge]$ for the vector product.) The analytical expression of this tensor of rank 2 is $[\left. \matrix{{\bf x} & =x^{i}{\bf e}_{i} \cr {\bf y} & =y\hskip1pt^{j}{\bf e}_{j} \cr}\right\} \quad \Longrightarrow \quad{\bf x}\bigwedge {\bf y} = (x^{i}y\hskip1pt^{j}- y^{i}x\hskip1pt^{j})\, {\bf e}_{i} \otimes {\bf e}_{j}.]$

The components $[p^{ij} = x^{i}y\hskip1pt^{j}- y^{i}x\hskip1pt^{j}]$ of this tensor satisfy the properties $[p^{ij}= - p\hskip1pt^{ji} ;\quad p^{ii}= 0.]$ It is an antisymmetric tensor of rank 2.

1.1.3.7.2. Vector product

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Consider the so-called permutation tensor of rank 3 (it is actually an axial tensor – see Section 1.1.4.5.3 ) defined by $[\left\{ \matrix{\varepsilon _{ijk} = + 1 &\hbox{ if the permutation } ijk \hbox{ is even}\hfill\cr \varepsilon _{ijk} = - 1 &\hbox{ if the permutation } ijk \hbox{ is odd}\hfill\cr \varepsilon _{ijk} = 0\hfill &\hbox{ if at least two of the three indices are equal}\cr} \right.]$ and let us form the contracted product $[z_{k}= \textstyle{1\over 2} \varepsilon_{ijk}p^{ij}=\varepsilon_{ijk}x^iy^j. \eqno(1.1.3.4)]$ It is easy to check that $[\left\{\matrix{z_{1} = x^{2} y^{3} - y^{2} x^{3}\cr z_{2} = x^{3} y^{1} - y^{3} x^{1}\cr z_{3} = x^{1} y^{2} - y^{2} x^{1}.\cr}\right.]$

One recognizes the coordinates of the vector product.

1.1.3.7.3. Properties of the vector product

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Expression (1.1.3.4) of the vector product shows that it is of a covariant nature. This is indeed correct, and it is well known that the vector product of two vectors of the direct lattice is a vector of the reciprocal lattice [see Section 1.1.4 of Volume B of International Tables for Crystallography (2001)].

The vector product is a very particular vector which it is better not to call a vector: sometimes it is called a pseudovector or an axial vector in contrast to normal vectors or polar vectors. The components of the vector product are the independent components of the antisymmetric tensor $[p_{ij}]$ . In the space of n dimensions, one would write $[v_{i_{3}i_{4}\ldots i_{n}} = \textstyle{1 \over 2} \varepsilon_{i_{1}i_{2}\ldots i_{n}} p^{i_{1}i_{2}}.]$

The number of independent components of $[p^{ij}]$ is equal to $[(n^{2} - n)/2]$ or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of $[p^{ij}]$ are not the components of a vector in the space of four dimensions.

Let us also consider the behaviour of the vector product under the change of axes represented by the matrix $[\pmatrix{\bar{1} & 0 & 0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}.]$

This is a symmetry with respect to a point that transforms a right-handed set of axes into a left-handed set and reciprocally. In such a change, the components of a normal vector change sign. Those of the vector product, on the contrary, remain unchanged, indicating – as one well knows – that the orientation of the vector product has changed and that it is not, therefore, a vector in the normal sense, i.e. independent of the system of axes.

1.1.3.8. Tensor derivatives

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1.1.3.8.1. Interpretation of the coefficients of the matrix – change of coordinates

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We have under a change of axes: $[x^{\prime i}= A^i_{j}x\hskip1pt^{j}.]$ This shows that the new components, $[x'^{i}]$ , can be considered linear functions of the old components, $[x\hskip1pt^{j}]$ , and one can write $[A^i_{j}= \partial x'^{i}/\partial x\hskip1pt^{j}= \partial _{j} x'^{i}.]$ It should be noted that the covariance has been increased.

1.1.3.8.2. Generalization

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Consider a field of tensors $[t\hskip1pt^{j}_{i}]$ that are functions of space variables. In a change of coordinate system, one has $[t\hskip1pt^j_{i}= A^\alpha _{i}B^j_{\beta } t'^{\beta }_{\alpha }.]$ Differentiate with respect to $[x^{k}]$ : $[\eqalignno{{\partial t\hskip1pt^j_{i}\over \partial x^{k}} = \partial _{k}t\hskip1pt^j_{i} & = A^\alpha_{i}B^j_{\beta } {\partial t'^{\beta}_{\alpha }\over\partial x'^{\gamma }}{\partial x'^{\gamma }\over \partial x^{k}} &\cr \partial _{k}t\hskip1pt^j_{i} & =A^\alpha_{i}B^j_{\beta }A^\gamma_{k}\partial _{\gamma }t'^{\beta}_{\alpha }. &\cr}]$ It can be seen that the partial derivatives $[\partial _{k}t\hskip1pt^j_{i}]$ behave under a change of axes like a tensor of rank 3 whose covariance has been increased by 1 with respect to that of the tensor $[t\hskip1pt^j_{i}]$ . It is therefore possible to introduce a tensor of rank 1, $[{\boldnabla}]$ (nabla), of which the components are the operators given by the partial derivatives $[\partial /\partial x^{i}]$ .

1.1.3.8.3. Differential operators

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If one applies the operator nabla to a scalar φ, one obtains $[\hbox{grad } \varphi = {\boldnabla }\varphi.]$ This is a covariant vector in reciprocal space.

Now let us form the tensor product of $[{\boldnabla }]$ by a vector v of variable components. We then have $[{\boldnabla} \otimes {\bf v} = {\partial v\hskip1pt^{j}\over\partial x^{i}}{\bf e}_{i}\otimes {\bf e}\hskip1pt^{j}.]$

The quantities $[\partial _{i}v\hskip1pt^{j}]$ form a tensor of rank 2. If we contract it, we obtain the divergence of v: $[\hbox{div } {\bf v} = \partial _{i}v^{i}.]$ Taking the vector product, we get $[\hbox{curl } {\bf v} = {\boldnabla } \wedge {\bf v}.]$ The curl is then an axial vector .

1.1.3.8.4. Development of a vector function in a Taylor series

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Let $[{\bf u}({\bf r})]$ be a vector function. Its development as a Taylor series is written $[u^{i}({\bf r} + \hbox{d}{\bf r}) = u^{i}({\bf r}) + {\partial u^i \over \partial x^j}\,\, \hbox{d}x\hskip1pt^{j} + {\textstyle{1 \over 2}} {\partial^2 u^i \over \partial x^j \partial x^k}\,\, \hbox{d}x\hskip1pt^{j}\,\, \hbox{d}x^{k} + \ldots . \eqno (1.1.3.5)]$ The coefficients of the expansion, $[\partial u^{i}/\partial x\hskip1pt^{j}]$ , $[\partial ^{2}u^{i}/\partial x\hskip1pt^{j}\partial x^{k}, \ldots]$ are tensors of rank $[2, 3, \ldots]$ .

An example is given by the relation between displacement and electric field: $[D^{i}= \varepsilon^i _{j}E\hskip1pt^{j}+ \chi ^{i}_{jk}E\hskip1pt^{j}E^{k}+ \ldots ]$ (see Sections 1.6.2 and 1.7.2 ).

We see that the linear relation usually employed is in reality a development that is arrested at the first term. The second term corresponds to nonlinear optics. In general, it is very small but is not negligible in ferroelectric crystals in the neighbourhood of the ferroelectric–paraelectric transition. Nonlinear optics are studied in Chapter 1.7 .

1.1.4. Symmetry properties

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For the symmetry properties of the tensors used in physics, the reader may also consult Bhagavantam (1966), Billings (1969), Mason (1966), Nowick (1995), Nye (1985), Paufler (1986), Shuvalov (1988), Sirotin & Shaskol'skaya (1982), and Wooster (1973).

1.1.4.1. Introduction – Neumann's principle

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We saw in Section 1.1.1 that physical properties express in general the response of a medium to an impetus. It has been known for a long time that symmetry considerations play an important role in the study of physical phenomena. These considerations are often very fruitful and have led, for instance, to the discovery of piezoelectricity by the Curie brothers in 1880 (Curie & Curie, 1880 , 1881 ). It is not unusual for physical properties to be related to asymmetries. This is the case in electrical polarization, optical activity etc. The first to codify this role was the German physicist and crystallographer F. E. Neumann, who expressed in 1833 the symmetry principle, now called Neumann's principle: if a crystal is invariant with respect to certain symmetry elements, any of its physical properties must also be invariant with respect to the same symmetry elements (Neumann, 1885 ).

This principle may be illustrated by considering the optical properties of a crystal. In an anisotropic medium, the index of refraction depends on direction. For a given wave normal, two waves may propagate, with different velocities; this is the double refraction effect. The indices of refraction of the two waves vary with direction and can be found by using the index ellipsoid known as the optical indicatrix (see Section 1.6.3.2 ). Consider the central section of the ellipsoid perpendicular to the direction of propagation of the wave. It is an ellipse. The indices of the two waves that may propagate along this direction are equal to the semi-axes of that ellipse. There are two directions for which the central section is circular, and therefore two wave directions for which there is no double refraction. These directions are called optic axes, and the medium is said to be biaxial. If the medium is invariant with respect to a threefold, a fourfold or a sixfold axis (as in a trigonal, tetragonal or hexagonal crystal, for instance), its ellipsoid must also be invariant with respect to the same axis, according to Neumann's principle. As an ellipsoid can only be ordinary or of revolution, the indicatrix of a trigonal, tetragonal or hexagonal crystal is necessarily an ellipsoid of revolution that has only one circular central section and one optic axis. These crystals are said to be uniaxial. In a cubic crystal that has four threefold axes, the indicatrix must have several axes of revolution, it is therefore a sphere, and cubic media behave as isotropic media for properties represented by a tensor of rank 2.

1.1.4.2. Curie laws

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The example given above shows that the symmetry of the property may possess a higher symmetry than the medium. The property is represented in that case by the indicatrix. The symmetry of an ellipsoid is $[\eqalignno{{A_{2} \over M} {A'_{2} \over M'} {A''_{2} \over M''} C &= mmm \hbox{ for any ellipsoid}\cr&\quad \hbox{(orthorhombic symmetry)}\cr {A_{\infty} \over M} {\infty A_{2} \over \infty M} C &= {\infty \over m} m \hbox{ for an ellipsoid of revolution}\cr&\quad{\hbox{(cylindrical symmetry)}}\cr \infty {A_{\infty} \over M} C &= \infty {\infty \over m} \hbox{ for a sphere}\cr&\quad\hbox{(spherical symmetry)}.\cr}]$ [Axes $[A_{\infty}]$ are axes of revolution, or axes of isotropy, introduced by Curie (1884 , 1894 ), cf. International Tables for Crystallography (2005), Vol. A, Table 10.1.4.2 .]

The symmetry of the indicatrix is identical to that of the medium if the crystal belongs to the orthorhombic holohedry and is higher in all other cases.

This remark is the basis of the generalization of the symmetry principle by P. Curie (1859–1906). He stated that (Curie, 1894 ) :

(i) the symmetry characteristic of a phenomenon is the highest compatible with the existence of the phenomenon;
(ii) the phenomenon may exist in a medium that possesses that symmetry or that of a subgroup of that symmetry;

and concludes that some symmetry elements may coexist with the phenomenon but that their presence is not necessary. On the contrary, what is necessary is the absence of certain symmetry elements: `asymmetry creates the phenomenon' (`C'est la dissymétrie qui crée le phénomène'; Curie, 1894 , p. 400). Noting that physical phenomena usually express relations between a cause and an effect (an influence and a response), P. Curie restated the two above propositions in the following way, now known as Curie laws, although they are not, properly speaking, laws:

(i) the asymmetry of the effects must pre-exist in the causes;
(ii) the effects may be more symmetric than the causes.

The application of the Curie laws enable one to determine the symmetry characteristic of a phenomenon. Let us consider the phenomenon first as an effect. If Φ is the symmetry of the phenomenon and C the symmetry of the cause that produces it, $[C \leq \Phi.]$

Let us now consider the phenomenon as a cause producing a certain effect with symmetry E: $[\Phi \leq E.]$ We can therefore conclude that $[C \leq \Phi \leq E.]$

If we choose among the various possible causes the most symmetric one, and among the various possible effects the one with the lowest symmetry, we can then determine the symmetry that characterizes the phenomenon.

As an example, let us determine the symmetry associated with a mechanical force. A force can be considered as the result of a traction effort, the symmetry of which is $[A_{\infty} \infty M]$ . If considered as a cause, its effect may be the motion of a sphere in a given direction (for example, a spherical ball falling under its own weight). Again, the symmetry is $[A_{\infty} \infty M]$ . The symmetries associated with the force considered as a cause and as an effect being the same, we may conclude that $[A_{\infty} \infty M]$ is its characteristic symmetry.

1.1.4.3. Symmetries associated with an electric field and with magnetic induction (flux density)

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1.1.4.3.1. Symmetry of an electric field

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Considered as an effect, an electric field may have been produced by two circular coaxial electrodes, the first one carrying positive electric charges, the other one negative charges (Fig. 1.1.4.1 ). The cause possesses an axis of revolution and an infinity of mirrors parallel to it, $[A_{\infty }\infty M]$ . Considered as a cause, the electric field induces for instance the motion of a spherical electric charge parallel to itself. The associated symmetry is the same in each case, and the symmetry of the electric field is identical to that of a force, $[A_{\infty }\infty M]$ . The electric polarization or the electric displacement have the same symmetry.

