An introduction to crystal physics
Description of the physical properties of crystals
Ervin Hartmann
Introduction
Most monographs on physics discussing the physical properties of matter usually proceed from isotropic materials and as a generalization include a more or less limited description of the behaviour of crystalline bodies. This way of presentation is doubtless advantageous, however, it implies a separate discussion of the various properties, inevitably obscuring the general principles and methods applicable in the theory of crystal properties. The purpose of the present work is to discuss and summarize in a unified treatment the physical properties of crystals, and to illustrate by some typical examples the principles and methods underlying a uniform description of these properties.
For an introduction, in order to develop a clear crystal physical picture, let us investigate the problem of the dielectric susceptibility in an isotropic and an anisotropic medium respectively.
In isotropic insulators the dipoles generated in the course of dielectric polarization are parallel with the electric field (Fig. 1a), consequently the relation between the vector of the electric polarization () and the electric field (), that is the susceptibility, can be characterized by one single value ().
(1.1) 
In an anisotropic medium, however, the dipoles formed during the dielectric polarization are generally not parallel with the electric field. Fig. 1b depicts the relatively simple case in which the dipoles though arranged in a plane parallel with the electric field (represented by the plane of the drawing) have a different direction. In this case the vector of the electric polarization () due to the vertical electric field ( = )) has not only a vertical () but also a horizontal component (). Consequently in the case of Fig. 1b two quantities (; )are necessary to describe the relation between the vertical electric field and the electric polarization, and the components of the two vectors in question are connected by the equations
(1.2)  
Generally, when investigating the dielectric polarization in the three dimensional space one finds that, disregarding the higher order effects, in an x_{1}, x_{2}, x_{3} coordinate system the following equations hold between the components of the vector of the electric field ( = [E_{1}, E_{2}, E_{3}]) and the components of the electric polarization ( = [P_{1}, P_{2}, P_{3}]):
(1.3) 
This means that altogether nine data are necessary to describe the relation between the electric field and the electric polarization. By means of vector algebra it can be shown that in case of an orthogonal coordinate transformation the nine coefficients in the eq. (1.3) transform as the products of the components of two vectors, i.e. they are components of a secondrank tensor. Consequently the dielectric susceptibility in an anisotropic medium can be described by a secondrank tensor.
Equations (1.3) can be rewritten in an abbreviated form
(1.4) 
With Einstein's notation the symbol can be omitted if in the same term a suffix occurs twice. Accordingly eq. (1.4) takes the following form
(1.5) 
In the forthcoming discussions the Einstein convention will be used.
2. Physical properties as tensors
It has been demonstrated in the introduction that the dielectric susceptibility of an anisotropic medium can be described with a secondrank tensor which expresses the relation between two physical quantities i.e. the relation between the vector of polarization and the vector of the electric field. Similarly the greater part of the various physical properties may be described with a tensor which establishes the relation existing between measurable physical tensor quantities. Every scalar is a zerorank, and every vector a firstrank tensor. Generally in crystal physics a set of 3^{r} quantities with r indices transforming under transition from the old coordinates to the new ones as the products of the components of r vectors is called a polar (or true) tensor of rank r.
Accordingly if the [B_{ijk...n}] and [A_{pqr...u}] tensors represent physical quantities the general form of the relation between these quantities may be written (in firstorder approximation) using the Einstein's convention as follows
(2.1) 
It follows from the tensor algebra that if [A_{pqr...u}] denotes an frank and [B_{ijk...n}] a grank tensor the [a_{ijk...npqr...u}], denoting the physical property, must be an (f + g)rank tensor.
Let us consider some examples. In a given state the density of matter expresses the relationship between its mass and volume, they are represented by 0rank tensors, consequently the density is represented by a 0rank tensor (i.e. a scalar). The pyroelectric properties of crystals are described by a firstrank tensor. The pyroelectric tensor, (essentially a vector) represents the relation between a firstrank tensor (the vector of electric polarization) and a zerorank tensor (the temperature). Besides the dielectric susceptibility, the electrical conductivity, the heat conductivity, the thermal expansion and so on may be represented by a secondrank tensor. Further examples including properties which can be expressed with higherrank tensors are summarized in Table 1.
