International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 333
https://doi.org/10.1107/97809553602060000628 Chapter 1.1. Introduction to the properties of tensors^{a}Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France 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. 
Physical laws express in general the response of a medium to a certain influence. Most physical properties may therefore be defined by a relation coupling two or more measurable quantities. For instance, the specific heat characterizes the relation between a variation of temperature and a variation of entropy at a given temperature in a given medium, the dielectric susceptibility the relation between electric field and electric polarization, the elastic constants the relation between an applied stress and the resulting strain etc. These relations are between quantities of the same nature: thermal, electrical and mechanical, respectively. But there are also cross effects, for instance:
The physical quantities that are involved in these relations can be divided into two categories:

Each of the quantities mentioned in the preceding section is represented by a mathematical expression. Some are direction independent and are represented by scalars: specific mass, specific heat, volume, pressure, entropy, temperature, quantity of electricity, electric potential. Others are direction dependent and are represented by vectors: force, electric field, electric displacement, the gradient of a scalar quantity. Still others cannot be represented by scalars or vectors and are represented by more complicated mathematical expressions. Magnetic quantities are represented by axial vectors (or pseudovectors), which are a particular kind of tensor (see Section 1.1.4.5.3). A few examples will show the necessity of using tensors in physics and Section 1.1.3 will present elementary mathematical properties of tensors.
Remark. Of the four examples given above, the first three (thermal expansion, dielectric constant, stressed rod) are related to physical property tensors (also called material tensors), which are characteristic of the medium and whose components have the same value everywhere in the medium if the latter is homogeneous, while the fourth one (expansion in Taylor series of a field of vectors) is related to a field tensor whose components vary at every point of the medium. This is the case, for instance, for the strain and for the stress tensors (see Sections 1.3.1 and 1.3.2 ).
Each extensive parameter is in principle a function of all the intensive parameters. For a variation of a particular intensive parameter, there will be a variation of every extensive parameter. One may therefore writeThe summation is over all the intensive parameters that have varied.
One may use a matrix notation to write the equations relating the variations of each extensive parameter to the variations of all the intensive parameters: where the intensive and extensive parameters are arranged in column matrices, (di) and (de), respectively. In a similar way, one could write the relations between intensive and extensive parameters asMatrices (C) and (R) are inverse matrices. Their leading diagonal terms relate an extensive parameter and the associated intensive parameter (their product has the dimensions of energy), e.g. the elastic constants, the dielectric constant, the specific heat etc. The corresponding physical properties are called principal properties. If one only of the intensive parameters, , varies, a variation of this parameter is the cause of which the effect is a variation, (without summation), of each of the extensive parameters. The matrix coefficients may therefore be considered as partial differentials:
The parameters that relate causes and effects represent physical properties and matrix (C) is called the matrix of physical properties. Let us consider the following intensive parameters: T stress, E electric field, H magnetic field, Θ temperature and the associated extensive parameters: S strain, P electric polarization, B magnetic induction, σ entropy, respectively. Matrix equation (1.1.1.4) may then be written:
The various intensive and extensive parameters are represented by scalars, vectors or tensors of higher rank, and each has several components. The terms of matrix (C) are therefore actually submatrices containing all the coefficients relating all the components of a given extensive parameter to the components of an intensive parameter. The leading diagonal terms, , , , , correspond to the principal physical properties, which are elasticity, dielectric susceptibility, magnetic susceptibility and specific heat, respectively. The nondiagonal terms are also associated with physical properties, but they relate intensive and extensive parameters whose products do not have the dimension of energy. They may be coupled in pairs symmetrically with respect to the main diagonal:
It is important to note that equation (1.1.1.6) is of a thermodynamic nature and simply provides a general framework. It indicates the possibility for a given physical property to exist, but in no way states that a given material will exhibit it. Curie laws, which will be described in Section 1.1.4.2, show for instance that certain properties such as pyroelectricity or piezoelectricity may only appear in crystals that belong to certain point groups.