Figure 1.1.4.1 | top | pdf |

Symmetry of an electric field.

1.1.4.3.2. Symmetry of magnetic induction

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The determination of the symmetry of magnetic quantities is more delicate. Considered as an effect, magnetic induction may be obtained by passing an electric current in a loop (Fig. 1.1.4.2 ). The corresponding symmetry is that of a cylinder rotating around its axis, $[(A_{\infty }/ M) C]$ . Conversely, the variation of the flux of magnetic induction through a loop induces an electric current in the loop. If the magnetic induction is considered as a cause, its effect has the same symmetry. The symmetry associated with the magnetic induction is therefore $[(A_{\infty }/ M) C]$ .

Figure 1.1.4.2 | top | pdf |

Symmetry of magnetic induction.

This symmetry is completely different from that of the electric field. This difference can be understood by reference to Maxwell's equations, which relate electric and magnetic quantities: $[\hbox{curl }{\bf E} = {\boldnabla}\wedge {\bf E} = - {\partial {\bf B} \over \partial t};\quad \hbox{curl } {\bf H} = {\boldnabla }\wedge {\bf H} = {\partial {\bf D} \over \partial t}.]$

It was seen in Section 1.1.3.8.3 that the curl is an axial vector because it is a vector product. Maxwell's equations thus show that if the electric quantities (E, D) are polar vectors, the magnetic quantities (B, H) are axial vectors and vice versa; the equations of Maxwell are, in effect, perfectly symmetrical on this point. Indeed, one could have been tempted to determine the symmetry of the magnetic field by considering interactions between magnets, which would have led to the symmetry $[A_{\infty }\infty M]$ for the magnetic quantities. However, in the world where we live and where the origin of magnetism is in the spin of the electron, the magnetic field is an axial vector of symmetry $[(A_{\infty }/ M) C]$ while the electric field is a polar vector of symmetry $[A_{\infty }\infty M]$ .

1.1.4.4. Superposition of several causes in the same medium – pyroelectricity and piezolectricity

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1.1.4.4.1. Introduction

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Let us now consider a phenomenon resulting from the superposition of several causes in the same medium. The symmetry of the global cause is the intersection of the groups of symmetry of the various causes: the asymmetries add up (Curie, 1894 ). This remark can be applied to the determination of the point groups where physical properties such as pyroelectricity or piezoelectricity are possible.

1.1.4.4.2. Pyroelectricity

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Pyroelectricity is the property presented by certain materials that exhibit electric polarization when the temperature is changed uniformly. Actually, this property appears in crystals for which the centres of gravity of the positive and negative charges do not coincide in the unit cell. They present therefore a spontaneous polarization that varies with temperature because, owing to thermal expansion, the distances between these centres of gravity are temperature dependent. A very important case is that of the ferroelectric crystals where the direction of the polarization can be changed under the application of an external electric field.

From the viewpoint of symmetry, pyroelectricity can be considered as the superposition of two causes, namely the crystal with its symmetry on one hand and the increase of temperature, which is isotropic, on the other. The intersection of the groups of symmetry of the two causes is in this case identical to the group of symmetry of the crystal. The symmetry associated with the effect is that of the electric polarization that is produced, $[A_{\infty }\infty M]$ . Since the asymmetry of the cause must pre-exist in the causes, the latter may not possess more than one axis of symmetry nor mirrors other than those parallel to the single axis. The only crystal point groups compatible with this condition are [1, 2, 3, 4, 6, m, 2mm, 3m, 4mm, 6mm.] There are therefore only ten crystallographic groups that are compatible with the pyroelectric effect. For instance, tourmaline, in which the effect was first observed, belongs to 3m.

1.1.4.4.3. Piezoelectricity

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Piezoelectricity, discovered by the Curie brothers (Curie & Curie, 1880 ), is the property presented by certain materials that exhibit an electric polarization when submitted to an applied mechanical stress such as a uniaxial compression (see, for instance, Cady, 1964 ; Ikeda, 1990 ). Conversely, their shape changes when they are submitted to an external electric field; this is the converse piezoelectric effect. The physical interpretation of piezoelectricity is the following: under the action of the applied stress, the centres of gravity of negative and positive charges move to different positions in the unit cell, which produces an electric polarization.

From the viewpoint of symmetry, piezoelectricity can be considered as the superposition of two causes, the crystal with its own symmetry and the applied stress. The symmetry associated with a uniaxial compression is that of two equal and opposite forces, namely $[A_\infty /M\ \ \infty A_2/\infty M C]$ . The effect is an electric polarization, of symmetry $[A_{\infty }\infty M]$ , which must be higher than or equal to the intersection of the symmetries of the two causes: $[{A_\infty \over M} {\infty A_{2} \over \infty M} C \bigcap S_{{\rm crystal}} \leq A_{\infty }\infty M,]$ where $[S_{\rm crystal}]$ denotes the symmetry of the crystal.

It may be noted that the effect does not possess a centre of symmetry. The crystal point groups compatible with the property of piezoelectricity are therefore among the 21 noncentrosymmetric point groups. More elaborate symmetry considerations show further that group 432 is also not compatible with piezoelectricity. This will be proved in Section 1.1.4.10.4 using the symmetry properties of tensors. There are therefore 20 point groups compatible with piezoelectricity: $[\eqalignno{&1, \ 2, \ m, \ 222, \ 2mm, &\cr &3, \ 32, \ 3m, \ 4, \ \bar{4}, \ 422, \ 4mm, \ \bar{4}2m, \ 6, \ \bar{6}, \ 622, \ 6mm, \ \bar{6}2m &\cr &23, \ \bar{4}3m. &\cr}]$ The intersection of the symmetries of the crystal and of the applied stress depend of course on the orientation of this stress relative to the crystallographic axes. Let us take, for instance, a crystal of quartz, which belongs to group $[32 = A_{3}3A_{2}]$ . The above condition becomes $[{A_\infty \over M} {\infty A_2 \over \infty M} C \bigcap A_33A_2 \leq A_{\infty }\infty M.]$ If the applied compression is parallel to the threefold axis, the intersection is identical to the symmetry of the crystal, $[A_{3}3A_{2}]$ , which possesses symmetry elements that do not exist in the effect, and piezoelectricity cannot appear. This is of course obvious because the threefold axis is not polar. For all other directions, piezoelectricity may appear.

1.1.4.5. Intrinsic symmetry of tensors

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1.1.4.5.1. Introduction

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The symmetry of a tensor representing a physical property or a physical quantity may be due either to its own nature or to the symmetry of the medium. The former case is called intrinsic symmetry. It is a property that can be exhibited both by physical property tensors or by field tensors. The latter case is the consequence of Neumann's principle and will be discussed in Section 1.1.4.6 . It applies to physical property tensors.

1.1.4.5.2. Symmetric tensors

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1.1.4.5.2.1. Tensors of rank 2

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A bilinear form is symmetric if $[T({\bf x},{\bf y}) = T({\bf y},{\bf x}).]$ Its components satisfy the relations $[t_{ij}= t_{ji}.]$

The associated matrix, T, is therefore equal to its transpose $[T^{T}]$ : $[T = \pmatrix { t_{11} & t_{12} & t_{13}\cr t_{21} & t_{22} & t_{23}\cr t_{31} & t_{32} & t_{33}\cr } = T^T = \pmatrix { t_{11} & t_{21} & t_{31}\cr t_{12} & t_{22} & t_{32}\cr t_{13} & t_{23} & t_{33}\cr}.]$ In a space with n dimensions, the number of independent components is equal to $[(n^{2}- n)/2 + n = (n^{2} + n)/2.]$

Examples

(1) The metric tensor (Section 1.1.2.2 ) is symmetric because the scalar product is commutative.
(2) The tensors representing one of the physical properties associated with the leading diagonal of the matrix of physical properties (Section 1.1.1.4 ), such as the dielectric constant . Let us take up again the demonstration of this case and consider a capacitor being charged. The variation of the stored energy per unit volume for a variation dD of the displacement is $[\hbox{d}W = {\bf E}\cdot\hbox{d}{\bf D},]$ where [equation (1.1.3.3)] $[D^{i}= \varepsilon ^i_{j}E\hskip1pt^{j}.]$

Since both $[D^{i}]$ and $[E\hskip1pt^{j}]$ are expressed through contravariant components, the expression for the energy should be written $[\hbox{d}W = g_{ij}E\hskip1pt^{j}\,\,\hbox{d}D^{i}.]$ If we replace $[D^{i}]$ by its expression, we obtain $[\hbox{d}W = g_{ij}\varepsilon ^i_{k}E\hskip1pt^{j} \,\,\hbox{d}E^{k} = \varepsilon _{jk}E\hskip1pt^{j}\,\,\hbox{d}E^{k},]$ where we have introduced the doubly covariant form of the dieletric constant tensor, $[\varepsilon _{jk}]$ . Differentiating twice gives $[{\partial ^{2}W\over \partial E^{k}\partial E\hskip1pt^{j}} = \varepsilon _{jk}.]$

If one can assume, as one usually does in physics, that the energy is a `good' function and that the order of the derivatives is of little importance, then one can write $[{\partial ^{2}W\over\partial E^{k}\partial E\hskip1pt^{j}} = {\partial ^{2}W\over \partial E\hskip1pt^{j}\partial E^{k}}.]$ As one can exchange the role of the dummy indices, one has $[\partial ^{2}W/(\partial E\hskip1pt^{j}\partial E^{k}) = \varepsilon _{kj}.]$ Hence one deduces that $[\varepsilon _{jk}= \varepsilon _{kj}.]$

The dielectric constant tensor is therefore symmetric. One notes that the symmetry is conveyed on two indices of the same variance. One could show in a similar way that the tensor representing magnetic susceptibility is symmetric.
(3) There are other possible causes for the symmetry of a tensor of rank 2. The strain tensor (Section 1.3.1 ), which is a field tensor, is symmetric because one does not take into account the rotative part of the deformation; the stress tensor, also a field tensor (Section 1.3.1 ), is symmetric because one neglects body torques (couples per unit volume); the thermal conductivity tensor is symmetric because circulating flows do not produce any detectable effects etc.

1.1.4.5.2.2. Tensors of higher rank

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A tensor of rank higher than 2 may be symmetric with respect to the indices of one or more couples of indices. For instance, by its very nature, the demonstration given in Section 1.1.1.4 shows that the tensors representing principal physical properties are of even rank. If n is the rank of the associated square matrix, the number of independent components is equal to $[(n^{2} + n)/2]$ . In the case of a tensor of rank 4, such as the tensor of elastic constants relating the strain and stress tensors (Section 1.3.3.2.1 ), the number of components of the tensor is $[3^{4} = 81]$ . The associated matrix is a $[ 9 \times 9]$ one, and the number of independent components is equal to 45.

1.1.4.5.3. Antisymmetric tensors – axial tensors

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1.1.4.5.3.1. Tensors of rank 2

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A bilinear form is said to be antisymmetric if $[T({\bf x},{\bf y}) = - T({\bf y},{\bf x}).]$ Its components satisfy the relations $[t_{ij}= - t_{ji}.]$ The associated matrix, T, is therefore also antisymmetric: $[T = - T^{T}= \pmatrix { 0 & t_{12} & t_{13}\cr - t_{12} & 0 & t_{23}\cr - t_{13} & - t_{23} & 0\cr}.]$ The number of independent components is equal to $[(n^{2} - n)/2]$ , where n is the number of dimensions of the space. It is equal to 3 in a three-dimensional space, and one can consider these components as those of a pseudovector or axial vector . It must never be forgotten that under a change of basis the components of an axial vector transform like those of a tensor of rank 2.

Every tensor can be decomposed into the sum of two tensors, one symmetric and the other one antisymmetric: [T = S + A,] with $[ S = ( T + T^{T})/2]$ and $[A = (T - T^{T})/2]$ .

Example. As shown in Section 1.1.3.7.2 , the components of the vector product of two vectors, x and y, $[z_{k}= \varepsilon _{ijk}x^{i}y\hskip1pt^{j},]$ are really the independent components of an antisymmetric tensor of rank 2. The magnetic quantities, B, H (Section 1.1.4.3.2 ), the tensor representing the pyromagnetic effect (Section 1.1.1.3 ) etc. are axial tensors.

1.1.4.5.3.2. Tensors of higher rank

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If the rank of the tensor is higher than 2, the tensor may be antisymmetric with respect to the indices of one or several couples of indices.