Crystals have further on also some anisotropic properties which cannot be directly represented by tensors, such properties  not to be discussed in this paper  are for instance the tensile strength, flow stress, surface energy, rate of growth and dissolution, and so on.
Property or effect  Tensor notation  Tensor rank  Maximum no. of independent components  Defining equation  Physical quantities in the defining equation  
Density  1  mass  volume  
Specific heat  c  1  entropy  T temperature  
Pyroelectricity  [p_{i}]  1  3  [P_{i}] dielectric polarization  T temperature  
Electrocaloric effect  [p_{i}]  1  3  entropy  [E_{i}] electric field  
Dielectric permittivity  2  6  [D_{i}] electric displacement  [E_{j}] electric field  
Magnetic permeability  2  6  [B_{i}] magnetic induction  [H_{j}] magnetic field  
Electrical conductivity  2  6  [j_{i}] current density  [E_{k}] electric field  
Electrical resistivity  2  6  [E_{i}] electric field  [j_{k}] current density  
Thermal conductivity  [k_{ij}]  2  6  [h_{i}] heat flux  temperature gradient  
Thermal expansion  2  6  strain  T temperature  
Seebeck effect  2  9  [E_{i}] electric field  temperature gradient  
Peltier effect  2  9  [h_{i}] heat flux  [j_{k}] current density  
Hall effect  3  9  [E_{i}] electric field  [j_{k}] current density [H_{l}] magnetic field  
Direct piezoelectric effect  [d_{ijk}]  3  18  [P_{i}] dielectric polarization  stress  
Converse piezoelectric effect  [d_{ijk}]  3  18  strain  [E_{i}] electric field  
Piezomagnetic effect  [q_{lij}]  3  18  [M_{l}] magnetic polarization  stress  
Electrooptical effect  [r_{ijk}]  3  18  [a_{ij}] dielectric impermeability  [E_{k}] electric field  
Second harmonic generation  [d_{ijk}]  3  18  dielectric polarization at frequency  electric field  
Secondorder elastic stiffnesses  [c_{ijkl}]  4  21  stress  strain  
Secondorder elastic compliances  [s_{ijkl}]  4  21  strain  stress  
Piezooptic effect  4  36  [a_{ij}] dielectric impermeability  stress  
Quadratic electrooptic effect  [R_{ijkl}]  4  36  [a_{ij}] dielectric impermeability  electric field  
Electrostriction  4  36  strain  electric field  
Thirdorder elastic stresses  [c_{ijklmn}]  6  56  energy of deformation [c_{ijkl}] secondorder stiffnesses  the Lagrange finite strain components 
3. The intrinsic symmetry of the physical properties
The rank of the tensors determines the numbers of the tensor components. The number of the components of the 0,1,2,3,4,5,6rank tensors are 1,3,9,27,81,243,729. However, certain symmetries considerably reduce the number of the independent components. These may be intrinsic symmetries, inherent in the physical property, or crystal symmetries, whose effect on the number of the independent components will be discussed in the next section.
In some instances the intrinsic symmetries follow from the definition of the physical property in question. Thus for example in the case of elasticity it follows from the symmetry of the stress and deformation tensors that also the fourthrank tensor of the secondorder elastic stiffnesses [c_{ijkl}] (see Table 1) is symmetric with respect to the (ij) and (kl) permutations. This way the number of the independent coefficients of the fourthrank elastic tensor decreases from 81 to 36. Further on from the symmetry of the deformation tensor follows the symmetry of the [d_{ijk}] piezoelectric tensor (see Table 1) with respect to the commutability of the j and k suffixes, which means that the piezoelectric tensor has not more than 18 independent components.
The symmetries inherent in the physical properties may be in most cases found by thermodynamical reasoning. In case of equilibrium properties, i.e. properties which refer to thermodynamically reversible changes, the intrinsic symmetry of the properties can be disclosed by investigating the thermodynamical potentials. For physical properties characteristic for transport processes the intrinsic symmetry is the consequence of Onsager's principle.