If parameter varies by , the specific energy varies by du, which is equal to We have, therefore and, using (1.1.1.5), Since the energy is a state variable with a perfect differential, one can interchange the order of the differentiations: Since p and q are dummy indices, they may be exchanged and the last term of this equation is equal to . It follows thatMatrices and are therefore symmetric. We may draw two important conclusions from this result:
Let us now consider systems that are in steady state and not in thermodynamic equilibrium. The intensive and extensive parameters are time dependent and relation (1.1.1.3) can be written where the intensive parameters are, for instance, a temperature gradient, a concentration gradient, a gradient of electric potential. The corresponding extensive parameters are the heat flow, the diffusion of matter and the current density. The diagonal terms of matrix correspond to thermal conductivity (Fourier's law), diffusion coefficients (Fick's law) and electric conductivity (Ohm's law), respectively. Nondiagonal terms correspond to cross effects such as the thermoelectric effect, thermal diffusion etc. All the properties corresponding to these examples are represented by tensors of rank 2. The case of secondrank axial tensors where the symmetrical part of the tensors changes sign on time reversal was discussed by Zheludev (1986).
The Onsager reciprocity relations (Onsager, 1931a,b) express the symmetry of matrix . They are justified by considerations of statistical thermodynamics and are not as obvious as those expressing the symmetry of matrix (). For instance, the symmetry of the tensor of rank 2 representing thermal conductivity is associated with the fact that a circulating flow is undetectable.
Transport properties are described in Chapter 1.8 of this volume.
[The reader may also refer to Section 1.1.4 of Volume B of International Tables for Crystallography (2001).]
Let us consider a vector space spanned by the set of n basis vectors , , . The decomposition of a vector using this basis is written using the Einstein convention. The interpretation of the position of the indices is given below. For the present, we shall use the simple rules:
Let us now consider a second basis, . The vector x is independent of the choice of basis and it can be decomposed also in the second basis:
If and are the transformation matrices between the bases and , the following relations hold between the two bases: (summations over j and i, respectively). The matrices and are inverse matrices: (Kronecker symbol: if if ).
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 and 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.
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 and is Let us put The nine components 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
The table of the components is therefore symmetrical. One of the definition properties of the scalar product is that if for all x, then . This is translated as
In order that only the trivial solution exists, it is necessary that the determinant constructed from the 's is different from zero: This important property will be used in Section 1.1.2.4.1.
An orthonormal coordinate frame is characterized by the fact that One deduces from this that the scalar product is written simply as
Let us consider a change of basis between two orthonormal systems of coordinates: Multiplying the two sides of this relation by , it follows that 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
The matrix coefficients, , are the direction cosines of with respect to the basis vectors. Similarly, we have so that where ^{T} indicates transpose. It follows that so that The matrices A and B are unitary matrices or matrices of rotation and
One can write for the coefficients giving six relations between the nine coefficients . There are thus three independent coefficients of the matrix A.
Using the developments (1.1.2.1) and (1.1.2.5), the scalar products of a vector x and of the basis vectors can be written The n quantities 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 are the unknowns. One can solve it since (see the end of Section 1.1.2.2). It follows that with
The table of the 's is the inverse of the table of the 's. Let us now take up the development of x with respect to the basis :
Let us replace by the expression (1.1.2.10): and let us introduce the set of n vectors which span the space . 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
The 's are the components of x in the basis . 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
It can be shown that the basis vectors transform in a change of basis like the components of the physical space. They are therefore contravariant. In a similar way, the components of a vector x with respect to the basis transform in a change of basis like the basis vectors in direct space, ; they are therefore covariant:
Let us take the scalar products of a covariant vector and a contravariant vector : [using expressions (1.1.2.5), (1.1.2.11) and (1.1.2.13)].
The relation we obtain, , is identical to the relations defining the reciprocal lattice in crystallography; the reciprocal basis then is identical to the dual basis .
In a change of basis, following (1.1.2.3) and (1.1.2.5), the 's transform according to Let us now consider the scalar products, , of two contravariant basis vectors. Using (1.1.2.11) and (1.1.2.13), it can be shown that
In a change of basis, following (1.1.2.16), the 's transform according to The volumes V ′ and V of the cells built on the basis vectors and , respectively, are given by the triple scalar products of these two sets of basis vectors and are related by where 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
If the basis is orthonormal, and V are equal to one, is equal to the volume V ′ of the cell built on the basis vectors and This relation is actually general and one can remove the prime index:
In the same way, we have for the corresponding reciprocal basiswhere is the volume of the reciprocal cell. Since the tables of the 's and of the '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.
