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

International Tables for Crystallography (2006). Vol. D, ch. 1.1, pp. 10-31

Section 1.1.4. Symmetry properties

A. Authiera*

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

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)[link], Billings (1969)[link], Mason (1966)[link], Nowick (1995)[link], Nye (1985)[link], Paufler (1986)[link], Shuvalov (1988)[link], Sirotin & Shaskol'skaya (1982)[link], and Wooster (1973)[link].

1.1.4.1. Introduction – Neumann's principle

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We saw in Section 1.1.1[link] 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[link], 1881[link]). 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[link]).

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[link] ). 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[link], 1894[link]), cf. International Tables for Crystallography (2005)[link], Vol. A, Table 10.1.4.2[link] .]

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[link]) :

  • (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[link], 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[link]). 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]

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[link]). 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]

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[link] 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[link]). 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[link]), 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[link]; Ikeda, 1990[link]). 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[link] 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[link]. 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[link]) 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[link]), 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)[link]] [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[link] ), 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[link] ), 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[link] 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[link] ), 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[link], 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[link]), the tensor representing the pyromagnetic effect (Section 1.1.1.3[link]) 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[link]), [(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[link] ) [\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[link] ). 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[link] ) 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[link]). 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[link], 1952a[link],b[link],c[link], 1987[link]), Fumi & Ripamonti (1980a[link],b[link]), Nowick (1995)[link], Nye (1957[link], 1985[link]), Sands (1995)[link], Sirotin & Shaskol'skaya (1982)[link], and Wooster (1973)[link]. There are three main methods for this derivation: the matrix method (described in Section 1.1.4.6.1[link]), the direct inspection method (described in Section 1.1.4.6.3[link]) and the group-theoretical method (described in Section 1.2.4[link] and used in the accompanying software, see Section 1.2.7.4[link] ).

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)[link] 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)[link] 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)[link] 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)[link] 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

  • [p = 2], [r - s \leq 2]: the result of the reduction will be the same for any [q \geq 3];

  • [p = 3], [r - s \leq 3]: the result of the reduction will be the same for any [q \geq 4];

  • [p = 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)[link] 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[link],b[link], 1987[link]), 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[link]).

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: [111, 122, 133, 211, 222, 133, 112, 121, 212, 221, 323, 331, 332, 313.]

  • (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[link]; 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[link], 1985[link]): [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[link] 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}], [3]; [4/m], [{\bar 4}], [4]; [6/m], [{\bar 6}], [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)[link], 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], [32], [3m]; [4/mm], [422], [4mm], [{\bar {4}}{2}m]; [{6}/mm], [622], [6mm], [{\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 [Ox_2])

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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 [1]

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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 [2]

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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 [2/m]

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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 [222]

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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 [mm2]

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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 [mmm]

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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 [3]

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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 [32] 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 [3m] 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 [4]

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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 [422]

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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 [4mm]

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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 [4/m]

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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}]: [Scheme scheme26]

    There are 6 independent components.

  • (ii) Mirror perpendicular to [Ox_{1}] (the twofold axis is at [45 ^{\circ}]) [Scheme scheme27]

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

1.1.4.8.5.7. Group [4/mm]

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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 [6], [A_{\infty}], [622], [A_{\infty} \infty A_{2}], [6mm] and [A_{\infty} \infty M]

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It was shown in Section 1.1.4.6.2.3[link] 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 [6/m], [(A_{\infty}/M)C], [6/mm] 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 [23]

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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 [432] 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}], [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 [2/m], [2], 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 [mmm], [2mm], [222])

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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 [3] 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], [32], [3m], 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 [4/m], [4], [{\bar 4}]

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[Scheme scheme105]

There are 21 independent components.

1.1.4.9.5.2. Groups [4/m m], [422], [4mm], [{\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 [6/m], [{\bar 6}], [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 [6/mm], [622], [6mm], [{\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 [23], [{\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], [432], [{\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 [3] and [{\bar 3}] [Scheme scheme109] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 15 independent components.