In order to illustrate the thermodynamical discussion of equilibrium properties let us consider a more complicated example from which not only the symmetry of the tensors representing individual physical properties (tensors of elasticity, electric and magnetic susceptibility) but also the relationship among the tensors representing various properties becomes obvious. In the example the elastic, thermo, electric and magnetic effects are investigated simultaneously. Independent variables should be the stress [], the electric field [E_{k}], the magnetic field [H_{l}] and temperature [T] whereas the deformation [], the polarization [P_{i}], the magnetization [M_{i}] and the entropy [S] are selected as dependent variables. The differentials of the former quantities are obviously connected with the following relationships:
(3.1) 
The partial derivatives are characteristic of the following effects:
1. Elastic deformation.
2. Reciprocal (or converse) piezoelectric effect.
3. Reciprocal (or converse) piezomagnetic effect.
4. Thermal dilatation.
5. Piezoelectric effect.
6. Electric polarization.
7. Magnetoelectric polarization.
8. Pyroelectricity.
9. Piezomagnetic effect.
10. Reciprocal (or converse) magnetoelectric polarization.
11. Magnetic polarization.
12. Pyromagnetism.
13. Piezocaloric effect.
14. Electrocaloric effect.
15. Magnetocaloric effect.
16. Heat transmission.
In order to recognize the relationships among the partial derivatives of the equationsystem (3.1) let us discuss the Gibb's potential of the system
(3.2) 
(3.3) 
(3.4) 
(3.5) 
(3.6) 
(3.7) 
(3.8) 
(3.9) 
Investigating the second partial derivatives of the Gibbs' potential and taking into consideration the commutability of the sequence of the partial differentiations one comes to the conclusion that the elastic [s_{ijkl}], dielectric [] and diamagnetic susceptibility [] tensors, as defined by eqs. (3.10)(3.12) below, are symmetrical
(3.10) 
(3.11) 
(3.12) 
(3.13) 
(3.14) 
(3.15) 
(3.16) 
(3.17) 
(3.18) 
(a) the components of the tensors representing the piezoelectric and reciprocal piezoelectric effect (eq. 3.13)),
(b) the components of the tensors representing the piezomagnetic and reciprocal piezomagnetic effect (eq. (3.14)),
(c) the components of the tensors representing the magnetoelectric polarization and reciprocal magnetoelectric polarization (eq. (3.15)),
(d) the components of the tensors representing the piezocaloric effect and the thermal dilatation (eq. (3.16)),
(e) the components of the tensors representing the pyroelectric and electrocaloric effect (eq. (3.17)),
(f) the components of the tensors representing the pyromagnetic and magneto caloric effect (eq. (3.18)).
Integrating equations (3.1) taking into consideration the above statements, and restricting only to the firstorder effects, the following system of equations is obtained
(3.19) 
It is perhaps worthwhile to draw the reader's attention to the fact that the system of equations (3.19) represents 16 = 9 + 3 + 3 + 1 equations, the right side of these equations contains 16 terms, since the suffixes occurring twice in each term imply summation according to the Einstein convention. Furthermore the deformation tensor [] and the stress tensor []are symmetrical, consequently the system of equation (3.19) contains altogether 13 independent equations with 13 independent variables.
As has been pointed out the intrinsic symmetry characteristic for the transport processes is the consequence of Onsager's reciprocal relations. However, it is important to stress that this relation is valid only if the fluxes and the thermodynamical forces connected with them are suitably selected. For simplicity let us study the case of the electrical conductivity. The thermodynamical force [X_{k}] attached to the electrical current density [j_{i}] is
(3.20) 
In this case the law of linear current flow is
(3.21) 
(3.22) 
L_{ik} = L_{ki}.  (3.23) 
(3.24) 
(3.25) 
Discussing composite transport processes by some proper selection of the fluxes and thermodynamical forces corresponding to them not only the symmetry of the various tensors, but also the relationships among the tensors representing various properties follow from Onsager's reciprocity relations. For example when discussing the thermoelectric effects the symmetry of the electrical and thermal conductivity tensors follow from Onsager's principle as well as the relationships between the tensors representing the Seebeck effect []and the Peltier tensor [] (see Table 1).