For the mathematical definition of tensors, the reader may consult, for instance, Lichnerowicz (1947), Schwartz (1975) or Sands (1995).
A linear form in the space is written where is independent of the chosen basis and the 's are the coordinates of T in the dual basis. Let us consider now a bilinear form in the product space of two vector spaces with n and p dimensions, respectively:
The np quantities 's are, by definition, the components of a tensor of rank 2 and the form is invariant if one changes the basis in the space . The tensor is said to be twice covariant. It is also possible to construct a bilinear form by replacing the spaces and by their respective conjugates and . Thus, one writes where is the doubly contravariant form of the tensor, whereas and 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.
Let us consider two vector spaces, with n dimensions and with p dimensions, and let there be two linear forms, in and in . We shall associate with these forms a bilinear form called a tensor product which belongs to the product space with np dimensions, :
This correspondence possesses the following properties:
The analytical expression of the tensor product is then One deduces from this that
It is a tensor of rank 2. One can equally well envisage the tensor product of more than two spaces, for example, 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 with itself or with its conjugate . Thus, a tensor product of rank 3 will have components. The tensor product can be generalized as the product of multilinear forms. One can write, for example,
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
Let us put these expressions into the trilinear form (1.1.3.1):
Now we can equally well make the components of the tensor appear in the new basis:
As the decomposition is unique, one obtains
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 .
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′: so that Now the matrices A and B are inverses of one another: The symmetry operator is a tensor of rank 2, once covariant and once contravariant.
It is necessary that the tensors are of the same nature (same rank and same variance).
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.
Let us first consider the dielectric constant. In the introduction, we remarked that for an isotropic medium
If the medium is anisotropic, we have, for one of the components, This relation and the equivalent ones for the other components can also be written using the Einstein convention.
The scalar product of D by an arbitrary vector x is
The righthand member of this relation is a bilinear form that is invariant under a change of basis. The set of nine quantities constitutes therefore the set of components of a tensor of rank 2. Expression (1.1.3.3) is the contracted product of by .
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, , which appear in a medium submitted to a stress represented by the secondrank tensor are where the tensor nature of 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 , we obtain a linear form on the lefthand side, and a trilinear form on the righthand side, which shows that the coefficients are the components of a thirdrank tensor. Let us now consider the piezooptic (or photoelastic) effect (see Sections 1.1.4.10.5 and 1.6.7 ). The components of the variation of the dielectric impermeability due to an applied stress are
In a similar fashion, consider the contracted product of both sides of this relation by two vectors of covariant components and , respectively. We obtain a bilinear form on the lefthand side, and a quadrilinear form on the righthand side, showing that the coefficients are the components of a fourthrank tensor.
Let us consider a tensor represented in an orthonormal frame where variance is not important. The value of component in an arbitrary direction is given by where the , 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 .
The representation surfaces of secondrank tensors are quadrics. The directions of their principal axes are obtained as follows. Let be a secondrank tensor and let be a vector with coordinates . The doubly contracted product, , is a scalar. The locus of points M such that is a quadric. Its principal axes are along the directions of the eigenvectors of the matrix with elements . They are solutions of the set of equations where the associated quantities λ are the eigenvalues.
Let us take as axes the principal axes. The equation of the quadric reduces to
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:
Equation (1.1.2.17) describing the behaviour of the quantities 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 's transform under a change of basis like the product of two contravariant coordinates. The coefficients and 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.
Let us take a tensor product We know that It follows that is a tensor product of two vectors expressed in the dual space:
One can thus pass from the doubly covariant form to the doubly contravariant form of the tensor by means of the relation
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 or , according to the case. For instance,
Let us consider, for example, the force, F, which is a tensor quantity (tensor of rank 1). One can define it:
The tensor defined by is called the outer product of vectors x and y. (Note: The symbol is different from the symbol for the vector product.) The analytical expression of this tensor of rank 2 is
The components of this tensor satisfy the properties It is an antisymmetric tensor of rank 2.
Consider the socalled permutation tensor of rank 3 (it is actually an axial tensor – see Section 1.1.4.5.3) defined by and let us form the contracted product It is easy to check that
One recognizes the coordinates of the vector product.
