International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 3.2, pp. 737741

In the previous section (Section 3.2.1), the crystallographic and noncrystallographic point groups are treated under geometrical aspects only. In the present section the pointgroup symmetries of the physical properties are considered. Among the physical properties, those represented by tensors are accessible to a mathematical treatment of their symmetries, which are apparent by the invariance of tensor components under symmetry operations. For more details the reader is referred to Nye (1957, 1985), Paufler (1986), Schwarzenbach & Chapuis (2006) and Shuvalov (1988). A similar comprehensive treatment of nontensorial properties, such as cleavage, plasticity, hardness or crystal growth does not exist, but these properties are, of course, also governed by the crystallographic pointgroup symmetry.
Neumann's principle (Neumann, 1885) describes the relation between the symmetry of a physical property and the crystallographic point group of a crystal. It states that the symmetry of any physical property of a crystal is higher than, or at least equal to, its crystallographic pointgroup symmetry, or in the language of groups, the symmetry of any physical property of a crystal is a proper or improper supergroup of its crystallographic symmetry : . This is easily illustrated for polar secondrank tensors, which are represented by ellipsoids and hyperboloids. The representation surfaces of triclinic, monoclinic and orthorhombic crystals are general ellipsoids or hyperboloids of symmetry , which is a proper supergroup of all triclinic, monoclinic and orthorhombic point groups with the only exception of the orthorhombic holohedry , for which the two symmetries are the same. For the uniaxial crystals of the trigonal, tetragonal and hexagonal systems the representation ellipsoids and hyperboloids have rotation symmetries with the cylindrical point group = (see Table 3.2.1.6), which is a proper supergroup of all uniaxial crystallographic point groups. For cubic crystals, secondrank tensors are isotropic and represented by a sphere with pointgroup symmetry (), a proper supergroup of all cubic crystallographic point groups. For more details the reader is referred to Paufler (1986) and Authier (2014). A short resume is given by Klapper & Hahn (2005).
As a consequence of the invariance of tensor components under a symmetry operation (or alternatively: under a transformation of the coordinate system to a symmetryequivalent one), some of the tensor components are equal or even zero. The number of independent components decreases when the symmetry of the crystal increases. Thus, an increase of the pointgroup symmetry from 1 (triclinic) to (cubic) or to the sphere group (i.e. isotropy) reduces the number of tensor components for symmetrical secondrank tensors from 6 to 1. Even more drastic is this reduction for all tensors of odd rank (such as pyroelectricity and piezoelectricity) or axial tensors of second rank (e.g. optical activity): all components are zero if an inversion centre is present, i.e. properties described by these tensors do not exist in centrosymmetric crystals (see the textbooks of tensor physics mentioned above). These properties, which exist only in noncentrosymmetric crystals are, as a rule, the most important ones, not only for physical applications but also for structure determination, because they allow a proof of the absence of a symmetry centre.
For the description of noncentrosymmetric crystals and their specific properties, certain notions are of importance and these are explained in the following two sections.
Curie's principle (Curie, 1894) describes the crystallographic symmetry of a macroscopic crystal which is subject to an external influence F, for example to an electric field E, to uniaxial stress σ_{ii}, to a temperature change ΔT etc. For this treatment, the pointgroup symmetries of the external influences (Curie groups) are defined as follows (see Authier, 2014, p. 11; Paufler, 1986, p. 29):
According to Curie's principle, the pointgroup symmetry of the crystal under the external field F is the intersection symmetry of the two point groups: of the crystal without field and of the field without crystal: ; i.e. is a (proper or improper) subgroup of both groups and .
As examples we consider the effect of an electric field ( ) and of a uniaxial stress () along one of the (four) threefold rotoinversion axes of cubic crystals with point groups and .
Electric field parallel to [111]:
Uniaxial stress parallel to [111]:
Note that, in contrast to uniaxial stress, the electric field destroys centrosymmetry, leading to pyroelectricity and even optical gyration. The polarity and the gyration sense are reversed upon reversal of the electric field. If the electric field and the uniaxial stress were applied to an arbitrary (nonsymmetry) direction of the above cubic crystals, point groups 1 and would result in the two cases.
If a scalar influence () (centrosymmetric sphere group) is applied to a crystal, its symmetry is not changed, provided that no phase transition is induced.