(3.26) 
(3.27) 
4. Neumann's principle
It has been demonstrated in the previous section that the intrinsic symmetry of the physical properties decreases the number of the independent tensor components. Further reduction of the independent components of a physical property tensor, and the zero value of certain components follow from the fact that the crystal symmetry exerts some influence on the symmetry of the physical properties. This fact is expressed by Neuman's principle formulated already in the 19th century according to which the symmetry elements of any physical property of a crystal must include all the symmetry elements of the point group of the crystal :
(4.1) 
The wellknown equations of transformation from an orthogonal x_{1}, x_{2}, x_{3} system to another similarly orthogonal system are for first, second, third and fourthrank polar tensors according to their definition:
(4.2) 
(4.3) 
(4.4) 
(4.5) 
(4.6) 
(4.7) 
In some cases the tensor describing the physical properties is not polar, but axial (as for instance the tensor describing the optical activity or piezomagnetism). For axial (or pseudo) tensors the following transformation relation may be used as definition
(4.8) 
It is not difficult to find out whether the tensor representing any physical property is polar or axial, since this can be easily decided by the eq. (2.1) defining the physical property in question. If only one of the tensors [A_{pqr...u}] and [B_{ijk...n}] in eq. (2.1) is axial (for instance magnetic field is an axial tensor of rank one) also the property tensor [a_{ijk...npqr...u}] as defined by eq. (2.1) will be axial, in every other case the tensor is polar.
It should be remarked that if also the magnetic point groups are considered eq. (4.6) and eq. (4.8) expressing the transformation properties of the tensor components are valid only for conventional symmetry operations. If, however, the conventional symmetry operations are combined with timeinversion which actually happens in antisymmetry operations (see [1014]) the right sides of eqs. (4.6) and (4.8) respectively should be multiplied with (1) whenever eq. (2.1) defining the physical properties contains the magnetic vector quantities (magnetic field, magnetic induction, magnetization vector) odd times. Tensors representing this type of properties are called Ctensors.^{1} For a more detailed discussion of this problem the reader is referred to the literature.^{13}
Considering the equations of transformation (4.6) and (4.8) and with regard to the above remark, the relationships between the components of the polar and axial tensors for a given crystal class can now be defined, since the invariance of a tensor with regard to any symmetry operation requires the relationship
(4.9) 
(4.10) 
(4.11) 
For every tensor component an equation of the type (4.10) and (4.11) respectively should be valid so that the tensor components must satisfy a system of these equations. Since this holds for every symmetry operation of a given crystal class, the number of the systems of equations between the tensor components will be equal to the number of the symmetry operations which may be performed in the given crystal class. However, in order to obtain every relationship among the components of a tensor representing any physical property in case of a given crystal class, it is not necessary to write down for every symmetry operation the system of equations of the type (4.10) and (4.11) respectively. It is well known from the group theory that for various crystal classes every symmetry operation may be deduced from a few basic symmetry operations. The application of the matrices corresponding to these basic operations (the generating matrices) are sufficient to obtain the effect due to the symmetry of a crystal class on the given tensor in question. Tables 2 and 3 summarize a series of generating matrices for every conventional crystal class.
These interrelations appear to be at first instance somewhat complicated, a simple example, however, will help to obtain a better understanding. Let us consider the form of the pyroelectric tensor in the crystal class 3 of the trigonal system assuming that the x_{3} axis of the coordinate system is the threefold rotation axis. As one may see on consulting Tables 2 and 3 the coordinate transformation related to the symmetry operation can be described with the following matrix
(4.12) 
Taking into consideration the condition of invariance of the polar tensor as expressed in eq. (4.10) one obtains for the tensor components the following equations
(4.13) 
p_{3}=p_{3}.
It follows from these equations that p_{1} = p_{2} = 0 and only p_{3} can differ from zero, consequently the form of the pyroelectric tensor of crystal class 3 of the trigonal system will be p = [0; 0; p_{3}].