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 . In the space of n dimensions, one would write
The number of independent components of is equal to or 3 in the space of three dimensions and 6 in the space of four dimensions, and the independent components of 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
This is a symmetry with respect to a point that transforms a righthanded set of axes into a lefthanded 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.
We have under a change of axes: This shows that the new components, , can be considered linear functions of the old components, , and one can write It should be noted that the covariance has been increased.
Consider a field of tensors that are functions of space variables. In a change of coordinate system, one has Differentiate with respect to : It can be seen that the partial derivatives 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 . It is therefore possible to introduce a tensor of rank 1, (nabla), of which the components are the operators given by the partial derivatives .
If one applies the operator nabla to a scalar ϕ, one obtains This is a covariant vector in reciprocal space.
Now let us form the tensor product of by a vector v of variable components. We then have
The quantities form a tensor of rank 2. If we contract it, we obtain the divergence of v: Taking the vector product, we get The curl is then an axial vector.
Let be a vector function. Its development as a Taylor series is writtenThe coefficients of the expansion, , are tensors of rank .
An example is given by the relation between displacement and electric field: (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 .
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).
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 semiaxes 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.
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 [Axes 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) :
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:
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,
Let us now consider the phenomenon as a cause producing a certain effect with symmetry E: We can therefore conclude that
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 . 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 . The symmetries associated with the force considered as a cause and as an effect being the same, we may conclude that is its characteristic symmetry.
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, . 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, . The electric polarization or the electric displacement have the same symmetry.
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, . 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 .
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:
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 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 while the electric field is a polar vector of symmetry .
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.
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, . Since the asymmetry of the cause must preexist 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 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.
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 . The effect is an electric polarization, of symmetry , which must be higher than or equal to the intersection of the symmetries of the two causes: where 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: 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 . The above condition becomes If the applied compression is parallel to the threefold axis, the intersection is identical to the symmetry of the crystal, , 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.
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.
A bilinear form is symmetric if Its components satisfy the relations
The associated matrix, T, is therefore equal to its transpose : In a space with n dimensions, the number of independent components is equal to
Examples
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 . 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 . The associated matrix is a one, and the number of independent components is equal to 45.
A bilinear form is said to be antisymmetric if Its components satisfy the relations The associated matrix, T, is therefore also antisymmetric: The number of independent components is equal to , where n is the number of dimensions of the space. It is equal to 3 in a threedimensional 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: with and .
Example. As shown in Section 1.1.3.7.2, the components of the vector product of two vectors, x and y, 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.
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.
Examples
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.
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 grouptheoretical method (described in Section 1.2.4 and used in the accompanying software, see Section 1.2.7.4 ).
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 If A represents a symmetry operation, it is a unitary matrix:
Since the tensor is invariant under the action of the symmetry operator A, one has, according to Neumann's principle, and, therefore,
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 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 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.
If one takes as the system of axes the eigenvectors of the operator A, the matrix is written in the form where θ is the rotation angle, is taken parallel to the rotation axis and coefficient 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 (without any summation).
If the product (without summation) is equal to unity, equation (1.1.4.2) is trivial and there is significance in the component . On the contrary, if it is different from 1, the only solution for (1.1.4.2) is that . One then finds immediately that certain components of the tensor are zero and that others are unchanged.
All the diagonal components are in this case equal to −1. One thus has:
By replacing the matrix coefficients by their expression, (1.1.4.2) becomes, for a proper rotation, 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 is the rank of the tensor. The component is not affected by the symmetry operation if where K is an integer, and is equal to zero if
The angle of rotation θ can be put into the form , where q is the order of the axis. The condition for the component not to be zero is then
The condition is fulfilled differently depending on the rank of the tensor, p, and the order of the axis, q. Indeed, we have and
It follows that:
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 by associating with the tensor a matrix T: where B is the symmetry operation. Through identification of homologous coefficients in matrices T and , one obtains relations between components that enable the determination of the independent components.
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
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.
The reduction is given for each of the 11 Laue classes.
Groups , 1: no reduction, the tensor has 9 independent components. The result is represented in the following symbolic way (Nye, 1957, 1985): where the sign • represents a nonzero component.