(i) Tensors of rank 3 antisymmetric with respect to every couple of indices. A trilinear form $[T({\bf x},{\bf y},{\bf z}) = t_{ijk}x^{i}y\hskip1pt^{j}z^{k}]$ is said to be antisymmetric if it satifies the relations $[\left.\matrix{T({\bf x},{\bf y},{\bf z}) & = - T({\bf y},{\bf x},{\bf z})\hfill\cr & = - T({\bf x},{\bf z},{\bf y})\hfill\cr & = - T({\bf z},{\bf y},{\bf x}).\cr}\right\}]$

Tensor $[t_{ijk}]$ has 27 components. It is found that all of them are equal to zero, except $[t_{123}= t_{231}= t_{312}= - t_{213}= - t_{132}= - t_{321}.]$

The three-times contracted product with the permutations tensor (Section 1.1.3.7.2 ), $[(1/6)\varepsilon_{ijk}t_{ijk}]$ , is a pseudoscalar or axial scalar. It is not a usual scalar: the sign of this product changes when one changes the hand of the reference axes, change of basis represented by the matrix $[\pmatrix {\bar 1 & 0 & 0\cr 0 & \bar 1 & 0\cr 0 & 0 & \bar 1}.]$

Form $[T({\bf x},{\bf y},{\bf z})]$ can also be written $[T({\bf x},{\bf y},{\bf z}) = Pt_{123},]$ where $[P=\varepsilon_{ijk}x^iy^jz^k=\left|\matrix{x^1&x^2&x^3\cr y^1&y^2&y^3\cr z^1&z^2&z^3\cr}\right|]$ is the triple scalar product of the three vectors x, y, z: $[P=({\bf x}, {\bf y}, {\bf z})=({\bf x}\wedge{\bf y}\cdot{\bf z}).]$ It is also a pseudoscalar. The permutation tensor is not a real tensor of rank 3: if the hand of the axes is changed, the sign of P also changes; P is therefore not a trilinear form.

Another example of a pseudoscalar is given by the rotatory power of an optically active medium, which is expressed through the relation (see Section 1.6.5.4 ) $[\theta = \rho d,]$ where θ is the rotation angle of the light wave, d the distance traversed in the material and ρ is a pseudoscalar: if one takes the mirror image of this medium, the sign of the rotation of the light wave also changes.
(ii) Tensor of rank 3 antisymmetric with respect to one couple of indices. Let us consider a trilinear form such that $[T({\bf x},{\bf y},{\bf z}) = - T({\bf y},{\bf x},{\bf z}).]$ Its components satisfy the relation $[t\hskip1pt^{iil} = 0; \quad t\hskip1pt^{ijl} = -t\hskip1pt^{jil}.]$

The twice contracted product $[t\hskip1pt_{k}^{l} = \textstyle{1 \over 2}\varepsilon_{ijk}t\hskip1pt^{ijl}]$ is an axial tensor of rank 2 whose components are the independent components of the antisymmetric tensor of rank 3, $[t\hskip1pt^{ijl}]$ .

Examples

(1) Hall constant. The Hall effect is observed in semiconductors. If one takes a semiconductor crystal and applies a magnetic induction B and at the same time imposes a current density j at right angles to it, one observes an electric field E at right angles to the other two fields (see Section 1.8.3.4 ). The expression for the field can be written $[E_{i} = R_{H \ ikl}\hskip1pt j_{k}B_{l},]$ where $[R_{H \ ikl}]$ is the Hall constant, which is a tensor of rank 3. However, because the direction of the current density is imposed by the physical law (the set of vectors B, j, E constitutes a right-handed frame), one has $[R_{H \ ikl} = -R_{H \ kil},]$ which shows that $[R_{H \ ikl}]$ is an antisymmetric (axial) tensor of rank 3. As can be seen from its physical properties, only the components such that $[i \neq k \neq l]$ are different from zero. These are $[R_{H \ 123} = - R_{H \ 213}; \quad R_{H \ 132} = - R_{H \ 312}; \quad R_{H \ 312}; \quad R_{H \ 321}.]$
(2) Optical rotation. The gyration tensor used to describe the property of optical rotation presented by gyrotropic materials (see Section 1.6.5.4 ) is an axial tensor of rank 2, which is actually an antisymmetric tensor of rank 3.
(3) Acoustic activity. The acoustic gyrotropic tensor describes the rotation of the polarization plane of a transverse acoustic wave propagating along the acoustic axis (see for instance Kumaraswamy & Krishnamurthy, 1980 ). The elastic constants may be expanded as $[c_{ijkl}(\omega, {\bf k}) = c_{ijkl}(\omega) + \hbox{i} d_{ijklm}(\omega)k_{m} + \ldots,]$ where $[d_{ijklm}]$ is a fifth-rank tensor. Time-reversal invariance requires that $[d_{ijklm} =- d_{jiklm}]$ , which shows that it is an antisymmetric (axial) tensor.

1.1.4.5.3.3. Properties of axial tensors

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The two preceding sections have shown examples of axial tensors of ranks 0 (pseudoscalar), 1 (pseudovector ) and 2. They have in common that all their components change sign when the sign of the basis is changed, and this can be taken as the definition of an axial tensor. Their components are the components of an antisymmetric tensor of higher rank. It is important to bear in mind that in order to obtain their behaviour in a change of basis, one should first determine the behaviour of the components of this antisymmetric tensor.

1.1.4.6. Symmetry of tensors imposed by the crystalline medium

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Many papers have been devoted to the derivation of the invariant components of physical property tensors under the influence of the symmetry elements of the crystallographic point groups: see, for instance, Fumi (1951 , 1952a ,b ,c , 1987 ), Fumi & Ripamonti (1980a ,b ), Nowick (1995), Nye (1957 , 1985 ), Sands (1995), Sirotin & Shaskol'skaya (1982), and Wooster (1973). There are three main methods for this derivation: the matrix method (described in Section 1.1.4.6.1 ), the direct inspection method (described in Section 1.1.4.6.3 ) and the group-theoretical method (described in Section 1.2.4 and used in the accompanying software, see Section 1.2.7.4 ).

1.1.4.6.1. Matrix method – application of Neumann's principle

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An operation of symmetry turns back the crystalline edifice on itself; it allows the physical properties of the crystal and the tensors representing them to be invariant. An operation of symmetry is equivalent to a change of coordinate system. In a change of system, a tensor becomes $[t'^{\alpha \beta}_{\gamma \delta} = t^{ij}_{kl}A^{\alpha}_{i} A^{\beta}_{j}B^{k}_{\gamma}B^{l}_{\delta}.]$ If A represents a symmetry operation, it is a unitary matrix: $[A = B^{T} = B^{-1}.]$

Since the tensor is invariant under the action of the symmetry operator A, one has, according to Neumann's principle, $[t'^{\alpha \beta}_{\gamma \delta } = t^{\alpha \beta}_{\gamma \delta}]$ and, therefore, $[t^{\alpha \beta}_{\gamma \delta} = t^{ij}_{kl} A^{\alpha}_{i} A^{\beta}_{j}B^{k}_{\gamma}B^{l}_{\delta}. \eqno(1.1.4.1)]$

There are therefore a certain number of linear relations between the components of the tensor and the number of independent components is reduced. If there are p components and q relations between the components, there are [p-q] independent components. This number is independent of the system of axes. When applied to each of the 32 point groups, this reduction enables one to find the form of the tensor in each case. It depends on the rank of the tensor. In the present chapter, the reduction will be derived for tensors up to the fourth rank and for all crystallographic groups as well as for the isotropic groups. An orthonormal frame will be assumed in all cases, so that co- and contravariance will not be apparent and the positions of indices as subscripts or superscripts will not be meaningful. The $[Ox_{3}]$ axis will be chosen parallel to the threefold, fourfold or sixfold axis in the trigonal, tetragonal and hexagonal systems. The accompanying software to the present volume enables the reduction for tensors of any rank to be derived.

1.1.4.6.2. The operator A is in diagonal form

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1.1.4.6.2.1. Introduction

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If one takes as the system of axes the eigenvectors of the operator A, the matrix is written in the form $[\pmatrix { \exp {\rm i} \theta & 0 & 0\cr 0 & \exp -{\rm i} \theta & 0 \cr 0 & 0 & \pm 1 \cr },]$ where θ is the rotation angle, $[Ox_{3}]$ is taken parallel to the rotation axis and coefficient $[A_{3}]$ is equal to +1 or −1 depending on whether the rotation axis is direct or inverse (proper or improper operator).

The equations (1.1.4.1) can then be simplified and reduce to $[t^{ij}_{kl} = t^{ij}_{kl} A^\alpha_{i}A^\beta_{j}B^k_{\gamma }B^l_{\delta } \eqno{(1.1.4.2)}]$ (without any summation).

If the product $[A^{i}_{i}A\hskip1pt^{j}_{j}B^{k}_{k}B^{l}_{l}]$ (without summation) is equal to unity, equation (1.1.4.2) is trivial and there is significance in the component $[t_{kl}]$ . On the contrary, if it is different from 1, the only solution for (1.1.4.2) is that $[t^{ij}_{kl } = 0]$ . One then finds immediately that certain components of the tensor are zero and that others are unchanged.

1.1.4.6.2.2. Case of a centre of symmetry

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All the diagonal components are in this case equal to −1. One thus has:

(i) Tensors of even rank, $[t\hskip1pt^{ij\ldots}= (-1)^{2p}t\hskip1pt^{ij\ldots}]$ . The components are not affected by the presence of the centre of symmetry. The reduction of tensors of even rank is therefore the same in a centred group and in its noncentred subgroups, that is in any of the 11 Laue classes: $[\matrix{&\bar{{\bf 1}} \quad 1\hfill\cr &{\bf 2/{\bi m}} \quad 2,\ m \hfill\cr &{\bi mmm} \quad 222,\ 2mm \hfill\cr &\bar{{\bf 3}} \quad 3\hfill\cr &{\bar{\bf 3}}{\bi m} \quad 32, \ 3m\hfill\cr &{\bf 4/{\bi m}} \quad {\bar 4}, \ 4 \hfill\cr &{\bf 4/{\bi m m}} \quad {\bar 4}2m, \ 422, \ 4mm\hfill\cr &{\bf 6/{\bi m}} \quad {\bar 6}, \ 6 \hfill\cr &{\bf 6/{\bi m m}} \quad {\bar 6}2m, \ 622, \ 6mm\hfill\cr &{\bi m}{\bar{\bf 3}} \quad 23\hfill\cr &{\bi m}{\bar{\bf 3}}{\bi m} \quad 432, \ {\bar 4}32.\hfill\cr}]$ If a tensor is invariant with respect to two elements of symmetry, it is invariant with respect to their product. It is then sufficient to make the reduction for the generating elements of the group and (since this concerns a tensor of even rank) for the 11 Laue classes.
(ii) Tensors of odd rank, $[t\hskip1pt^{ij\ldots}= (-1)^{2p+1}t\hskip1pt^{ij\ldots}]$ . All the components are equal to zero. The physical properties represented by tensors of rank 3, such as piezoelectricity, piezomagnetism, nonlinear optics, for instance, will therefore not be present in a centrosymmetric crystal.

1.1.4.6.2.3. General case

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By replacing the matrix coefficients $[A^i_{i}]$ by their expression, (1.1.4.2) becomes, for a proper rotation, $[t\hskip1pt^{jk\ldots }= t\hskip1pt^{jk\ldots}\exp(ir\theta)\exp (-is\theta) (1)^{t}= t\hskip1pt^{jk\ldots} \exp i(r-s)\theta,]$ where r is the number of indices equal to 1, s is the number of indices equal to 2, t is the number of indices equal to 3 and [r + s + t = p] is the rank of the tensor. The component $[t\hskip1pt^{jk\ldots}]$ is not affected by the symmetry operation if $[(r - s)\theta = 2K\pi,]$ where K is an integer, and is equal to zero if $[(r - s)\theta \neq 2K\pi.]$

The angle of rotation θ can be put into the form $[2\pi /q]$ , where q is the order of the axis. The condition for the component not to be zero is then [r - s = Kq.]

The condition is fulfilled differently depending on the rank of the tensor, p, and the order of the axis, q. Indeed, we have $[r - s \leq p]$ and

, $[r - s \leq 2]$ : the result of the reduction will be the same for any $[q \geq 3]$ ;
, $[r - s \leq 3]$ : the result of the reduction will be the same for any $[q \geq 4]$ ;
, $[r - s \leq 4]$ : the result of the reduction will be the same for any $[q \geq 5]$ .

It follows that:

(i) for tensors of rank 2, the reduction will be the same for trigonal (threefold axis), tetragonal (fourfold axis) and hexagonal (sixfold axis) groups;
(ii) for tensors of rank 3, the reduction will be the same for tetragonal and hexagonal groups;
(iii) for tensors of rank 4, the reduction will be different for trigonal, tetragonal and hexagonal groups.

The inconvenience of the diagonalization method is that the vectors and eigenvalues are, in general, complex, so in practice one uses another method. For instance, we may note that equation (1.1.4.1) can be written in the case of [p = 2] by associating with the tensor a $[3 \times 3]$ matrix T: $[T = B T B^{T},]$ where B is the symmetry operation. Through identification of homologous coefficients in matrices T and $[B T B^{T}]$ , one obtains relations between components $[t_{ij}]$ that enable the determination of the independent components.

1.1.4.6.3. The method of direct inspection

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The method of `direct inspection', due to Fumi (1952a ,b , 1987 ), is very simple. It is based on the fundamental properties of tensors; the components transform under a change of basis like a product of vector components (Section 1.1.3.2 ).