For further reading on tensor properties of crystals the textbooks in the references and International Tables for Crystallography, Volume D (2014) are recommended.
All these terms refer to the same symmetry restriction, the absence of improper rotations (rotoinversions, rotoreflections) in a crystal or in a molecule. This implies in particular the absence of a centre of symmetry, , and of a mirror plane, , but also of a axis. As a consequence, such chiral crystals or molecules can occur in two different forms, which are related as a right and a left hand; hence, they are called righthanded and lefthanded forms. These two forms of a molecule or a crystal are mirrorrelated and not superimposable (not congruent). Thus, the only symmetry operations that are allowed for chiral objects are proper rotations. Such objects are also called dissymmetric, in contrast to asymmetric objects, which have no symmetry.
The terms enantiomerism and chirality are mainly used in chemistry and applied to molecules, whereas the term enantiomorphism is preferred in crystallography if reference is made to crystals.
Enantiomorphic crystals, as well as solutions or melts containing chiral molecules of one handedness, exhibit optical activity (cf. Section 3.2.2.4.2). Crystals and molecules of the other handedness show optical activity with the opposite sense of rotation. For this reason, two molecules of opposite chirality are also called optical isomers.
Chiral molecules form enantiomorphic crystals of the corresponding handedness. These crystals belong, therefore, to one of the 11 crystal classes allowing enantiomorphism (Table 3.2.2.1). Racemic mixtures (containing equal amounts of molecules of each chirality), however, may crystallize in nonenantiomorphic or even centrosymmetric crystal classes. Racemization (i.e. the switching of molecules from one chirality to the other) of an optically active melt or solution can occur in some cases during crystallization, leading to nonenantiomorphic crystals.

Enantiomorphic crystals can also be built up from achiral molecules or atom groups. In these cases, the achiral molecules or atom groups form chiral configurations in the structure. The best known examples are quartz and NaClO_{3}. For details, reference should be made to Rogers (1975).
A direction is called polar if its two directional senses are geometrically or physically different. A polar symmetry direction of a crystal is called a polar axis. Only proper rotation or screw axes can be polar. The polar and nonpolar directions in the 21 noncentrosymmetric point groups are listed in Table 3.2.2.2.

The terms polar point group or polar crystal class are used in two different meanings. In crystal physics, a crystal class is considered as polar if it allows the existence of a permanent dipole moment, i.e. if it is capable of pyroelectricity (cf. Section 3.2.2.5). In crystallography, however, the term polar crystal class is frequently used synonymously with noncentrosymmetric crystal class. The synonymous use of polar and acentric, however, can be misleading, as is shown by the following example. Consider an optically active liquid. Its symmetry can be represented as a righthanded or a lefthanded sphere (cf. Sections 3.2.1.4 and 3.2.2.4). The optical activity is isotropic, i.e. magnitude and rotation sense are the same in any direction and its counterdirection. Thus, no polar direction exists, although the liquid is noncentrosymmetric with respect to optical activity.
According to Neumann's principle, information about the point group of a crystal may be obtained by the observation of physical effects. Here, the term `physical properties' includes crystal morphology and etch figures. The use of any of the techniques described below does not necessarily result in the complete determination of symmetry but, when used in conjunction with other methods, much information may be obtained. It is important to realize that the evidence from these methods is often negative, i.e. that symmetry conclusions drawn from such evidence must be considered as only provisional.
In the following sections, the physical properties suitable for the determination of symmetry are outlined briefly. For more details, reference should be made to the monographs by Bhagavantam (1966), Nye (1957) and Wooster (1973).
If a crystal shows well developed faces, information on its symmetry may be derived from the external form of the crystal. By means of the optical goniometer, faces related by symmetry can be determined even for crystals far below 1 mm in diameter. By this procedure, however, only the eigensymmetry (cf. Section 3.2.1.2.2) of the crystal morphology (which may consist of a single form or a combination of forms) can be established. The determination of the point group is unique in all cases where the observed eigensymmetry group is compatible with only one generating group.
Column 6 in Table 3.2.1.3 lists all point groups for which a given crystal form (characterized by its name and eigensymmetry) can occur. In 19 cases, the point group can be uniquely determined because only one entry appears in column 6. These crystal forms are always characteristic general forms, for which eigensymmetry and generating pointgroup symmetry are identical. They belong to the 19 point groups with more than one symmetry direction.