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 , one has
The other components are not affected. The result is represented as
There are 5 independent components. If the twofold axis is taken along axis , which is the usual case in crystallography, the table of independent components becomes
Groups mmm, 2mm, 222: the reduction is obtained by considering two perpendicular twofold axes, parallel to and to , respectively. One obtains
There are 3 independent components.
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.
If we consider a fourfold axis parallel to represented by the matrix given in (1.1.4.3), by applying the direct inspection method one finds 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.
The cubic system is characterized by the presence of threefold axes along the directions. The action of a threefold axis along [111] on the components of a vector results in a permutation of these components, which become, respectively, and then . One deduces that the components of a tensor of rank 2 satisfy the relations
The cubic groups all include as a subgroup the group 23 of which the generating elements are a twofold axis along and a threefold axis along [111]. If one combines the corresponding results, one deduces that which can be summarized by
There is a single independent component and the medium behaves like a property represented by a tensor of rank 2, like an isotropic medium.
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.
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.
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.
There are 3 independent components. The quadric is any quadric, the axes of which coincide with the crystallographic axes. Only three parameters are required.
There are 2 independent components. The quadric is of revolution. It is characterized by two parameters: the lengths of its two axes.
Choosing the twofold axis parallel to and applying the direct inspection method, one finds
There are 13 independent components. If the twofold axis is parallel to , one finds
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:
There are 14 independent components. If the mirror axis is normal to , one finds
There are three orthonormal twofold axes. The reduction is obtained by combining the results associated with two twofold axes, parallel to and , respectively.
There are 6 independent components.
The method of direct inspection can be applied for a fourfold axis. One finds
There are 7 independent components.
The matrix corresponding to axis is and the form of the matrix is
There are 6 independent components.
One combines either the reductions for groups and 222, or the reductions for groups and 2mm.
The number of independent components is of course the same, 6.
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.
One combines the reductions for the groups corresponding to a threefold axis parallel to and to a mirror perpendicular to :
There are 2 independent components.
One combines the reductions corresponding to a twofold axis parallel to and to a threefold axis parallel to [111]:
There are 2 independent components.
There is no reduction; all the components are independent. Their number is equal to 81. They are usually represented as a matrix, where components are replaced by ijkl, for brevity: This matrix can be represented symbolically by where the matrix has been subdivided for clarity in to nine submatrices.
The reduction is obtained by the method of direct inspection. For a twofold axis parallel to , one finds
There are 41 independent components.
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.
Many tensors representing physical properties or physical quantities appear in relations involving symmetric tensors. Consider, for instance, the strain resulting from the application of an electric field E (the piezoelectric effect): where the firstorder terms represent the components of the thirdrank converse piezoelectric tensor and the secondorder terms represent the components of the fourthrank electrostriction tensor. In a similar way, the direct piezoelectric effect corresponds to the appearance of an electric polarization P when a stress is applied to a crystal:
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 , 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 thirdrank tensors and in Section 1.1.4.10.5 for fourthrank tensors.
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 oneindex notation.
1.1.4.10.3. Reduction of the number of independent components of thirdrank polar tensors due to the symmetry of the strain and stress tensors
Equation (1.1.4.5) can be written
The sums for have a definite physical meaning, but it is impossible to devise an experiment that permits and to be measured separately. It is therefore usual to set them equal:
It was seen in Section 1.1.4.8.1 that the components of a thirdrank tensor can be represented as a matrix which can be subdivided into three submatrices:
Relation (1.1.4.7) shows that submatrices 1 and 2 are identical.
One puts, introducing a twoindex notation, Relation (1.1.4.7) becomes
The coefficients may be written as a matrix: This matrix is constituted by two submatrices. The lefthand one is identical to the submatrix 1, and the righthand one is equal to the sum of the two submatrices 2 and 3:
The inverse piezoelectric effect expresses the strain in a crystal submitted to an applied electric field: where the matrix associated with the coefficients is a 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 are then given by This relation can be written simply as where the matrix of the coefficients is a matrix which is the transpose of the matrix.
There is another set of piezoelectric constants (see Section 1.1.5) which relates the stress, , and the electric field, , which are both intensive parameters: where a new piezoelectric tensor is introduced, . Its components can be represented as a matrix:
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
Submatrices 2 and 3 are equal. One introduces here a twoindex notation through the relation , and the matrix can be written
The relation between the full and the reduced matrix is therefore different for the and the 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, , , ).