Crystal system  Class symbol  Generating matrices  No. of symmetry elements  The choice of x_{1}, x_{2}, x_{3} crystal physical axes in relation to the symmetry axes  


International  Schoenflies  


Triclinic  1  C_{1}  M_{0}  1  
S_{2}=C_{i}  M_{1}  2  
Monoclinic  2  C_{2}  M_{2}  2  
m  C_{1h}=C_{s}  M_{3}  2  
2/m  C_{2h}  M_{2}, M_{3}  4  
Orthorhombic  222  V=D_{2}  M_{4}, M_{2}  4  
mm2  C_{2v}  M_{5}, M_{2}  4  
mmm  V_{h}=D_{2h}  M_{5}, M_{6}, M_{3}  8  
Tetragonal  4  C_{4}  M_{7}  4  
S_{4}  M_{8}  4  
422  D_{4}  M_{7}, M_{4}  8  
4/m  C_{4h}  M_{7}, M_{3}  8  
4mm  C_{4v}  M_{7}, M_{5}  8  
V_{d}=D_{2d}  M_{8}, M_{4}  8  
4/mmm  D_{4h}  M_{7}, M_{3}, M_{5}  16  
Trigonal  3  C_{3}  M_{9}  3  
S_{6}=C_{3i}  M_{10}  6  
32  D_{3}  M_{9}, M_{4}  6  
3m  C_{3v}  M_{9}, M_{5}  6  
D_{3d}  M_{10}, M_{5}  12  
Hexagonal  6  C_{6}  M_{11}  6  
C_{3h}  M_{12}  6  
D_{3h}  M_{12}, M_{5}  12  
622  D_{6}  M_{11}, M_{4}  12  
6/m  C_{6h}  M_{11}, M_{3}  12  
6/mm  C_{6v}  M_{11}, M_{5}  12  
6/mmm  D_{6h}  M_{11}, M_{5}, M_{3}  24  
Cubic  23  T  M_{13}, M_{2}  12  
m3  T_{h}  M_{14}, M_{2}  24  
432  O  M_{13}, M_{7}  24  
T_{d}  M_{13}, M_{8}  24  
m3m  O_{h}  M_{14}, M_{7}  48  



identity  


inversion  fourfold inversionrotation about x_{3} axis  
twofold rotation about x_{3} axis  threefold rotation about x_{3} axis  
reflection in x_{1}x_{2} plane  threefold inversionrotation about x_{3} axis  
twofold rotation about x_{1} axis  sixfold rotation about x_{3} axis  
reflection in x_{2}x_{3} plane  sixfold inversionrotation about x_{3} axis  
reflection in x_{1}x_{3} plane  threefold rotation about [111] direction  
fourfold rotation about x_{3} axis  threefold inversionrotation about [111] direction  

The method used in this example may be applied in every case, though with higher rank tensors it may be in many cases rather tiresome.
Considerable time can be saved (with the exception of the trigonal and hexagonal classes) by the direct inspection method worked out by Fumi,^{4}, which though in principle not differing from the previous treatment leads to results in a relatively short time. Fumi's method is based on the fact that in an orthogonal coordinate system the polar tensor components transform in the same way as the products of the corresponding coordinates (see eq. (4.6)). One must, however, be careful not to commute the sequence of the factors, thus for instance instead of the product x_{1}x_{2} one cannot write x_{2}x_{1}.
Fumi's method may be studied by a simple example considering the form of a polar secondrank tensor, e.g. the dielectric susceptibility tensor of the crystal class 2 of the monoclinic system. Assuming that the twofold symmetry axis coincides with the x_{2} coordinate axis the symmetry operator will transform the coordinates in the following way
(4.14) 
(4.15) 
(4.16) 
(4.17) 
(4.18) 
(4.19) 
The forms of tensors or matrices composed of tensor components for the various crystal classes can be found properly tabulated in the special literature, see for example [1, 2, 57, 16, 17].