  • (ii) Groups [{\bar 3}m], [32], [3m] [Scheme scheme110] 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 [4/m], [4], [{\bar 4}] [Scheme scheme111]

    There are 13 independent components.

  • (ii) Groups [4/mm], [422], [4mm], [{\bar 4}2m] [Scheme scheme112]

    There are 9 independent components.

1.1.4.9.9.6. Hexagonal and cylindrical systems

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  • (i) Groups [6/m], [{\bar 6}], [6]; [(A_{\infty }/M)C, A_{\infty}] [Scheme scheme113] with[t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 12 independent components.

  • (ii) Groups [6/mm], [622], [6mm], [{\bar 6}2m]; [(A_{\infty }/M) \infty (A_{2}/M)C], [A_{\infty} \infty A_{2}] [Scheme scheme114] 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 [23], [{\bar 3}m] [Scheme scheme42] with [t_{1111} - t_{1122} = t_{1212} + t_{1221}.]

    There are 5 independent components.

  • (ii) Groups [m{\bar 3}m], [432], [{\bar 4}3m], and spherical system: the reduced tensors are already symmetric (see Sections 1.1.4.9.7[link] and 1.1.4.9.8[link]).

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[link] and 1.3.2[link] ) 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[link] for third-rank tensors and in Section 1.1.4.10.5[link] 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[link] and 1.3.2[link] ). 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[link] ). 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[link] ).

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)[link] 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[link] 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)[link] 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)[link] 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)[link], as shown in Section 1.1.1.4[link].

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[link]) 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)[link] 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)[link], 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}]: [Scheme scheme47]

    There are 8 independent components.

  • (ii) Group m: [Scheme scheme48]

    There are 10 independent components.

  • (iii) Group [2/m]: all the components are equal to zero.

1.1.4.10.4.3. Orthorhombic system

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  • (i) Group 222: [Scheme scheme49]

    There are 3 independent components.

  • (ii) Group [mm2]: [Scheme scheme50]

    There are 5 independent components.

  • (iii) Group [mmm]: all the components are equal to zero.

1.1.4.10.4.4. Trigonal system

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  • (i) Group 3: [Scheme scheme51] 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}]: [Scheme scheme52] with the same conventions. There are 4 independent components.

  • (iii) Group [3m], mirror perpendicular to [Ox_{1}]: [Scheme scheme53] 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}]: [Scheme scheme54]

    There are 4 independent components.

  • (ii) Groups 422, 622 and [A_{\infty}\infty A_{2}]: [Scheme scheme55]

    There is 1 independent component.

  • (iii) Groups [4mm], [6mm] and [A_{\infty }\infty M]: [Scheme scheme56]

    There are 3 independent components.

  • (iv) Groups [4/m], [6/m] and [(A_{\infty }/M)C]: all the components are equal to zero.

  • (v) Group [\bar{4}]: [Scheme scheme57]

    There are 4 independent components.

  • (vi) Group [\bar{6} = 3/m]: [Scheme scheme58] with the same conventions as for group 3. There are 2 independent components.

  • (vii) Group [\bar{4}2m] – twofold axis parallel to [Ox_{1}]: [Scheme scheme59]

    There are 2 independent components.

  • (viii) Group [\bar{4}2m] – mirror perpendicular to [Ox_{1}] (twofold axis at [45^{\circ}]): [Scheme scheme60]

    The number of independent components is of course the same.

  • (ix) Group [\bar{6}2/m]: [Scheme scheme61] with the same conventions as for group 3. There is 1 independent component.

  • (x) Groups [4/mm], [6/mm] 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]: [Scheme scheme62]

    There is 1 independent component.