Examples

(1) Let us consider a tensor of rank 3 invariant with respect to a twofold axis parallel to $[Ox_{3}]$ . The matrix representing this operator is $[\pmatrix{{\bar 1} &0 &0\cr 0 &{\bar 1} &0\cr 0 &0 &1\cr}.]$

The component $[t_{ijk}]$ behaves under a change of axes like the product of the components $[x_{i}, x_{j}, x_{k}]$ . The components $[x_{1}, x_{2}, x_{3}]$ of a vector become, respectively, $[-x_{1}]$ , $[ -x_{2}]$ , $[x_{3}]$ . To simplify the notation, we shall denote the components of the tensor simply by ijk. If, amongst the indices i, j and k, there is an even number (including the number zero) of indices that are equal to 3, the product $[x_{i}x_{j}x_{k}]$ will become $[-x_{i}x_{j}x_{k}]$ under the rotation. As the component `ijk' remains invariant and is also equal to its opposite, it must be zero. 14 components will thus be equal to zero:
(2) Let us now consider that the same tensor of rank 3 is invariant with respect to a fourfold axis parallel to $[Ox_{3}]$ . The matrix representing this operator and its action on a vector of coordinates $[x_{1}, x_{2}, x_{3}]$ is given by $[\pmatrix{\phantom{-}x_{2}\cr -x_{1}\cr \phantom{-}x_{3}\cr} = \pmatrix{0 &1 &0\cr {\bar 1} &0 &0\cr 0 &0 &1\cr} \pmatrix{x_{1}\cr x_{2}\cr x_{3}\cr}. \eqno(1.1.4.3)]$ Coordinate $[x_{1}]$ becomes $[x_{2}]$ , $[x_{2}]$ becomes $[-x_{1}]$ and $[x_{3}]$ becomes $[x_{3}]$ . Component ijk transforms like product $[x^{i}x\hskip1pt^{j}x^{k}]$ according to the rule given above. Since the twofold axis parallel to $[Ox_{3}]$ is a subgroup of the fourfold axis, we can start from the corresponding reduction. We find $[\matrix{311 &\Longleftrightarrow &\phantom{-}322 &: &t_{311} = &\phantom{-}t_{322}\phantom{.}\cr 123 &\Longleftrightarrow &-(213) &: &t_{123} = &-t_{213}\phantom{.}\cr 113 &\Longleftrightarrow &\phantom{-}223 &: &t_{113} = &\phantom{-}t_{223}\phantom{.}\cr 333 &\Longleftrightarrow &\phantom{-}333 &: &t_{333} = &\phantom{-}t_{333}\phantom{.}\cr 132 &\Longleftrightarrow &-(231) &: &t_{132} = &-t_{231}\phantom{.}\cr 131 &\Longleftrightarrow &\phantom{-}232 &: &t_{131} = &\phantom{-}t_{232}\phantom{.}\cr 312 &\Longleftrightarrow &-(321) &: &t_{312} = &-t_{321}.\cr}]$ All the other components are equal to zero.

It is not possible to apply the method of direct inspection for point group 3. One must in this case use the matrix method described in Section 1.1.4.6.2 ; once this result is assumed, the method can be applied to all other point groups.

1.1.4.7. Reduction of the components of a tensor of rank 2

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The reduction is given for each of the 11 Laue classes.

1.1.4.7.1. Triclinic system

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Groups $[{\bar 1}]$ , 1: no reduction, the tensor has 9 independent components. The result is represented in the following symbolic way (Nye, 1957 , 1985 ): [Scheme scheme1] where the sign • represents a nonzero component.

1.1.4.7.2. Monoclinic system

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Groups 2m, 2, m: it is sufficient to consider the twofold axis or the mirror. As the representative matrix is diagonal, the calculation is immediate. Taking the twofold axis to be parallel to $[Ox_{3}]$ , one has $[t^{1}_{3} = t^{3}_{1} = t^{2}_{3} = t^{3}_{2} = 0.]$

The other components are not affected. The result is represented as [Scheme scheme2]

There are 5 independent components. If the twofold axis is taken along axis $[Ox_{2}]$ , which is the usual case in crystallography, the table of independent components becomes [Scheme scheme3]

1.1.4.7.3. Orthorhombic system

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Groups mmm, 2mm, 222: the reduction is obtained by considering two perpendicular twofold axes, parallel to $[Ox_{3}]$ and to $[Ox_{2}]$ , respectively. One obtains [Scheme scheme4]

There are 3 independent components.

1.1.4.7.4. Trigonal, tetragonal, hexagonal and cylindrical systems

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We remarked in Section 1.1.4.6.2.3 that, in the case of tensors of rank 2, the reduction is the same for threefold, fourfold or sixfold axes. It suffices therefore to perform the reduction for the tetragonal groups. That for the other systems follows automatically.

1.1.4.7.4.1. Groups $[{\bar 3}]$ , ; , $[{\bar 4}]$ , ; , $[{\bar 6}]$ , ; $[(A_{\infty}/M)C]$ , $[A_{\infty}]$

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If we consider a fourfold axis parallel to $[Ox_{3}]$ represented by the matrix given in (1.1.4.3), by applying the direct inspection method one finds [Scheme scheme5] where the symbol ⊖ means that the corresponding component is numerically equal to that to which it is linked, but of opposite sign. There are 3 independent components.

1.1.4.7.4.2. Groups $[{\bar 3}m]$ , , ; , , , $[{\bar {4}}{2}m]$ ; , , , $[{\bar{6}}{2}m]$ ; $[(A_{\infty}/M) \infty (A_{2}/M)C]$ , $[A_{\infty}\infty A_{2}]$

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The result is obtained by combining the preceding result and that corresponding to a twofold axis normal to the fourfold axis. One finds [Scheme scheme6]

There are 2 independent components.

1.1.4.7.5. Cubic and spherical systems

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The cubic system is characterized by the presence of threefold axes along the $[\langle 111 \rangle]$ directions. The action of a threefold axis along [111] on the components $[x_{1}, x_{2}, x_{3}]$ of a vector results in a permutation of these components, which become, respectively, $[x_{2}, x_{3}, x_{1}]$ and then $[x_{3}, x_{1}, x_{2}]$ . One deduces that the components of a tensor of rank 2 satisfy the relations $[t^{1}_{1} = t^{2}_{2} = t^{3}_{3}.]$

The cubic groups all include as a subgroup the group 23 of which the generating elements are a twofold axis along $[Ox_{3}]$ and a threefold axis along [111]. If one combines the corresponding results, one deduces that $[t^{2}_{1} = t^{3}_{2} = t^{1}_{3} = t^{3}_{1} = t^{1}_{2} = t^{2}_{3} = 0,]$ which can be summarized by [Scheme scheme7]

There is a single independent component and the medium behaves like a property represented by a tensor of rank 2, like an isotropic medium.

1.1.4.7.6. Symmetric tensors of rank 2

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If the tensor is symmetric, the number of independent components is still reduced. One obtains the following, representing the nonzero components for the leading diagonal and for one half of the others.

1.1.4.7.6.1. Triclinic system

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[Scheme scheme8]

There are 6 independent components. It is possible to interpret the number of independent components of a tensor of rank 2 by considering the associated quadric, for instance the optical indicatrix. In the triclinic system, the quadric is any quadric. It is characterized by six parameters: the lengths of the three axes and the orientation of these axes relative to the crystallographic axes.

1.1.4.7.6.2. Monoclinic system (twofold axis parallel to )

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[Scheme scheme9]

There are 4 independent components. The quadric is still any quadric, but one of its axes coincides with the twofold axis of the monoclinic lattice. Four parameters are required: the lengths of the axes and one angle.

1.1.4.7.6.3. Orthorhombic system

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[Scheme scheme10]

There are 3 independent components. The quadric is any quadric, the axes of which coincide with the crystallographic axes. Only three parameters are required.

1.1.4.7.6.4. Trigonal, tetragonal and hexagonal systems, isotropic groups

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[Scheme scheme11]

There are 2 independent components. The quadric is of revolution. It is characterized by two parameters: the lengths of its two axes.

1.1.4.7.6.5. Cubic system

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[Scheme scheme12]

There is 1 independent component. The associated quadric is a sphere.

1.1.4.8. Reduction of the components of a tensor of rank 3

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1.1.4.8.1. Triclinic system

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1.1.4.8.1.1. Group

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All the components are independent. Their number is equal to 27. They are usually represented as a $[3\times 9]$ matrix which can be subdivided into three $[3\times 3]$ submatrices: $[\left(\matrix{111 &122 &133\cr \noalign{\vskip3pt} 211 &222 &233\cr \noalign{\vskip3pt} 311 &322 &333\cr} \left|\matrix{123 &131 &112\cr \noalign{\vskip3pt} 223 &231 &212\cr \noalign{\vskip3pt} 323 &331 &312\cr}\right| \matrix{132 &113 &121\cr \noalign{\vskip3pt} 232 &213 &221\cr \noalign{\vskip3pt} 332 &313 &321\cr}\right).]$

1.1.4.8.1.2. Group $[{\bar 1}]$

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All the components are equal to zero.

1.1.4.8.2. Monoclinic system

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1.1.4.8.2.1. Group

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Choosing the twofold axis parallel to $[Ox_{3}]$ and applying the direct inspection method, one finds [Scheme scheme13]

There are 13 independent components. If the twofold axis is parallel to $[Ox_{2}]$ , one finds [Scheme scheme14]

1.1.4.8.2.2. Group m

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One obtains the matrix representing the operator m by multiplying by −1 the coefficients of the matrix representing a twofold axis. The result of the reduction will then be exactly complementary: the components of the tensor which include an odd number of 3's are now equal to zero. One writes the result as follows: [Scheme scheme15]

There are 14 independent components. If the mirror axis is normal to $[Ox_{2}]$ , one finds [Scheme scheme16]

1.1.4.8.2.3. Group

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All the components are equal to zero.

1.1.4.8.3. Orthorhombic system

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1.1.4.8.3.1. Group

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There are three orthonormal twofold axes. The reduction is obtained by combining the results associated with two twofold axes, parallel to $[Ox_{3}]$ and $[Ox_{2}]$ , respectively. [Scheme scheme17]

There are 6 independent components.

1.1.4.8.3.2. Group

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The reduction is obtained by combining the results associated with a twofold axis parallel to $[Ox_{3}]$ and with a mirror normal to $[Ox_{2}]$ : [Scheme scheme18]

There are 7 independent components.

1.1.4.8.3.3. Group

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All the components are equal to zero.

1.1.4.8.4. Trigonal system

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1.1.4.8.4.1. Group

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The threefold axis is parallel to $[Ox_{3}]$ . The matrix method should be used here. One finds [Scheme scheme19]

There are 9 independent components.

1.1.4.8.4.2. Group with a twofold axis parallel to $[Ox_{1}]$

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[Scheme scheme20]

There are 4 independent components.

1.1.4.8.4.3. Group with a mirror normal to $[Ox_{1}]$

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[Scheme scheme21]

There are 4 independent components.

1.1.4.8.4.4. Groups $[\bar{3}]$ and $[\bar{3}m]$

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All the components are equal to zero.

1.1.4.8.5. Tetragonal system

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1.1.4.8.5.1. Group

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The method of direct inspection can be applied for a fourfold axis. One finds [Scheme scheme22]

There are 7 independent components.

1.1.4.8.5.2. Group

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One combines the reductions for groups 4 and 222: [Scheme scheme23]

There are 3 independent components.

1.1.4.8.5.3. Group

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One combines the reductions for groups 4 and 2m: [Scheme scheme24]

There are 4 independent components.

1.1.4.8.5.4. Group

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All the components are equal to zero.

1.1.4.8.5.5. Group $[\bar{4}]$

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The matrix corresponding to axis $[\bar{4}]$ is $[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$ and the form of the $[3 \times 9]$ matrix is [Scheme scheme25]

There are 6 independent components.

1.1.4.8.5.6. Group $[\bar{4}2m]$

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One combines either the reductions for groups $[\bar{4}]$ and 222, or the reductions for groups $[\bar{4}]$ and 2mm.

(i) Twofold axis parallel to $[Ox_{1}]$ :

There are 6 independent components.
(ii) Mirror perpendicular to $[Ox_{1}]$ (the twofold axis is at $[45 ^{\circ}]$ )

The number of independent components is of course the same, 6.

1.1.4.8.5.7. Group

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All the components are equal to zero.

1.1.4.8.6. Hexagonal and cylindrical systems

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1.1.4.8.6.1. Groups , $[A_{\infty}]$ , , $[A_{\infty} \infty A_{2}]$ , and $[A_{\infty} \infty M]$

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It was shown in Section 1.1.4.6.2.3 that, in the case of tensors of rank 3, the reduction is the same for axes of order 4, 6 or higher. The reduction will then be the same as for the tetragonal system.

1.1.4.8.6.2. Group $[\bar{6} = 3/m]$

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One combines the reductions for the groups corresponding to a threefold axis parallel to $[Ox_{3}]$ and to a mirror perpendicular to $[Ox_{3}]$ : [Scheme scheme28]

There are 2 independent components.

1.1.4.8.6.3. Group $[\bar{6}2m]$

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One combines the reductions for groups 6 and 2mm: [Scheme scheme29]

There is 1 independent component.

1.1.4.8.6.4. Groups , $[(A_{\infty}/M)C]$ , and $[(A_{\infty}/M) \infty (A_{2}/M)C]$

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All the components are equal to zero.