Finally it should be observed that in some cases simple geometric considerations enable the determination of the independent components of the tensors representing the physical properties. It is easy to see that no pyroelectric effect can exist in a crystal possessing a centre of symmetry. This means that for these crystals every component of the pyroelectric tensor is zero, p = [0, 0, 0], because in these crystals if the vector of polarization were pointed in a given direction the vector should appear also in the opposite direction as a result of Neumann's principle, consequently its value can only be zero.
5. The value of a physical property in a given direction
One frequently finds data in the literature which give the value of some physical property in a given direction. In this chapter the concept of the magnitude of a physical property in a given direction, and also the relationship between this value and the respective tensor components will be illustrated on the examples of the direction dependence of the electrical conductivity and Young's modulus respectively.
The specific conductivity in the direction of the electric field is defined as the ratio of the component parallel with the electric field of the current density () and the magnitude of the electric field (E), i.e. . Let the components of the electric field be E_{i} = En_{i}, where n_{i} denotes the ith component of the unit vector () pointing into the direction of the electric field. The component of the j current density parallel with is in tensor notation
(5.1) 
(5.2) 
(5.3) 
Equation (5.3) may be applied in two ways. One possibility is to calculate the tensor components from the measured conductivity values and the corresponding direction cosines. For this purpose one should measure the electrical conductivity in different directions, which are not connected by symmetry, as many times as the number of the independent components. Another possibility of applying eq. (5.3) is quite opposite to the first one. With the aid of the already known tensor components the conductivity value can be computed for any direction.
Equation (5.3) becomes considerably simplified for crystals of the tetragonal, trigonal and hexagonal systems which have only two independent tensorcomponents ( and )
(5.4) 
(5.5) 
(5.6) 
As another example we will study the direction dependence of Young's modulus. To begin with it should be stated that Young's modulus in the pulling direction is defined as the ratio of the longitudinal stress () and the longitudinal strain (. If the axis of the coordinate system is placed in the direction of the unit vector Young's modulus in this direction will apparently be
(5.7) 
(5.8) 
(5.9) 
(5.10) 
(5.11) 
(5.12) 
6. Higher order effects
The relationship between two physical properties is not necessarily linear. The relation between the dependent and independent physical variables can be often expressed with Taylor expansion. Thus for instance the electric field dependence of the electric polarization in a strong field is described with expansion in power series
(6.1) 
There is some freedom in deciding the order of an effect, which depends upon the aspect the effect is studied. Thus in the above example if instead of the tensor [] the differential quotient with respect to the electric field of the electric polarization vector (i.e. the [] tensor) is considered as dielectric susceptibility, the previously secondorder effect may be regarded as a firstorder effect, which describes the electric field dependence of the dielectric susceptibility. This becomes obvious from the equation
(6.2) 
(6.3) 
Finally it should be noted that in the theory of elasticity the coefficients of the secondorder effect are called thirdorder elastic stiffnesses, because it is more suitable to start the discussion of the nonlinear stressdeformation relationship with the energy function whose thirdorder derivatives supply the coefficients of the principally secondorder effects.
7. Description of the physical properties in matrix notation
It has been shown in the previous sections that the various physical properties of the crystals can be described quite suitably with tensors. However, especially with higher rank tensors the great number of suffixes may become inconvenient. In many cases the introduction of a new notation, the matrix notation, is suitable to reduce the number of suffixes. The introduction of the matrix notation may be encouraged by the usually considerable reduction, due to intrinsic symmetries, of the number of independent components of the tensors representing the physical properties.
Let us consider first a simple example. For some purpose the following () matrices are formed from the components of the symmetrical second rank [] tensor
(7.1) 
(7.2) 
(7.3) 
This way by introducing the matrix notation the number of the suffixes is reduced on the condition that the new suffixes beside 1, 2 and 3 can now take also the values 4, 5 and 6. Although the relationship between the suffixes of the tensor components and matrix elements is unambiguous, the relationship between the tensor components and the matrix elements is defined to the extent of a multiplication factor. A few examples explaining the role of this factor are given below.