1.1.4.10.4. Independent components of the matrix associated with a thirdrank polar tensor according to the following point groups
1.1.4.10.5. Reduction of the number of independent components of fourthrank polar tensors due to the symmetry of the strain and stress tensors
Let us consider five examples of fourthrank tensors:
In each of the equations from (1.1.4.9) to (1.1.4.10), the contracted product of a fourthrank tensor by a symmetric secondrank tensor is equal to a symmetric secondrank tensor. As in the case of the thirdrank 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: where the secondrank tensors are represented by column matrices, which can each be subdivided into three submatrices and the matrix associated with the fourthrank tensors is subdivided into nine submatrices, as shown in Section 1.1.4.9.1. The symmetry of the secondrank 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
The sums for have a definite physical meaning, but it is impossible to devise an experiment permitting and to be measured separately. It is therefore usual to set them equal in order to avoid an unnecessary constant:
Furthermore, the lefthand term of (1.1.4.11) remains unchanged if we interchange the indices i and j. The terms on the righthand side therefore also remain unchanged, whatever the value of or . It follows that Similar relations hold for , , and : 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: We shall now introduce a twoindex notation for the elastic compliances, according to the following conventions: We have thus associated with the fourthrank tensor a square matrix with 36 coefficients:
One can translate relation (1.1.4.12) using the matrix representing by adding term by term the coefficients of submatrices 2 and 3, 4 and 7 and 5, 6, 8 and 9, respectively:
Using the twoindex notation, equation (1.1.4.9) becomes
A similar development can be applied to the other fourthrank tensors , which will be replaced by matrices with 36 coefficients, according to the following rules.
1.1.4.10.6. Independent components of the matrix associated with a fourthrank tensor according to the following point groups
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 is given by The secondrank axial tensor associated with this tensor is defined by
For instance, the piezomagnetic coefficients that give the magnetic moment due to an applied stress are the components of a secondrank axial tensor, (see Section 1.5.7.1 ):
1.1.4.10.7.2. Independent components of symmetric axial tensors according to the following point groups
Some axial tensors are also symmetric. For instance, the optical rotatory power of a gyrotropic crystal in a given direction of direction cosines is proportional to a quantity G defined by (see Section 1.6.5.4 ) where the gyration tensor is an axial tensor. This expression shows that only the symmetric part of is relevant. This leads to a further reduction of the number of independent components:
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'.
[The reader may also consult Mason (1966), Nye (1985) or Sirotin & Shaskol'skaya (1982).]
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 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 (see Section 1.3.2.8 ). Let us consider the Gibbs freeenergy function defined by Differentiation of gives The extensive parameters are therefore partial derivatives of the free energy: Each of these quantities may be expanded by performing a further differentiation in terms of the intensive parameters, , and Θ. We have, to the first order, To a first approximation, the partial derivatives may be considered as constants, and the above relations may be integrated: 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:
In a similar way,
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 circumstancesRemark. Equations (1.1.5.1) are, as has been said, firstorder 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 higherorder terms as, for instance, in:
We use here another Gibbs function, the electric Gibbs function, , defined by
Differentiation of givesIt follows that and a set of relations analogous to (1.1.5.1):where the components are the isothermal elastic stiffnesses at constant field and constant temperature, are the piezoelectric stress coefficients at constant strain and constant temperature, are the temperaturestress constants andare the components of the pyroelectric effect at constant strain.
The relations between these coefficients and the usual coefficients are easily obtained:
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, , and the pyroelectric coefficients at constant strain, , also called real pyroelectric coefficients, . Let us put and in the first equation of (1.1.5.1). For a given variation of temperature, , the observed strain is From the second equations of (1.1.5.1) and (1.1.5.2), it follows that Substituting the expression and eliminating , it follows that
This relation shows that part of the pyroelectric effect is actually due to the piezoelectric effect.
Piezoelectric resonators usually operate at a high frequency where there are no heat exchanges, and therefore in an adiabatic regime . From the third equation of (1.1.5.1), we obtain a relation between the temperature variation, the applied stress and the electric field:
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: By comparing these expressions with (1.1.5.1), we obtain the following relations between the adiabatic and the isothermal coefficients:

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