1.1.4.8.7. Cubic and spherical systems

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1.1.4.8.7.1. Group

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One combines the reductions corresponding to a twofold axis parallel to $[Ox_{3}]$ and to a threefold axis parallel to [111]: [Scheme scheme30]

There are 2 independent components.

1.1.4.8.7.2. Groups and $[\infty A_{\infty}/M]$

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One combines the reductions corresponding to groups 422 and 23: [Scheme scheme31]

There is 1 independent component.

1.1.4.8.7.3. Group $[{\bar 4}3m]$

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One combines the reductions corresponding to groups $[{\bar 4}2m]$ and 23: [Scheme scheme32]

There is 1 independent component.

1.1.4.8.7.4. Groups $[m{\bar 3}]$ , $[m{\bar 3}m]$ and $[\infty(A_{\infty}/M)C]$

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All the components are equal to zero.

1.1.4.9. Reduction of the components of a tensor of rank 4

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1.1.4.9.1. Triclinic system (groups $[{\bar 1}]$ , )

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There is no reduction; all the components are independent. Their number is equal to 81. They are usually represented as a $[9\times 9]$ matrix, where components $[t_{ijkl}]$ are replaced by ijkl, for brevity: [Scheme scheme102] This matrix can be represented symbolically by [Scheme scheme33] where the $[9\times 9]$ matrix has been subdivided for clarity in to nine $[3\times 3]$ submatrices.

1.1.4.9.2. Monoclinic system (groups , , m)

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The reduction is obtained by the method of direct inspection. For a twofold axis parallel to $[Ox_{2}]$ , one finds [Scheme scheme34]

There are 41 independent components.

1.1.4.9.3. Orthorhombic system (groups , , )

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[Scheme scheme35]

There are 21 independent components.

1.1.4.9.4. Trigonal system

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1.1.4.9.4.1. Groups and $[{\bar 3}]$

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The reduction is first applied in the system of axes tied to the eigenvectors of the operator representing a threefold axis. The system of axes is then changed to a system of orthonormal axes with $[Ox_{3}]$ parallel to the threefold axis: [Scheme scheme103] with $[\left. \matrix{t_{1111} - t_{1122} = &t_{1212} + t_{1221}\cr t_{1112} + t_{1121} = &- (t_{1211} + t_{2111}).\cr}\right\}]$

There are 27 independent components.

1.1.4.9.4.2. Groups $[{\bar 3}m]$ , , , with the twofold axis parallel to $[Ox_{1}]$

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[Scheme scheme104] with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 14 independent components.

1.1.4.9.5. Tetragonal system

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1.1.4.9.5.1. Groups , , $[{\bar 4}]$

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[Scheme scheme105]

There are 21 independent components.

1.1.4.9.5.2. Groups , , , $[{\bar 4}2m]$

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[Scheme scheme106]

There are 11 independent components.

1.1.4.9.6. Hexagonal and cylindrical systems

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1.1.4.9.6.1. Groups , $[{\bar 6}]$ , ; $[(A_{\infty}/M)C]$ , $[A_{\infty}]$

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[Scheme scheme107] with $[\left. \matrix{t_{1111} - t_{1122} = t_{1212} + t_{1221}\hfill\cr t_{1112} + t_{1121} = - (t_{1211}+ t_{2111}).\cr}\right\}]$

There are 19 independent components.

1.1.4.9.6.2. Groups , , , $[{\bar 6}2m]$ ; $[(A_{\infty}/M) \infty]$ ; $[(A_{2}/M)C]$ , $[A_{\infty}\infty A_{2}]$

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[Scheme scheme108] with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 11 independent components.

1.1.4.9.7. Cubic system

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1.1.4.9.7.1. Groups , $[{\bar 3}m]$

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[Scheme scheme36]

There are 7 independent components.

1.1.4.9.7.2. Groups $[m{\bar 3}m]$ , , $[{\bar 4}3m]$

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[Scheme scheme37]

There are 4 independent components. The tensor is symmetric.

1.1.4.9.8. Spherical system

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1.1.4.9.8.1. Groups $[\infty (A_{\infty}/M)C]$ and $[\infty A_{\infty}]$

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[Scheme scheme38] with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 3 independent components. The tensor is symmetric.

1.1.4.9.9. Symmetric tensors of rank 4

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For symmetric tensors such as those representing principal properties, one finds the following, representing the nonzero components for the leading diagonal and for one half of the others.

1.1.4.9.9.1. Triclinic system

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[Scheme scheme39]

There are 45 independent coefficients.

1.1.4.9.9.2. Monoclinic system

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[Scheme scheme40]

There are 25 independent coefficients.

1.1.4.9.9.3. Orthorhombic system

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[Scheme scheme41]

There are 15 independent coefficients.

1.1.4.9.9.4. Trigonal system

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(i) Groups and $[{\bar 3}]$ with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 15 independent components.
(ii) Groups $[{\bar 3}m]$ , , with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 11 independent components.

1.1.4.9.9.5. Tetragonal system

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(i) Groups , , $[{\bar 4}]$

There are 13 independent components.
(ii) Groups , , , $[{\bar 4}2m]$

There are 9 independent components.

1.1.4.9.9.6. Hexagonal and cylindrical systems

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(i) Groups , $[{\bar 6}]$ , ; $[(A_{\infty }/M)C, A_{\infty}]$ with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 12 independent components.
(ii) Groups , , , $[{\bar 6}2m]$ ; $[(A_{\infty }/M) \infty (A_{2}/M)C]$ , $[A_{\infty} \infty A_{2}]$ with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 10 independent components.

1.1.4.9.9.7. Cubic system

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(i) Groups , $[{\bar 3}m]$ with $[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]$

There are 5 independent components.
(ii) Groups $[m{\bar 3}m]$ , , $[{\bar 4}3m]$ , and spherical system: the reduced tensors are already symmetric (see Sections 1.1.4.9.7 and 1.1.4.9.8 ).

1.1.4.10. Reduced form of polar and axial tensors – matrix representation

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1.1.4.10.1. Introduction

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Many tensors representing physical properties or physical quantities appear in relations involving symmetric tensors. Consider, for instance, the strain $[S_{ij}]$ resulting from the application of an electric field E (the piezoelectric effect): $[S_{ij} = d_{ijk}E_{k} + Q_{ijkl}E_{k}E_{l}, \eqno(1.1.4.4)]$ where the first-order terms $[d_{ijk}]$ represent the components of the third-rank converse piezoelectric tensor and the second-order terms $[Q_{ijkl}]$ represent the components of the fourth-rank electrostriction tensor. In a similar way, the direct piezoelectric effect corresponds to the appearance of an electric polarization P when a stress $[T_{jk}]$ is applied to a crystal: $[P_{i} = d_{ijk}T_{jk}. \eqno(1.1.4.5)]$

Owing to the symmetry properties of the strain and stress tensors (see Sections 1.3.1 and 1.3.2 ) and of the tensor product $[E_{k}E_{l}]$ , there occurs a further reduction of the number of independent components of the tensors which are engaged in a contracted product with them, as is shown in Section 1.1.4.10.3 for third-rank tensors and in Section 1.1.4.10.5 for fourth-rank tensors.

1.1.4.10.2. Stress and strain tensors – Voigt matrices

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The stress and strain tensors are symmetric because body torques and rotations are not taken into account, respectively (see Sections 1.3.1 and 1.3.2 ). Their components are usually represented using Voigt's one-index notation.

(i) Strain tensor $[\left. \matrix{S_{1} = S_{11}\semi\hfill &S_{2} = S_{22}\semi\hfill &S_{3} = S_{33}\semi\hfill \cr S_{4} = S_{23} + S_{32}\semi\hfill &S_{5} = S_{31} + S_{13}\semi\hfill &S_{6} = S_{12} + S_{21}\semi\hfill \cr S_{4} = 2S_{23} = 2S_{32}\semi &S_{5} = 2S_{31} = 2S_{13}\semi &S_{6} = 2S_{12} = 2S_{21}.\cr}\right\} \eqno(1.1.4.6)]$ The Voigt components $[S_{\alpha}]$ form a Voigt matrix: $[\pmatrix{S_{1} &S_{6} &S_{5}\cr &S_{2} &S_{4}\cr & &S_{3}\cr}.]$ The terms of the leading diagonal represent the elongations (see Section 1.3.1 ). It is important to note that the non-diagonal terms, which represent the shears, are here equal to twice the corresponding components of the strain tensor. The components $[S_{\alpha}]$ of the Voigt strain matrix are therefore not the components of a tensor.
(ii) Stress tensor $[\left. \matrix{T_{1}= T_{11}\semi\hfill &T_{2} = T_{22}\semi\hfill &T_{3} = T_{33}\semi\hfill\cr T_{4} = T_{23} = T_{32}\semi &T_{5} = T_{31} = T_{13}\semi &T_{6} = T_{12} =T_{21}.\cr}\right\}]$ The Voigt components $[T_{\alpha}]$ form a Voigt matrix: $[\pmatrix{T_{1} &T_{6} &T_{5}\cr &T_{2} &T_{4}\cr & & T_{3}\cr}.]$ The terms of the leading diagonal correspond to principal normal constraints and the non-diagonal terms to shears (see Section 1.3.2 ).

1.1.4.10.3. Reduction of the number of independent components of third-rank polar tensors due to the symmetry of the strain and stress tensors

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Equation (1.1.4.5) can be written $[P_{i} = \textstyle\sum\limits_{j} d_{ijj}T_{jj} + \textstyle\sum\limits_{j\neq k} (d_{ijk}+ d_{ikj})T_{jk}.]$

The sums $[(d_{ijk} + d_{ikj})]$ for $[j \neq k]$ have a definite physical meaning, but it is impossible to devise an experiment that permits $[d_{ijk}]$ and $[d_{ikj}]$ to be measured separately. It is therefore usual to set them equal: $[d_{ijk} = d_{ikj}. \eqno(1.1.4.7)]$

It was seen in Section 1.1.4.8.1 that the components of a third-rank tensor can be represented as a $[9 \times 3]$ matrix which can be subdivided into three $[3 \times 3]$ submatrices: $[\pmatrix{&{\bf 1}&|&{\bf 2}&|&{\bf 3}&\cr}.]$

Relation (1.1.4.7) shows that submatrices 1 and 2 are identical.

One puts, introducing a two-index notation, $[\left. \matrix{\hfill d_{ijj} = d_{i\alpha}\quad (\alpha = 1, 2, 3){\phantom.}\cr d_{ijk} + d_{ikj}\,\, (j\neq k) = d_{i\alpha}\quad (\alpha = 4, 5, 6).\cr}\right\}]$ Relation (1.1.4.7) becomes $[P_{i} = d_{i\alpha} T_{\alpha}.]$

The coefficients $[d_{i\alpha}]$ may be written as a $[3 \times 6]$ matrix: $[\left(\matrix{11 &12 &13 \cr 21 &22 &23\cr 31 &32 &33\cr}\right | \left. \matrix{14 &15 &16\cr 24 &25 &26\cr 34 &35 &36\cr}\right).]$ This matrix is constituted by two $[3 \times 3]$ submatrices. The left-hand one is identical to the submatrix 1, and the right-hand one is equal to the sum of the two submatrices 2 and 3: $[\pmatrix{&{\bf 1}&|&{\bf 2} + {\bf 3}&\cr}.]$

The inverse piezoelectric effect expresses the strain in a crystal submitted to an applied electric field: $[S_{ij} = d_{ijk}E_{k},]$ where the matrix associated with the coefficients $[d_{ijk}]$ is a $[9 \times 3]$ matrix which is the transpose of that of the coefficients used in equation (1.1.4.5), as shown in Section 1.1.1.4 .

The components of the Voigt strain matrix $[S_{\alpha}]$ are then given by $[\left. \matrix{S_{\alpha } = d_{iik}E_{k} \hfill &(\alpha = 1,2,3)\cr S_{\alpha} = S_{ij} + S_{ji} = (d_{ijk} + d_{jik})E_{k} &(\alpha = 4,5,6).\cr}\right\}]$ This relation can be written simply as $[S_{\alpha} = d_{\alpha k}E_{k},]$ where the matrix of the coefficients $[d_{\alpha k}]$ is a $[6 \times 3]$ matrix which is the transpose of the $[d_{i\alpha}]$ matrix.

There is another set of piezoelectric constants (see Section 1.1.5 ) which relates the stress, $[T_{ij}]$ , and the electric field, $[E_{k}]$ , which are both intensive parameters: $[T_{ij} = e_{ijk}E_{k}, \eqno(1.1.4.8)]$ where a new piezoelectric tensor is introduced, $[e_{ijk}]$ . Its components can be represented as a $[3 \times 9]$ matrix: $[\pmatrix{{\bf 1}\cr-\cr{\bf 2}\cr-\cr{\bf 3}\cr}.]$

Both sides of relation (1.1.4.8) remain unchanged if the indices i and j are interchanged, on account of the symmetry of the stress tensor. This shows that $[e_{ijk} = e_{jik}.]$

Submatrices 2 and 3 are equal. One introduces here a two-index notation through the relation $[e_{\alpha k} = e_{ijk}]$ , and the $[e_{\alpha k}]$ matrix can be written $[\left({{\bf 1}\over{{\bf 2}+{\bf 3}}}\right).]$

The relation between the full and the reduced matrix is therefore different for the $[d_{ijk}]$ and the $[e_{kij}]$ tensors. This is due to the particular property of the strain Voigt matrix (1.1.4.6), and as a consequence the relations between nonzero components of the reduced matrices are different for certain point groups (3, 32, [3m] , $[\bar{6}]$ , $[\bar{6}2m]$ ).