If a crystal exhibits a combination of forms which by themselves do not permit unambiguous determination of the point group, those generating point groups are possible that are common to all crystal forms of the combination. The mutual orientation of the forms, if variable, has to be taken into account, too.
Example
Two tetragonal pyramids, each of eigensymmetry 4mm, rotated with respect to each other by an angle , determine the point group 4 uniquely because the eigensymmetry of the combination is only 4.
In practice, however, unequal or incomplete development of the faces of a form often simulates a symmetry that is lower than the actual crystal symmetry. In such cases, or when the morphological analysis is ambiguous, the crystallization of a small amount of the compound on a seed crystal, ground to a sphere, is useful. By this procedure, faces of additional forms (and often of the characteristic general form) appear as small facets on the sphere and their interfacial angles can be measured.
In favourable cases, even the space group can be derived from the morphology of a crystal: this is based on the socalled Bravais–Donnay–Harker principle. A textbook description is given by Phillips (1971, ch. 13).
Furthermore, measurements of the interfacial angles by means of the optical goniometer permit the determination of the relative dimensions of a `morphological unit cell' with good accuracy. Thus, for instance, the interaxial angles α, β, γ and the axial ratio a:b:c of a triclinic crystal may be derived. The ratio a:b:c, however, may contain an uncertainty by an integral factor with respect to the actual cell edges of the crystal. This means that any one unit length may have to be multiplied by an integer in order to obtain correspondence to the `structural' unit cell.
Additional information on the point group of a crystal can be gained from the face symmetry, which is usually determined by observation of etch figures, striations and other face markings. Etch pits are produced by heating the crystal in vacuum (evaporation from the surface) or by attacking it with an appropriate reagent, which should not be optically active. The etch pits generally appear at the end points of dislocation lines on the face. They exhibit the symmetry of one of the ten twodimensional point groups which, in general,^{22} corresponds to the symmetry of the crystal face under investigation.
The observation of etch figures is important when the morphological analysis is ambiguous (cf. Section 3.2.2.2). For instance, a tetragonal pyramid, which is compatible with point groups 4 and 4mm, can be uniquely attributed to point group 4 if its face symmetry is found to be 1. For face symmetry m, group 4mm would result. The (oriented) face symmetries of the 47 crystal forms in the various point groups are listed in column 6 of Table 3.2.1.3 and in column 3 of Table 3.2.3.2.
In noncentrosymmetric crystals, the etch pits on parallel but opposite faces, even though they have the same symmetry, may be of different size or shape, thus proving the absence of a symmetry centre. Note that the orientation of etch pits with respect to the edges of the face is significant (cf. Buerger, 1956), as well as the mutual arrangement of etch pits on opposite faces. Thus, for a pinacoid with face symmetry 1, the possible point groups , 2 and m of the crystal can be distinguished by the mutual orientation of etch pits on the two faces. Moreover, twinning by merohedry and the true symmetry of the two (or more) twin partners (`twin domains') may be detected.
The method of etching can be applied not only to growth faces but also to cleavage faces or arbitrarily cut faces.
Optical studies provide good facilities with which to determine the symmetry of transparent crystals. The following optical properties may be used.
The dependence of the refractive index on the vibration direction of a planepolarized light wave travelling through the crystal can be obtained from the optical indicatrix. This surface is an ellipsoid, which can degenerate into a rotation ellipsoid or even into a sphere. Thus, the lowest symmetry of the property `refraction' is 2/m 2/m 2/m, the point group of the general ellipsoid. According to the three different forms of the indicatrix, three categories of crystal systems have to be distinguished (Table 3.2.2.3).

The orientation of the indicatrix is related to the symmetry directions of the crystal. In tetragonal, trigonal and hexagonal crystals, the rotation axis of the indicatrix (which is the unique optic axis) is parallel to the main symmetry axis. For orthorhombic crystals, the three principal axes of the indicatrix are oriented parallel to the three symmetry directions of the crystal. In the monoclinic system, one of the axes of the indicatrix coincides with the monoclinic symmetry direction, whereas in the triclinic case, the indicatrix can, in principle, have any orientation relative to a chosen reference system. Thus, in triclinic and, with restrictions, in monoclinic crystals, the orientation of the indicatrix can change with wavelength λ and temperature T (orientation dispersion). In any system, the size of the indicatrix and, in all but the cubic system, its shape can also vary with λ and T.