The advantage of the matrix notation becomes obvious with the third and higher rank tensors. A third rank tensor, occurring rather frequently in crystal physics, is the piezoelectric tensor, which describes the relationship between the stresses effective on the crystal and the resulting electric polarization according to the equation
(7.4) 
(7.5) 
(7.6) 
(7.7) 
It is important to notice that some authors do not introduce the multiplication factor two in the interpretation of the n = 4, 5 and 6 elements of the piezoelectric matrix. This, however, excludes the compact description of the relationship between the vector of the electric polarization and the mechanical stress presented in eq. (7.7).
Similarly other matrices representing various physical properties may be defined by different authors in different ways. Therefore when trying to use the numerical values as published in the literature the various definitions used by the authors should be taken into account.
Finally in order to demonstrate the advantage of the matrix notation let us investigate the elasticity of crystals. Hooke's law takes in tensor notation the following form
(7.8) 
Now, since the components of the [c_{ijkl}] tensor are symmetrical with regard to the first two and last two suffixes, respectively, these may be substituted each with one new suffix according to (7.3).
Further on the elements of the (c_{mn}) matrix can be defined by the equation
(7.9) 
(7.10) 
(7.11) 
Expressing the strains in terms of the stresses one obtains the equation
(7.12) 
(7.13) 
Simple calculation shows that between the elements of the above (s_{mn}) matrix and the components of the [s_{ijkl}] tensor expressing the relationship between the components of the [] strain and the [] stress respectively the following correspondence holds
(7.14) 
Finally it should be noted that with the matrix notation, if applied, the number of the suffixes does not refer to the transformation formula of the matrix elements. Thus for example in case of a coordinate transformation the elements of the piezoelectric matrix (d_{ij}) transform differently from the matrix elements (c_{ij}) of the elasticity.
8. Curie's principle
The crystal symmetry depends upon the state of the crystal. If, due to some external influence, there is a change in the state of the crystal, there may also be a change in the crystal symmetry. The symmetry of a given state of a crystal may be determined using the Curie principle from the symmetry of the crystal free of any external influence and from the symmetry of the external influence.
According to Curie when various natural phenomena are piled upon each other forming a system, the dissymmetries are added up leaving only those elements which separately, in each phenomenon regarded in itself, were present. By dissymmetry Curie meant the sum of the absent symmetry elements.^{8} Curie's principle in itself may be formulated in the physics of crystals as follows: the symmetry group of a crystal under an external influence () is given by the greatest common subgroup of the symmetry group of the crystal without the influence (K) and of the symmetry group of the external influence (G) considering also the mutual position of the symmetry elements of these groups:^{9}
(8.1) 
As an example let us investigate the change of symmetry in the ADP (ammonium dihydrogen phosphate) crystals in an electric field of various directions. The ground state symmetry of these crystals is , i.e. it has a fourfold inversion axis (which contains in itself also a twofold rotation axis). The fourfold inversion axis lies in the line of intersection of two mutually perpendicular planes of symmetry. Two diad axes are perpendicular to the fourfold inversion axis and at 45 to the planes of symmetry. This is demonstrated in axionometric and stereographic representation respectively in Figs. 2 and 3. The symmetry of the electric field is , i.e. it corresponds to the symmetry of a cone, which has an infinitefold rotation axis containing every possible rotation axes of lower symmetry including also the twofold axes, further on the infinitefold rotation axis lies in the line of intersection of an infinite number of mirror planes.