  • (ii) Groups 432 and [\infty A_{\infty }]: it was seen in Section 1.1.4.8.6[link] 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[link] ): [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[link] ): [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[link] ): [\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)[link]: [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)[link] to (1.1.4.10)[link], 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[link]. 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)[link] 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)[link] 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)[link] 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)[link] 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)[link] 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}]: [Scheme scheme65] where [\eqalignno{c_{\alpha \beta} &=c_{ijkl}\cr p_{\alpha \beta} &=p_{ijkl}.\cr}]

  • (ii) Piezo-optic coefficients, [\pi_{ijkl}]: [Scheme scheme66] 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}], [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}]: [Scheme scheme70a] [Scheme scheme71] 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, [3m], [{\bar 3}m]: [Scheme scheme72] [Scheme scheme73] with the same conventions.

1.1.4.10.6.5. Tetragonal system

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  • (i) Groups 4, [{\bar 4}] and [4/m]: [Scheme scheme74]

  • (ii) Groups 422, [4mm], [{\bar 4}2m] and [4/mm]: [Scheme scheme75]

1.1.4.10.6.6. Hexagonal system

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  • (i) Groups 6, [{\bar 6}] and [6/m]: [Scheme scheme76] [Scheme scheme77]

  • (ii) Groups 622, [6mm], [{\bar 6}2m] and [6/mm]: [Scheme scheme78]

1.1.4.10.6.7. Cubic system

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  • (i) Groups 23 and [3m]: [Scheme scheme79]

  • (ii) Groups 432, [{\bar 4}3m] and [m{\bar 3}m]: [Scheme scheme80]

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[link] 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[link] ): [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: [Scheme scheme82]

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

  • (ii) Monoclinic system

    • (a) Group 2: [Scheme scheme83]

    • (b) Group m: [Scheme scheme84]

    • (c) Group [2/m]: all components are equal to zero.

  • (iii) Orthorhombic system

    • (a) Group 222: [Scheme scheme85]

    • (b) Group [mm2]: [Scheme scheme86]

    • (c) Group [mmm]: all components are equal to zero.

  • (iv) Trigonal, tetragonal, hexagonal and cylindrical systems

    • (a) Groups 3, 4, 6 and [A_{\infty}]: [Scheme scheme87]

    • (b) Groups 32, 42, 62 and [A_{\infty}\infty A_2]: [Scheme scheme88]

    • (c) Groups [3m], [4m], [6m] and [A_{\infty}\infty M]: [Scheme scheme89]

    • (d) Group [{\bar 4}]: [Scheme scheme90]

    • (e) Group [\bar{4}2m]: [Scheme scheme91]

    • (f) Groups [\bar{3}], [4/m], [\bar{6}2m], [\bar{3}m], [4/mm] and [6/mm]: all components are equal to zero.

  • (v) Cubic and spherical systems

    • (a) Groups 23, 432 and [\infty A_{\infty}]: [Scheme scheme92]

      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[link] ) [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: [Scheme scheme93]

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

  • (ii) Monoclinic system

    • (a) Group 2: [Scheme scheme94]

    • (b) Group m: [Scheme scheme95]

    • (c) Group [2/m]: all components are equal to zero.

  • (iii) Orthorhombic system

    • (a) Group 222: [Scheme scheme96]

    • (b) Group [mm2]: [Scheme scheme97]

    • (c) Group [mmm]: all components are equal to zero.

  • (iv) Trigonal, tetragonal and hexagonal systems

    • (a) Groups 3, 32, 4, 42, 6, 62: [Scheme scheme98]

    • (b) Group [\bar{4}]: [Scheme scheme99]

    • (c) Group [\bar{4}2m]: [Scheme scheme100]

    • (d) Groups [\bar{3}], [3m], [\bar{3}m], [4/m], [4mm], [4/mm], [\bar{6}], [\bar{6}2m] and [6/mm]: all components are equal to zero.

  • (v) Cubic and spherical systems

    • (a) Groups 23, 432 and [A_{\infty} \infty A_{2}]: [Scheme scheme101]

    • (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[link],b[link]) was the first to establish the distinction between `molecular dissymmetry' and `crystalline dissymetry'.

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