Section 10.2.1. General restrictions on physical properties imposed by symmetry

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 right-handed and left-handed forms. These two forms of a molecule or a crystal are mirror-related 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 10.2.4.2). 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. The crystals belong, therefore, to one of the 11 crystal classes allowing enantiomorphism (Table 10.2.1.1). Racemic mixtures (containing equal amounts of molecules of each chirality), however, may crystallize in non-enantiomorphic 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 non-enantiomorphic 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 NaClO₃. 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 10.2.1.2.

Table 10.2.1.2 | top | pdf | Polar axes and nonpolar directions in the 21 noncentrosymmetric crystal classes All directions normal to an evenfold rotation axis and along rotoinversion axes are nonpolar. All directions other than those in the column `Nonpolar directions' are polar. A symbol like [u0w] refers to the set of directions obtained for all possible values of u and w, in this case to all directions normal to the b axis, i.e. parallel to the plane (010). Symmetrically equivalent sets of nonpolar directions are placed between semicolons; the sequence of these sets follows the sequence of the symmetry directions in Table 2.2.4.1 . System Class Polar (symmetry) axes Nonpolar directions Triclinic 1 None None Monoclinic 2 [010] [u0w] Unique axis b m None [010] Monoclinic 2 [001] [uv0] Unique axis c m None [001] Orthorhombic 222 None [0vw]; [u0w]; [uv0] mm 2 [001] [uv0] Tetragonal 4 [001] [uv0] None [001]; [uv0] 422 None [uv0]; [0vw] [u0w];     [uuw] [] 4mm [001] [uv0] None [uv0]; [0vw] [u0w] None [uv0]; [uuw] [] Trigonal (Hexagonal axes) 3 [001] None 321 [100], [010], [u2uw] 312 , [120], [uuw] 3m1 [001] [100] [010] 31m [001] [120] Trigonal (Rhombohedral axes) 3 [111] None 32 , , [uuw] [uvv] [uvu] 3m [111] Hexagonal 6 [001] [uv0] None [001] 622 None [u2uw] []     [uuw] [] 6mm [001] [uv0] , [120], [uuw] [] [100], [010], [u2uw] Cubic 432

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 10.2.5). 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 right-handed or a left-handed sphere (cf. Sections 10.1.4 and 10.2.4). 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).