First let us investigate the case when the electric field points in the [001] direction, which means that the vector of the electric field is parallel with the fourfold inversion axis of the crystal. The symmetry elements of the electric field do not include the inversion axis, therefore according to the Curie principle the resulting symmetry elements do not contain this axis. However, it should be observed that the fourfold inversion axis contains also a twofold rotation axis which is a symmetry element of the electric field, consequently the symmetry elements of the crystal in an electric field of the direction [001] will also contain this twofold axis. Of the basic symmetry elements the two mirror planes are also symmetry elements of the electric field, thus they are conserved in the crystal too. The twofold rotation axes perpendicular to the line of intersection of the mirror planes do not belong to the symmetry elements of the electric field, consequently they will disappear. Summing up the common symmetry elements of the electric field and the crystal in this field we have two mirror planes perpendicular to each other and a twofold rotation axis in the line of intersection of the mirror planes. Thus the symmetry of the ADP crystal in the electric field of the [001] direction is reduced to the symmetry of the orthorhombic mm2 point group (Figs. 2 and 3a). If the electric field acts in the [] direction, i.e. along a twofold rotation axis perpendicular to the fourfold inversion axis no mirror plane of the crystal coincides with the mirror planes of the electric field which results in the disappearance of the mirror planes. Further on also the fourfold inversion axis (together with the twofold rotation axis connected with the inversion axis), and also from the two other twofold axes one axis will disappear, since the electric field has only in one direction a rotation symmetry. This way only the twofold rotation axis (along which the electric field is effective) remains conserved with the result that the symmetry of the crystal is reduced to the monoclinic point group 2 (as depicted in Fig. 3b). If the electric field influences the crystal in the [] direction, i.e. in one mirror plane, the common symmetry element will be one mirror plane: the crystal symmetry is reduced to the point group m of the monoclinic system (Fig. 3c). Finally if the electric field points in an arbitrary [hkl] direction, different from the directions already discussed, no symmetry element of the crystal and the electric field coincides. Consequently no symmetry element is preserved. In this case the symmetry of the crystal is reduced to the trivial point group 1 of the triclinic system (Fig. 3d).
It follows from the foregoing that the originally optically uniaxial ADP crystal will under the influence of an electric field behave like an optically biaxial crystal.
It should be noted that the Curie principle constitutes only a special case of the great principle of the superposition of the symmetry groups. A detailed discussion of this subject, however, would go beyond the scope of this paper, one can refer to the book of Shubnikov and Koptsik.^{10}
Acknowledgments
The author would like to thank Prof. M. P. Shaskolskaya, Dr. N. V. Perelomova and Dr. M. M. Tagieva for useful advice and valuable discussions. Further he thanks his colleagues for reading the manuscript and helpful comments.
References
1. Birss, R. R., Symmetry and Magnetism, Amsterdam, NorthHolland Publishing Company (1964).
2. Bhagavantam, S., Crystal Symmetry and Physical Properties, London, New York, Academic Press (1966).
3. Sirotin, Yu. I. and Shaskolskaya, M. P., Fundamentals of Crystal Physics, Moscow, Mir Publishers (1982).
4. Fumi, F. G., Acta Cryst. 5 (1952) 44.
5. Nye, J. F., Physical Properties of Crystals, Oxford, Clarendon Press (1957).
6. Smith, A. C., Janak, J. F., Adler, R. B., Electronic Conduction in Solids, New York, McGrawHill Book Company (1967).
7. Hellwege, K. H. (ed.), LandoltBörnstein, New Series, Group III., Vol. 11, Berlin, Heidelberg, New York, SpringerVerlag (1979).
8. Curie, P., Oeuvres, pp. 118, Paris, Société Français de Physique (1908).
9. Koptsik, V. A., Sov. Phys. Cryst. 2 (1957) 99.
10. Shubnikov, A. V. and Koptsik, V. A., Symmetry in Science and Art, New York and London, Plenum Press (1977).
11. Shubnikov, A. V. and Belov, N. V., Colored Symmetry, Oxford, Pergamon Press (1964).
12. Juretschke, Hellmut, J., Crystal Physics, Massachusetts, W. A. Benjamin, Inc. Reading (1974).
13. Koptsik, V. A., Shubnikov's groups (In Russian), Moscow, Moscow University Press (1966).
14. Opechowski, W. and Guccione, R., Magnetic symmetry, G. T. Rado and H. Suhl (eds.), Magnetism, Vol. II/A, p. 105, New York, London, Academic Press (1965).
15. Shuvalov, L. A. (ed.), Modern Crystallography IV. Physical Properties of Crystals. Springer Series in SolidState Sciences, Vol. 37. Springer Verlag, in preparation.
16. Wooster, W. A., Tensors and Group Theory for the Physical Properties of Crystals, Oxford, Clarendon Press (1973).
17. Perelomova, N. V. and Tagieva, N. M., Problems in Crystal Physics with solutions, Moscow, Mir Publishers (1983).
Many further references are found in each of the cited books.