1.1.4.10.4. Independent components of the matrix associated with a third-rank polar tensor according to the following point groups

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1.1.4.10.4.1. Triclinic system

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(i) Group 1: all the components are independent. There are 18 components.
(ii) Group $[\bar{1}]$ : all the components are equal to zero.

1.1.4.10.4.2. Monoclinic system

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(i) Group 2: twofold axis parallel to $[Ox_{2}]$ :

There are 8 independent components.
(ii) Group m:

There are 10 independent components.
(iii) Group : all the components are equal to zero.

1.1.4.10.4.3. Orthorhombic system

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(i) Group 222:

There are 3 independent components.
(ii) Group :

There are 5 independent components.
(iii) Group : all the components are equal to zero.

1.1.4.10.4.4. Trigonal system

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(i) Group 3: where the symbol ⊖ means that the corresponding component is equal to the opposite of that to which it is linked, ⊙ means that the component is equal to twice minus the value of the component to which it is linked for $[d_{ijk}]$ and to minus the value of the component to which it is linked for $[e_{ijk}]$ . There are 6 independent components.
(ii) Group 32, twofold axis parallel to $[Ox_{1}]$ : with the same conventions. There are 4 independent components.
(iii) Group , mirror perpendicular to $[Ox_{1}]$ : with the same conventions. There are 4 independent components.
(iv) Groups $[\bar{3}]$ and $[\bar{3}m]$ : all the components are equal to zero.

1.1.4.10.4.5. Tetragonal, hexagonal and cylindrical systems

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(i) Groups 4, 6 and $[A_{\infty}]$ :

There are 4 independent components.
(ii) Groups 422, 622 and $[A_{\infty}\infty A_{2}]$ :

There is 1 independent component.
(iii) Groups , and $[A_{\infty }\infty M]$ :

There are 3 independent components.
(iv) Groups , and $[(A_{\infty }/M)C]$ : all the components are equal to zero.
(v) Group $[\bar{4}]$ :

There are 4 independent components.
(vi) Group $[\bar{6} = 3/m]$ : with the same conventions as for group 3. There are 2 independent components.
(vii) Group $[\bar{4}2m]$ – twofold axis parallel to $[Ox_{1}]$ :

There are 2 independent components.
(viii) Group $[\bar{4}2m]$ – mirror perpendicular to $[Ox_{1}]$ (twofold axis at $[45^{\circ}]$ ):

The number of independent components is of course the same.
(ix) Group $[\bar{6}2/m]$ : with the same conventions as for group 3. There is 1 independent component.
(x) Groups , and $[(A_{\infty }/M) \infty (A_{2}/M)C]$ : all the components are equal to zero.

1.1.4.10.4.6. Cubic and spherical systems

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(i) Groups 23 and $[{\bar 4}3m]$ :

There is 1 independent component.
(ii) Groups 432 and $[\infty A_{\infty }]$ : it was seen in Section 1.1.4.8.6 that we have in this case $[d_{123}= - d_{132}.]$ It follows that $[d_{14} = 0]$ , all the components are equal to zero.
(iii) Groups $[m{\bar 3}]$ , $[m{\bar 3}m]$ and $[\infty (A_{\infty }/M)C]$ : all the components are equal to zero.

1.1.4.10.5. Reduction of the number of independent components of fourth-rank polar tensors due to the symmetry of the strain and stress tensors

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Let us consider five examples of fourth-rank tensors:

(i) Elastic compliances, $[s_{ijkl }]$ , relating the resulting strain tensor $[S_{ij}]$ to an applied stress $[T_{ij}]$ (see Section 1.3.3.2 ): $[S_{ij} = s_{ijkl}T_{kl}, \eqno{(1.1.4.9)}]$ where the compliances $[s_{ijkl}]$ are the components of a tensor of rank 4.
(ii) Elastic stiffnesses, $[c_{ijkl }]$ (see Section 1.3.3.2 ): $[T_{ij}= c_{ijkl}S_{kl}.]$
(iii) Piezo-optic coefficients, $[\pi _{ijkl}]$ , relating the variation $[\Delta \eta _{ij}]$ of the dielectric impermeability to an applied stress $[T_{kl}]$ (photoelastic effect – see Section 1.6.7 ): $[\Delta \eta_{ij} = \pi_{ijkl}T_{kl}.]$
(iv) Elasto-optic coefficients, $[p_{ijkl}]$ , relating the variation $[\Delta \eta_{ij}]$ of the dielectric impermeability to the strain $[S_{kl}]$ : $[\Delta \eta _{ij}= p_{ijkl}S_{kl}.]$
(v) Electrostriction coefficients, $[Q_{ijkl}]$ , which appear in equation (1.1.4.4): $[S_{ij} = Q_{ijkl}E_{k}E_{l}, \eqno{(1.1.4.10)}]$ where only the second-order terms are considered.

In each of the equations from (1.1.4.9) to (1.1.4.10), the contracted product of a fourth-rank tensor by a symmetric second-rank tensor is equal to a symmetric second-rank tensor. As in the case of the third-rank tensors, this results in a reduction of the number of independent components, but because of the properties of the strain Voigt matrix, and because two of the tensors are endowed with intrinsic symmetry (the elastic tensors), the reduction is different for each of the five tensors. The above relations can be written in matrix form: [Scheme scheme63] where the second-rank tensors are represented by $[1 \times 9]$ column matrices, which can each be subdivided into three $[1 \times 3]$ submatrices and the $[9 \times 9]$ matrix associated with the fourth-rank tensors is subdivided into nine $[3 \times 3]$ submatrices, as shown in Section 1.1.4.9.1 . The symmetry of the second-rank tensors means that submatrices 2 and 3 which are associated with them are equal.

Let us first consider the reduction of the tensor of elastic compliances. As in the case of the piezoelectric tensor, equation (1.1.4.9) can be written $[S_{ij}= \textstyle\sum\limits_{l} s_{ijll}T_{ll} + \sum\limits_{k\ne l}(s_{ijkl}+ s_{ijlk})T_{kl}. \eqno(1.1.4.11)]$

The sums $[(s_{ijkl}+ s_{ijlk})]$ for $[k \neq l]$ have a definite physical meaning, but it is impossible to devise an experiment permitting $[s_{ijkl}]$ and $[s_{ijlk}]$ to be measured separately. It is therefore usual to set them equal in order to avoid an unnecessary constant: $[s_{ijkl} = s_{ijlk}.]$

Furthermore, the left-hand term of (1.1.4.11) remains unchanged if we interchange the indices i and j. The terms on the right-hand side therefore also remain unchanged, whatever the value of $[T_{ll}]$ or $[T_{kl}]$ . It follows that $[\eqalignno{s_{ijll} & = s_{jill}\cr s_{ijkl} & = s_{ijlk} = s_{jikl} = s_{jilk}.\cr}]$ Similar relations hold for $[c_{ijkl}]$ , $[Q_{ijkl}]$ , $[p_{ijkl}]$ and $[\pi_{ijkl}]$ : the submatrices 2 and 3, 4 and 7, 5, 6, 8 and 9, respectively, are equal.

Equation (1.4.1.11) can be rewritten, introducing the coefficients of the Voigt strain matrix: $[\eqalignno{S_{\alpha} = S_{ii} &= \textstyle\sum\limits_{l} s_{iill}T_{ll} + \sum\limits_{k\ne l}(s_{iikl} + s_{iilk})T_{kl} \quad (\alpha = 1,2,3)\cr S_{\alpha} = S_{ij} + S_{ji} &= \textstyle\sum\limits_{l} (s_{ijll} + s_{jill})T_{ll}\cr &\quad + \textstyle\sum\limits_{k\ne l}(s_{ijkl} + s_{ijlk} + s_{jikl} + s_{jilk})T_{kl} \quad (\alpha = 4,5,6).\cr}]$ We shall now introduce a two-index notation for the elastic compliances, according to the following conventions: $[\left.\matrix{i = j\semi &k = l\semi &s_{\alpha \beta} = s_{iill}\hfill\cr i = j\semi &k\neq l\semi &s_{\alpha \beta} = s_{iikl}+s_{iilk}\hfill\cr i \neq j\semi &k = l\semi &s_{\alpha \beta} = s_{ijkk}+s_{jikk}\hfill\cr i \neq j\semi &k \neq l\semi &s_{\alpha \beta} = s_{ijkl}+ s_{jikl} + s_{ijlk}+ s_{jilk}.\hfill\cr}\right\} \eqno(1.1.4.12)]$ We have thus associated with the fourth-rank tensor a square $[6 \times 6]$ matrix with 36 coefficients: [Scheme scheme115]

One can translate relation (1.1.4.12) using the $[9 \times 9]$ matrix representing $[s_{ijkl}]$ by adding term by term the coefficients of submatrices 2 and 3, 4 and 7 and 5, 6, 8 and 9, respectively: [Scheme scheme64]

Using the two-index notation, equation (1.1.4.9) becomes $[S_{\alpha} = s_{\alpha \beta}T_{\beta}. \eqno(1.1.4.13)]$

A similar development can be applied to the other fourth-rank tensors $[\pi_{ijkl}]$ , which will be replaced by $[6 \times 6]$ matrices with 36 coefficients, according to the following rules.

(i) Elastic stiffnesses, $[c_{ijkl}]$ and elasto-optic coefficients, $[p_{ijkl}]$ : where $[\eqalignno{c_{\alpha \beta} &=c_{ijkl}\cr p_{\alpha \beta} &=p_{ijkl}.\cr}]$
(ii) Piezo-optic coefficients, $[\pi_{ijkl}]$ : where $[\left. \matrix{i = j\semi &k = l\semi &\pi_{\alpha \beta} = \pi_{iil l}\hfill \cr i = j\semi &k \neq l\semi &\pi_{\alpha \beta} = \pi_{iikl} + \pi_{iilk}\hfill \cr i \neq j\semi &k = l\semi &\pi_{\alpha \beta} = \pi_{ijkk} = \pi_{jikk}\hfill \cr i \neq j\semi &k \neq l\semi &\pi_{\alpha \beta} = \pi_{ijkl} + \pi_{jikl} = \pi_{ijl k} + \pi_{jilk}.\hfill\cr}\right\}]$
(iii) Electrostriction coefficients, $[Q_{ijkl}]$ : same relation as for the elastic compliances.

1.1.4.10.6. Independent components of the matrix associated with a fourth-rank tensor according to the following point groups

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1.1.4.10.6.1. Triclinic system, groups $[{\bar 1}]$ ,

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[Scheme scheme67]

1.1.4.10.6.2. Monoclinic system

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Groups [2/m] , 2, m, twofold axis parallel to $[Ox_{2}]$ : [Scheme scheme68]

1.1.4.10.6.3. Orthorhombic system

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Groups [mmm] , [2mm] , 222: [Scheme scheme69]

1.1.4.10.6.4. Trigonal system

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(i) Groups 3, $[{\bar 3}]$ : where ⊖ is a component numerically equal but opposite in sign to the heavy dot component to which it is linked; ⊕ is a component equal to twice the heavy dot component to which it is linked; ⊙ is a component equal to minus twice the heavy dot component to which it is linked; ⊗ is equal to $[1/2(p_{11} - p_{12})]$ , $[(\pi_{11} - \pi_{12})]$ , $[2(Q_{11} - Q_{12})]$ , $[1/2(c_{11} - c_{12})]$ and $[2(s_{11} - s_{12})]$ , respectively.
(ii) Groups 32, , $[{\bar 3}m]$ : with the same conventions.

1.1.4.10.6.5. Tetragonal system

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(i) Groups 4, $[{\bar 4}]$ and :
(ii) Groups 422, , $[{\bar 4}2m]$ and :

1.1.4.10.6.6. Hexagonal system

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(i) Groups 6, $[{\bar 6}]$ and :
(ii) Groups 622, , $[{\bar 6}2m]$ and :

1.1.4.10.6.7. Cubic system

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(i) Groups 23 and :
(ii) Groups 432, $[{\bar 4}3m]$ and $[m{\bar 3}m]$ :

1.1.4.10.6.8. Spherical system

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For all tensors [Scheme scheme81]

1.1.4.10.7. Reduction of the number of independent components of axial tensors of rank 2

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It was shown in Section 1.1.4.5.3.2 that axial tensors of rank 2 are actually tensors of rank 3 antisymmetric with respect to two indices. The matrix of independent components of a tensor such that $[g_{ijk}= - g_{jik}]$ is given by $[\left(\matrix{&122 &133\cr -121 & &223\cr -131 &-232 &\cr}\left|\matrix{123 &131 &\cr &231 &-122\cr -233 & &-132\cr}\right|\matrix{132 & &121\cr 232 &-123 &\cr &-133 &-231\cr}\right).]$ The second-rank axial tensor $[g_{kl}]$ associated with this tensor is defined by $[g_{kl} = {\textstyle{1\over 2}} \varepsilon_{ijk} g_{ijl}.]$

For instance, the piezomagnetic coefficients that give the magnetic moment $[M_{i}]$ due to an applied stress $[T_{\alpha }]$ are the components of a second-rank axial tensor, $[\Lambda_{i\alpha }]$ (see Section 1.5.7.1 ): $[M_{i}= \Lambda _{i\alpha} T_{\alpha}.]$

1.1.4.10.7.1. Independent components according to the following point groups

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(i) Triclinic system

(a) Group 1:
(b) Group $[{\bar 1}]$ : all components are equal to zero.