When studying the symmetry of a crystal by optical means, note that strain can lower the apparent symmetry owing to the high sensitivity of optical properties to strain.
The symmetry information obtained from optical activity is quite different from that given by optical refraction. Optical activity is in principle confined to the 21 noncentrosymmetric classes but it can occur in only 15 of them (Table 3.2.2.1). In the 11 enantiomorphism classes, a single crystal is either right or lefthanded. In the four nonenantiomorphous classes and , optical activity may also occur; here directions of both right and lefthanded rotations of the plane of polarization exist in the same crystal. In the other six noncentrosymmetric classes, , , optical activity is not possible.
In the two cubic enantiomorphous classes 23 and 432, the optical activity is isotropic and can be observed along any direction.^{23} For the other optically active crystals, the rotation of the plane of polarization can, in practice, be detected only in directions parallel (or approximately parallel) to the optic axes. This is because of the dominating effect of double refraction. No optical activity, however, is present along an inversion axis or along a direction parallel or perpendicular to a mirror plane. Thus, no activity occurs along the optic axis in crystal classes and . In classes m and mm2, no activity can be present along the two optic axes if these axes lie in m. If they are not parallel to m, they show optical rotation(s) of opposite sense.
Light waves passing through a noncentrosymmetric crystal induce new waves of twice the incident frequency. This secondharmonic generation is due to the nonlinear optical susceptibility. The secondharmonic coefficients form a thirdrank tensor, which is subject to the same symmetry constraints as the piezoelectric tensor (see Section 3.2.2.6). Thus, only 20 noncentrosymmetric crystals, except those of class 432, can show the secondharmonic effect; cf. Table 3.2.2.1.
Secondharmonic generation is a powerful method of testing crystalline materials for the absence of a symmetry centre. With an appropriate experimental device, very small amounts (less than 10 mg) of powder are sufficient to detect the secondharmonic signals, even for crystals with small deviations from centrosymmetry (Dougherty & Kurtz, 1976).
In principle, pyroelectricity can only exist in crystals with a permanent electric dipole moment. This moment is changed by heating and cooling, thus giving rise to electric charges on certain crystal faces, which can be detected by simple experimental procedures.
An electric dipole moment can be present only along a polar direction that has no symmetryequivalent directions.^{24} Such polar directions occur in the following ten classes: 6mm, 4mm, and their subgroups 6, 4, 3m, 3, mm2, 2, m, 1 (cf. Table 3.2.2.1). In point groups with a rotation axis, the electric moment is along this axis. In class m, the electric moment is parallel to any direction in the mirror plane (direction [u0w]). In class 1, any direction [uvw] is possible. In point groups 1 and m, besides a change in magnitude, a directional variation of the electric moment can also occur during heating or cooling.
In practice, it is difficult to prevent strains from developing throughout the crystal as a result of temperature gradients in the sample. This gives rise to piezoelectrically induced charges superposed on the true pyroelectric effect. Consequently, when the development of electric charges by a change in temperature is observed, the only safe deduction is that the specimen must lack a centre of symmetry. Failure to detect pyroelectricity may be due to extreme weakness of the effect, although modern methods are very sensitive.
A crystal is ferroelectric if the direction of the permanent electric dipole moment can be changed by an electric field. Thus, ferroelectricity can only occur in the ten pyroelectric crystal classes, mentioned above.
In piezoelectric crystals, an electric dipole moment can be induced by compressional and torsional stress. For a uniaxial compression, the induced moment may be parallel, normal or inclined to the compression axis. These cases are called longitudinal, transverse or mixed compressional piezoeffect, respectively. Correspondingly, for torsional stress, the electric moment may be parallel, normal or inclined to the torsion axis.
The piezoelectricity is described by a thirdrank tensor, the moduli of which vanish for all centrosymmetric point groups. Additionally, in class 432, all piezoelectric moduli are zero owing to the high symmetry. Thus, piezoelectricity can only occur in 20 noncentrosymmetric crystal classes (Table 3.2.2.1).