(ii) Monoclinic system

(a) Group 2:
(b) Group m:
(c) Group : all components are equal to zero.

(iii) Orthorhombic system

(a) Group 222:
(b) Group :
(c) Group : all components are equal to zero.

(iv) Trigonal, tetragonal, hexagonal and cylindrical systems

(a) Groups 3, 4, 6 and $[A_{\infty}]$ :
(b) Groups 32, 42, 62 and $[A_{\infty}\infty A_2]$ :
(c) Groups , , and $[A_{\infty}\infty M]$ :
(d) Group $[{\bar 4}]$ :
(e) Group $[\bar{4}2m]$ :
(f) Groups $[\bar{3}]$ , , $[\bar{6}2m]$ , $[\bar{3}m]$ , and : all components are equal to zero.

(v) Cubic and spherical systems

(a) Groups 23, 432 and $[\infty A_{\infty}]$ :

The axial tensor is reduced to a pseudoscalar.
(b) Groups $[m\bar{3}]$ , $[\bar{4}3m]$ , $[m\bar{3}m]$ and $[\infty (A_{\infty}/M)C]$ : all components are equal to zero.

1.1.4.10.7.2. Independent components of symmetric axial tensors according to the following point groups

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Some axial tensors are also symmetric. For instance, the optical rotatory power of a gyrotropic crystal in a given direction of direction cosines $[\alpha_{1,} \alpha_{2}, \alpha_{3}]$ is proportional to a quantity G defined by (see Section 1.6.5.4 ) $[G = g_{ij}\alpha_{i}\alpha_{j},]$ where the gyration tensor $[g_{ij}]$ is an axial tensor. This expression shows that only the symmetric part of $[g_{ij}]$ is relevant. This leads to a further reduction of the number of independent components:

(i) Triclinic system

(a) Group 1:
(b) Group $[\bar{1}]$ : all components are equal to zero.

(ii) Monoclinic system

(a) Group 2:
(b) Group m:
(c) Group : all components are equal to zero.

(iii) Orthorhombic system

(a) Group 222:
(b) Group :
(c) Group : all components are equal to zero.

(iv) Trigonal, tetragonal and hexagonal systems

(a) Groups 3, 32, 4, 42, 6, 62:
(b) Group $[\bar{4}]$ :
(c) Group $[\bar{4}2m]$ :
(d) Groups $[\bar{3}]$ , , $[\bar{3}m]$ , , , , $[\bar{6}]$ , $[\bar{6}2m]$ and : all components are equal to zero.

(v) Cubic and spherical systems

(a) Groups 23, 432 and $[A_{\infty} \infty A_{2}]$ :
(b) Groups $[m\bar{3}]$ , $[\bar{4}3m]$ , $[m\bar{3}m]$ and $[\infty (A_{\infty}/M)C]$ : all components are equal to zero.

In practice, gyrotropic crystals are only found among the enantiomorphic groups: 1, 2, 222, 3, 32, 4, 422, 6, 622, 23, 432. Pasteur (1848a ,b ) was the first to establish the distinction between `molecular dissymmetry' and `crystalline dissymetry'.

1.1.5. Thermodynamic functions and physical property tensors

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[The reader may also consult Mason (1966), Nye (1985) or Sirotin & Shaskol'skaya (1982).]

1.1.5.1. Isothermal study

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The energy of a system is the sum of all the forms of energy: thermal, mechanical, electrical etc. Let us consider a system whose only variables are these three. For a small variation of the associated extensive parameters, the variation of the internal energy is $[\hbox{d}{\cal U} = E_{n}\ \hbox{d}D_{n} + T_{kl}\ \hbox{d}S_{kl} + \Theta \ \hbox{d}\sigma,]$ where Θ is the temperature and σ is the entropy; there is summation over all dummy indices; an orthonormal frame is assumed and variance is not apparent. The mechanical energy of deformation is given by $[T_{kl}\,\,{\rm d}S_{kl}]$ (see Section 1.3.2.8 ). Let us consider the Gibbs free-energy function $[{\cal G}]$ defined by $[{\cal G} = {\cal U} - E_{n}D_{n} - T_{kl}S_{kl} - \Theta \sigma.]$ Differentiation of $[{\cal G}]$ gives $[\hbox{d}{\cal G} = - D_{n}\ \hbox{d}E_{n} - S_{kl}\ \hbox{d}T_{kl}- \sigma \ \hbox{d}\Theta.]$ The extensive parameters are therefore partial derivatives of the free energy: $[S_{kl} = - {\partial {\cal G} \over \partial T_{kl}};\ D_{n} = - {\partial {\cal G} \over \partial E_{n}};\ \sigma = - {\partial {\cal G} \over \partial \Theta}.]$ Each of these quantities may be expanded by performing a further differentiation in terms of the intensive parameters, $[T_{kl}]$ , $[E_{n}]$ and Θ. We have, to the first order, $[\eqalignno{\hbox{d}S_{kl} &= \left[{\partial S_{kl} \over \partial T_{ij}}\right]_{E,\Theta} \hbox{d}T_{ij} + \left[{\partial S_{kl} \over \partial E_{n}}\right]_{T,\Theta}\hbox{d}E_{n} + \left[{\partial S_{kl} \over \partial \Theta}\right]_{E,T} \delta\Theta\cr \hbox{d}D_{n} &= \left[{\partial D_{n} \over \partial T_{kl}}\right]_{E,\Theta}\hbox{d}T_{kl} + \left[{\partial D_{n} \over \partial E_{m}}\right]_{T,\Theta}\hbox{d}E_{m} + \left[{\partial D_{n} \over \partial \Theta}\right]_{E,T}\delta\Theta\cr \hbox{d}\sigma &= \left[{\partial \sigma \over \partial T_{ij}}\right]_{E,\Theta}\hbox{d}T_{ij} + \left[{\partial \sigma \over \partial E_{m}}\right]_{T,\Theta}\hbox{d}E_{m} + \left[{\partial \sigma \over \partial \Theta}\right]_{E,T}\delta\Theta.\cr}]$ To a first approximation, the partial derivatives may be considered as constants, and the above relations may be integrated: $[\left. \matrix{S_{kl} &= \left(s_{klij}\right)^{E,\Theta} T_{ij} &+ \,\left(d_{kl n}\right)^{T,\Theta} E_{n} &+\, \left(\alpha_{kl}\right)^{E,T}\delta\Theta\hfill\cr D_{n} &= \left(d_{nkl}\right)^{E,\Theta} T_{kl} \hfill &+ \,\left(\varepsilon_{nm}\right)^{T,\Theta} E_{m} \hfill &+ \,\left(p_{n}\right)^{E,T}\delta\Theta \hfill \cr \delta \sigma &= \left(\alpha_{ij}\right)^{E} T_{ij} \hfill &+ \,\left(p_{m}\right)^{T} E_{m} \hfill &+ \,(\rho C^{E,T} / \Theta ) \delta\Theta. \hfill \cr}\right\} \eqno(1.1.5.1)]$ This set of equations is the equivalent of relation (1.1.1.6) of Section 1.1.1.3 , which gives the coefficients of the matrix of physical properties. These coefficients are:

(i) For the principal properties: $[\left(s_{klij}\right)^{E,\Theta}]$ : elastic compliances at constant temperature and field; $[\left(\varepsilon_{nm}\right)^{T,\Theta}]$ : dielectric constant at constant temperatures and stress; $[\rho C^{T,E}]$ : heat capacity per unit volume at constant stress and field ( $[\rho]$ is the specific mass and $[C^{T,E}]$ is the specific heat at constant stress and field).
(ii) For the other properties: $[\left(d_{kln}\right)^{T,\Theta}]$ and $[\left(d_{nkl}\right)^{E,\Theta}]$ are the components of the piezoelectric effect and of the converse effect. They are represented by $[3\times 9]$ and $[9\times 3]$ matrices, respectively. One may notice that $[d_{kln} = {\partial S_{kl} \over \partial E_{n}} = - {\partial^{2} {\cal G} \over \partial E_{n} \partial T_{kl}} = - {\partial^{2} {\cal G} \over \partial T_{kl}\partial E_{n}} = {\partial D_{n} \over \partial T_{kl}}=d_{nkl},]$ which shows again that the components of two properties that are symmetric with respect to the leading diagonal of the matrix of physical properties are equal (Section 1.1.1.4 ) and that the corresponding matrices are transpose to one another.

In a similar way,

(a) the matrices $[\left(\alpha_{kl}\right)^{E,T}]$ of the thermal expansion and $[(\alpha_{ij})^E]$ of the piezocalorific effect are transpose to one another;
(b) the components $[(p_{n})^{T}]$ of the pyroelectric and of the electrocalorific effects are equal.

Remark . The piezoelectric effect, namely the existence of an electric polarization P under an applied stress, is always measured at zero applied electric field and at constant temperature. The second equation of (1.1.5.1) becomes under these circumstances $[P_{n} = D_{n} = (d_{nkl})^{\Theta} T_{kl}.]$ Remark. Equations (1.1.5.1) are, as has been said, first-order approximations because we have assumed the partial derivatives to be constants. Actually, this approximation is not correct, and in many cases it is necessary to take into account the higher-order terms as, for instance, in:

(a) nonlinear elasticity (see Sections 1.3.6 and 1.3.7 );
(b) electrostriction;
(c) nonlinear optics (see Chapter 1.7 );
(d) electro-optic and piezo-optic effects (see Sections 1.6.6 and 1.6.7 ).

1.1.5.2. Other forms of the piezoelectric constants

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We use here another Gibbs function, the electric Gibbs function, $[{\cal G}_2]$ , defined by $[{\cal G}_{2} = {\cal U} - E_{n}D_{n} - \Theta \sigma.]$

Differentiation of $[\cal G]$ gives $[d{\cal G}_2=-D_n\,{\rm d}E_n+T_{ij}\,{\rm d}S_{ij}-\sigma\,{\rm d}\Theta.]$ It follows that $[T_{ij}={{\partial{\cal G}_2}\over{\partial S_{ij}}}\semi\quad D_n=-{{\partial{\cal G}_2}\over{\partial E_n}}\semi\quad\sigma=-{{\partial{\cal G}_2}\over{\partial\Theta}}]$ and a set of relations analogous to (1.1.5.1): $[\left.\matrix{T_{ij} = (c_{ijkl})^{E,\Theta} S_{kl}\hfill &-\,(e_{ijn})^{S,\Theta} E_{n}\hfill &-\, (\lambda_{ij})^{E,S}\,\,\delta\Theta \hfill\cr D_{n} = (e_{nij})^{E,\Theta} S_{ij}\hfill &+ \,(\varepsilon_{nm})^{S,\Theta} E_{m}\hfill &+\, (p_{n})^{S} \,\,\delta\Theta\hfill \cr \delta\sigma = (\lambda_{ij})^{E} S_{ij}\hfill &+\, (p_{n})^{S} E_{n}\hfill &+ \,\rho C^{E,S}\,\, \delta\Theta/\Theta, \hfill\cr}\right\}\eqno(1.1.5.2)]$ where the components $[\left(c_{ijkl}\right)^{E,\Theta}]$ are the isothermal elastic stiffnesses at constant field and constant temperature, $[e_{ijn}=-{\partial T_{ij}\over\partial E_n}=-{\partial^2{\cal G}_2\over\partial E_n\partial S_{ij}}=-{\partial^2{\cal G}_2\over\partial S_{ij}\partial E_n}={\partial D_n\over \partial S_{ij}}=e_{nij}]$ are the piezoelectric stress coefficients at constant strain and constant temperature, $[\lambda_{ij}=-{\partial T_{ij}\over \partial \Theta}=-{\partial^2{\cal G}_2\over \partial\Theta\partial S_{ij}}=-{\partial^2{\cal G}_2\over\partial S_{ij}\partial\Theta}={\partial\delta\sigma\over\partial S_{ij}}]$ are the temperature-stress constants and $[p_n={\partial D_n\over\partial\Theta}=-{\partial^2{\cal G}_2\over\partial\Theta\partial E_n}=-{\partial^2{\cal G}_2\over \partial E_n\partial\Theta}={\partial\delta\sigma\over\partial D_n}]$ are the components of the pyroelectric effect at constant strain.

The relations between these coefficients and the usual coefficients $[d_{kln}]$ are easily obtained:

(i) At constant temperature and strain: if one puts $[\delta\Theta = 0]$ and $[S_{kl} = 0]$ in the first equation of (1.1.5.1) and (1.1.5.2), one obtains, respectively, $[\eqalignno{0 &= s_{klij} T_{ij} + d_{kln} E_{n}\cr T_{ij} &= - e_{ijn} E_{n},\cr}]$ from which it follows that $[d_{kln} = s_{klij}e_{ijn}]$ at constant temperature and strain.
(ii) At constant temperature and stress: if one puts $[\delta\Theta = 0]$ and $[T_{ij} = 0]$ , one obtains in a similar way $[\eqalignno{S_{kl} &= d_{kln} E_{n}\cr 0 &= c_{ijkl} S_{kl} - e_{ijn} E_{n},\cr}]$ from which it follows that $[e_{ijn} = c_{ijkl}d_{kln}]$ at constant temperature and stress.

1.1.5.3. Relation between the pyroelectric coefficients at constant stress and at constant strain

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By combining relations (1.1.5.1) and (1.1.5.2), it is possible to obtain relations between the pyroelectric coefficients at constant stress, $[p_{n}^{T}]$ , and the pyroelectric coefficients at constant strain, $[p_{n}^{S}]$ , also called real pyroelectric coefficients, $[p_{n}^{S}]$ . Let us put $[T_{ij} = 0]$ and $[E_{n} = 0]$ in the first equation of (1.1.5.1). For a given variation of temperature, $[\delta\Theta]$ , the observed strain is $[S_{kl} = \Big[\alpha_{kl}\Big]^{E,T}\delta \Theta.]$ From the second equations of (1.1.5.1) and (1.1.5.2), it follows that $[\eqalignno{D_n &= p_n^T \delta\Theta\cr D_n &= e_{nkl} S_{kl} + p_n\delta\Theta.\cr}]$ Substituting the expression $[S_{kl}]$ and eliminating [D_n] , it follows that $[p_n^T = e_{nkl}\Bigl[\alpha_{kl}\Bigr]^{E,T}+ p_n^{S}. \eqno(1.1.5.3)]$

This relation shows that part of the pyroelectric effect is actually due to the piezoelectric effect.

1.1.5.4. Adiabatic study

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Piezoelectric resonators usually operate at a high frequency where there are no heat exchanges, and therefore in an adiabatic regime $[(\Theta \delta \sigma = 0)]$ . From the third equation of (1.1.5.1), we obtain a relation between the temperature variation, the applied stress and the electric field: $[(\alpha _{ij})^E T_{ij} + (p_{m})^T E_{m} + {\rho C^{T,E} \over \Theta} \delta \Theta = 0.]$

If we substitute this relation in the two other relations of (1.1.5.1), we obtain two equivalent relations, but in the adiabatic regime: $[\eqalignno{S_{kl} &= \left(s_{kl ij}\right)^{E,\sigma} T_{ij} + \left(d_{kl m}\right)^{T,\sigma} E_m\cr D_{n} &= \left(d_{nij}\right)^{E,\sigma} T_{ij} + \left(\varepsilon_{nm}\right)^{T,\sigma} E_m.\cr}]$ By comparing these expressions with (1.1.5.1), we obtain the following relations between the adiabatic and the isothermal coefficients: $[\eqalignno{(s_{ijkl})^{E,\sigma} &= (s_{ijkl})^{E,\theta} - {(\alpha_{ij})^E (\alpha_{kl})^E \Theta \over \rho C^{T,E}}\cr (d_{nij})^{E,\sigma} &= (d_{nij})^{E,\theta} - {(p_{n})^T (\alpha_{kl})^E \Theta \over \rho C^{T,E}}\cr (\varepsilon_{mn})^{T,\sigma} &= (\varepsilon_{mn})^{T,\theta} - {(p_{n})^T (p_{m})^T \Theta \over \rho C^{T,E}}.\cr}]$

1.1.6. Glossary

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$[{\bf e}_i]$	basis vectors in direct space (covariant)
$[{\bf e}^i]$	basis vectors in reciprocal space (contravariant)
	components of a vector in direct space (contravariant)
$[x_{i}]$	components of a vector in reciprocal space (covariant)
$[g_{ij}]$	components of the metric tensor
$[t_{i_{1} \ldots i_{p}}^{j_1 \ldots j_q}]$	components of a tensor of rank n, p times covariant and q times contravariant ()
	transpose of matrix A
$[\otimes]$	tensor product
$[\bigwedge]$	outer product
$[\wedge]$	vector product
$[\partial_i]$	partial derivative with respect to
$[\delta_i^j]$	Kronecker symbol
$[\varepsilon_{ijk}]$	permutation tensor
V	volume
p	pressure
$[u_{i}]$	components of the displacement vector
$[S_{ij}]$	components of the strain tensor
$[S_{\alpha}]$	components of the strain Voigt matrix
$[T_{ij}]$	components of the stress tensor
$[T_{\alpha}]$	components of the stress Voigt matrix
$[s_{ijkl}]$	elastic compliances
$[s_{\alpha\beta}]$	reduced elastic compliances
$[(s_{ijkl})^{\sigma}]$	adiabatic elastic compliances
$[c_{ijkl}]$	elastic stiffnesses
$[c_{\alpha\beta}]$	reduced elastic stiffnesses
ν	Poisson's ratio
E	Young's modulus
Θ	temperature
σ	entropy
$[\alpha_{ij}]$	thermal expansion
$[\lambda_{ij}]$	temperature-stress constant
$[{\cal U}]$	internal energy
$[{\cal G}]$	Gibbs free energy
$[C^{E,T}]$	specific heat at constant stress and applied electric field
E	electric field
D	electric displacement
H	magnetic field
B	magnetic induction
$[\varepsilon_o]$	permittivity of vacuum
ɛ	dielectric constant
$[\varepsilon_{ij}]$	dielectric tensor
$[(\varepsilon_{ij})^{\sigma}]$	adiabatic dielectric tensor
$[\chi_{e}]$	dielectric susceptibility
$[\eta_{ij}]$	dielectric impermeability
$[p_{i}]$	pyroelectric tensor
$[d_{ijk}]$	piezoelectric tensor
$[d_{i\alpha}]$	reduced piezoelectric tensor
$[d_{\alpha i}]$	reduced inverse piezoelectric tensor
$[(d_{ijk})^{\sigma}]$	adiabatic piezoelectric tensor
$[e_{ijk}]$	piezoelectric tensor at constant strain
$[Q_{ijkl}]$	electrostriction tensor
$[Q_{\alpha\beta}]$	reduced electrostriction tensor
$[\pi_{ijkl}]$	piezo-optic tensor
$[\pi_{\alpha\beta}]$	reduced piezo-optic tensor
$[p_{ijkl}]$	elasto-optic tensor
$[p_{\alpha\beta}]$	reduced elasto-optic tensor
$[R_{H\ \ ijk}]$	Hall constant

References

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https://doi.org/10.1107/97809553602060000628

Chapter 1.1. Introduction to the properties of tensors

1.1.1. The matrix of physical properties

1.1.1.1. Notion of extensive and intensive quantities

1.1.1.2. Notion of tensor in physics

1.1.1.3. The matrix of physical properties

1.1.1.4. Symmetry of the matrix of physical properties

1.1.1.5. Onsager relations

1.1.2. Basic properties of vector spaces

1.1.2.1. Change of basis

1.1.2.2. Metric tensor

1.1.2.3. Orthonormal frames of coordinates – rotation matrix

1.1.2.4. Covariant coordinates – dual or reciprocal space

1.1.2.4.1. Covariant coordinates

1.1.2.4.2. Reciprocal space

1.1.2.4.3. Properties of the metric tensor

1.1.3. Mathematical notion of tensor

1.1.3.1. Definition of a tensor

1.1.3.1.1. Linear forms

1.1.3.1.2. Tensor product

1.1.3.2. Behaviour under a change of basis

1.1.3.3. Operations on tensors

1.1.3.3.1. Addition

1.1.3.3.2. Multiplication by a scalar

1.1.3.3.3. Contracted product, contraction

1.1.3.4. Tensor nature of physical quantities

1.1.3.5. Representation surface of a tensor

1.1.3.5.1. Definition

1.1.3.5.2. Representation surfaces of second-rank tensors

1.1.3.5.3. Representation surfaces of higher-rank tensors

1.1.3.6. Change of variance of the components of a tensor

1.1.3.6.1. Tensor nature of the metric tensor

1.1.3.6.2. How to change the variance of the components of a tensor

1.1.3.6.3. Examples of the use in physics of different representations of the same quantity

1.1.3.7. Outer product

1.1.3.7.1. Definition

1.1.3.7.2. Vector product

1.1.3.7.3. Properties of the vector product

1.1.3.8. Tensor derivatives

1.1.3.8.1. Interpretation of the coefficients of the matrix – change of coordinates

1.1.3.8.2. Generalization

1.1.3.8.3. Differential operators

1.1.3.8.4. Development of a vector function in a Taylor series

1.1.4. Symmetry properties

1.1.4.1. Introduction – Neumann's principle

1.1.4.2. Curie laws

1.1.4.3. Symmetries associated with an electric field and with magnetic induction (flux density)

1.1.4.3.1. Symmetry of an electric field

1.1.4.3.2. Symmetry of magnetic induction

1.1.4.4. Superposition of several causes in the same medium – pyroelectricity and piezolectricity

1.1.4.4.1. Introduction

1.1.4.4.2. Pyroelectricity

1.1.4.4.3. Piezoelectricity

1.1.4.5. Intrinsic symmetry of tensors

1.1.4.5.1. Introduction

1.1.4.5.2. Symmetric tensors

1.1.4.5.2.1. Tensors of rank 2

Examples

1.1.4.5.2.2. Tensors of higher rank

1.1.4.5.3. Antisymmetric tensors – axial tensors

1.1.4.5.3.1. Tensors of rank 2

1.1.4.5.3.2. Tensors of higher rank

Examples

1.1.4.5.3.3. Properties of axial tensors

1.1.4.6. Symmetry of tensors imposed by the crystalline medium

1.1.4.6.1. Matrix method – application of Neumann's principle

1.1.4.6.2. The operator A is in diagonal form

1.1.4.6.2.1. Introduction

1.1.4.6.2.2. Case of a centre of symmetry

1.1.4.6.2.3. General case

1.1.4.6.3. The method of direct inspection

Examples

1.1.4.7. Reduction of the components of a tensor of rank 2

1.1.4.7.1. Triclinic system

1.1.4.7.2. Monoclinic system

1.1.4.7.3. Orthorhombic system

1.1.4.7.4. Trigonal, tetragonal, hexagonal and cylindrical systems

1.1.4.7.4.1. Groups , ; , , ; , , ; ,

1.1.4.7.4.2. Groups , , ; , , , ; , , , ; ,

1.1.4.7.5. Cubic and spherical systems

1.1.4.7.6. Symmetric tensors of rank 2

1.1.4.7.4.1. Groups $[{\bar 3}]$ , ; , $[{\bar 4}]$ , ; , $[{\bar 6}]$ , ; $[(A_{\infty}/M)C]$ , $[A_{\infty}]$

1.1.4.7.4.2. Groups $[{\bar 3}m]$ , , ; , , , $[{\bar {4}}{2}m]$ ; , , , $[{\bar{6}}{2}m]$ ; $[(A_{\infty}/M) \infty (A_{2}/M)C]$ , $[A_{\infty}\infty A_{2}]$

1.1.4.8.1.2. Group $[{\bar 1}]$

1.1.4.8.4.2. Group with a twofold axis parallel to $[Ox_{1}]$

1.1.4.8.4.3. Group with a mirror normal to $[Ox_{1}]$

1.1.4.8.4.4. Groups $[\bar{3}]$ and $[\bar{3}m]$

1.1.4.8.5.5. Group $[\bar{4}]$

1.1.4.8.5.6. Group $[\bar{4}2m]$

1.1.4.8.6.1. Groups , $[A_{\infty}]$ , , $[A_{\infty} \infty A_{2}]$ , and $[A_{\infty} \infty M]$

1.1.4.8.6.2. Group $[\bar{6} = 3/m]$

1.1.4.8.6.3. Group $[\bar{6}2m]$

1.1.4.8.6.4. Groups , $[(A_{\infty}/M)C]$ , and $[(A_{\infty}/M) \infty (A_{2}/M)C]$

1.1.4.8.7.2. Groups and $[\infty A_{\infty}/M]$

1.1.4.8.7.3. Group $[{\bar 4}3m]$

1.1.4.8.7.4. Groups $[m{\bar 3}]$ , $[m{\bar 3}m]$ and $[\infty(A_{\infty}/M)C]$

1.1.4.9.1. Triclinic system (groups $[{\bar 1}]$ , )

1.1.4.9.4.1. Groups and $[{\bar 3}]$

1.1.4.9.4.2. Groups $[{\bar 3}m]$ , , , with the twofold axis parallel to $[Ox_{1}]$

1.1.4.9.5.1. Groups , , $[{\bar 4}]$

1.1.4.9.5.2. Groups , , , $[{\bar 4}2m]$

1.1.4.9.6.1. Groups , $[{\bar 6}]$ , ; $[(A_{\infty}/M)C]$ , $[A_{\infty}]$

1.1.4.9.6.2. Groups , , , $[{\bar 6}2m]$ ; $[(A_{\infty}/M) \infty]$ ; $[(A_{2}/M)C]$ , $[A_{\infty}\infty A_{2}]$

1.1.4.9.7.1. Groups , $[{\bar 3}m]$

1.1.4.9.7.2. Groups $[m{\bar 3}m]$ , , $[{\bar 4}3m]$

1.1.4.9.8.1. Groups $[\infty (A_{\infty}/M)C]$ and $[\infty A_{\infty}]$