The piezoelectric point groups 422 and 622 show the following peculiarity: there is no direction for which a longitudinal component of the electric moment is induced under uniaxial compression. Thus, no longitudinal or mixed compressional effects occur. The moment is always normal to the compression axis (pure transverse compressional effect). This means that, with the compression pistons as electrodes, no electric charges can be found, since only transverse compressional or torsional piezoeffects occur. In all other piezoelectric classes, there exist directions in which both longitudinal and transverse components of the electric dipole moment are induced under uniaxial compression.
An electric moment can also develop under hydrostatic pressure. This kind of piezoelectricity, like pyroelectricity, can be represented by a firstrank tensor (vector), whereby the hydrostatic pressure is regarded as a scalar. Thus, piezoelectricity under hydrostatic pressure is subject to the same symmetry constraints as pyroelectricity.
Like `secondharmonic generation' (Section 3.2.2.4.3), the piezoelectric effect is very useful for testing crystals for the absence of a symmetry centre. There exist powerful methods for testing powder samples or even small single crystals. In the old technique of Giebe & Scheibe (cf. Wooster & Brenton, 1970), the absorption and emission of radiofrequency energy by electromechanical oscillations of piezoelectric particles are detected. In the more modern method of observing `polarization echoes', radiofrequency pulses are applied to powder samples. By this procedure, electromechanical vibration pulses are induced in piezoelectric particles, the echoes of which can be detected (cf. Melcher & Shiren, 1976).
References
International Tables for Crystallography (2014). Vol. D, Physical Properties of Crystals, 2nd ed., edited by A. Authier. Chichester: Wiley.Authier, A. (2014). Introduction to the properties of tensors. In International Tables for Crystallography, Vol. D, 2nd ed., edited by A. Authier, ch. 1.1, pp. 3–33. Chichester: Wiley.
Bhagavantam, S. (1966). Crystal Symmetry and Physical Properties. London: Academic Press.
Buerger, M. J. (1956). Elementary Crystallography, especially chs. 10 and 11. New York: Wiley.
Curie, P. (1894). Sur la symétrie dans les phénomènes physiques, symétrie d'un champ électrique et d'un champ magnétique. J. Phys. (Paris), 3, 393–415.
Dougherty, J. P. & Kurtz, S. K. (1976). A second harmonic analyzer for the detection of noncentrosymmetry. J. Appl. Cryst. 9, 145–158.
Klapper, H. & Hahn, Th. (2005). Point groups. In Encyclopedia of Condensed Matter Physics, Vol. 4, pp. 323–334. Amsterdam: Elsevier Academic Press.
Melcher, R. L. & Shiren, N. S. (1976). Polarization echoes and longtime storage in piezoelectric powders. Phys. Rev. Lett. 36, 888–891.
Neumann, F. (1885). Vorlesungen über die Theorie der Elastizität der festen Körper und des Lichtäthers, edited by O. E. Meyer. Leipzig: B. G. Teubner Verlag.
Nye, J. F. (1957). Physical Properties of Crystals. Oxford: Clarendon Press. [Revised edition (1985).]
Paufler, P. (1986). Physikalische Kristallographie. Berlin: Akademie Verlag, and Weinheim: Verlag Chemie (VCH).
Phillips, F. C. (1971). An Introduction to Crystallography, 4th ed., chs. 3, 4, 6 and 13. London: Longman.
Rogers, D. (1975). Some fundamental problems of relating tensorial properties to the chirality or polarity of crystals. In Anomalous Scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 231–250. Copenhagen: Munksgaard.
Schwarzenbach, D. & Chapuis, G. (2006). Cristallographie, 2nd ed. Lausanne: Presses Polytechniques et Universitaires Romandes. (In French.) [1st ed. in English (1996), Chichester: Wiley; in German (2001), Berlin: Springer].
Shuvalov, L. A. (1988). Editor. Modern Crystallography IV: Physical Properties of Crystals. Berlin: Springer.
Wooster, W. A. (1973). Tensors and Group Theory for the Physical Properties of Crystals. Oxford: Clarendon Press.
Wooster, W. A. & Brenton, A. (1970). Experimental Crystal Physics, especially ch. 6. Oxford: Clarendon Press.