International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, p. 739

Section 3.2.2.1.4. Polar directions, polar axes, polar point groups

H. Klappera and Th. Hahna

3.2.2.1.4. Polar directions, polar axes, polar point groups

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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[link].

Table 3.2.2.2| 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). Symmetry-equivalent sets of nonpolar directions are placed between semicolons; the sequence of these sets follows the sequence of the symmetry directions in Table 2.1.3.1[link] .

SystemClassPolar (symmetry) axesNonpolar 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]
mm2 [001] [uv0]
Tetragonal 4 [001] [uv0]
[\bar{4}] None [001]; [uv0]
422 None [uv0]; [0vw] [u0w];
    [uuw] [[u\bar{u}w]]
4mm [001] [uv0]
[\bar{4}2m] None [uv0]; [0vw] [u0w]
[\bar{4}m2] None [uv0]; [uuw] [[u\bar{u}w]]
Trigonal
(Hexagonal axes)
3 [001] None
321 [100], [010], [\hbox{[}\bar{1}\bar{1}0\hbox{]}] [u2uw] [\hbox{[}\overline{2u}\bar{u}w\hbox{]}\ \hbox{[}u\bar{u}w\hbox{]}]
312 [\hbox{[}1\bar{1}0\hbox{]}], [120], [\hbox{[}\bar{2}\bar{1}0\hbox{]}] [uuw] [\hbox{[}\bar{u}0w\hbox{]}\ \hbox{[}0\bar{v}w\hbox{]}]
3m1 [001] [100] [010] [\hbox{[}\bar{1}\bar{1}0\hbox{]}]
31m [001] [\hbox{[}1\bar{1}0\hbox{]}] [120] [\hbox{[}\bar{2}\bar{1}0\hbox{]}]
Trigonal
(Rhombohedral axes)
3 [111] None
32 [\hbox{[}1\bar{1}0\hbox{]}], [\hbox{[}01\bar{1}\hbox{]}], [\hbox{[}\bar{1}01\hbox{]}] [uuw] [uvv] [uvu]
3m [111] [\hbox{[}1\bar{1}0\hbox{]}\ \hbox{[}01\bar{1}\hbox{]}\ \hbox{[}\bar{1}01\hbox{]}]
Hexagonal 6 [001] [uv0]
[\bar{6}] None [001]
622 None [u2uw] [[\overline{2u}\bar{u}w\hbox {] [}u\bar{u}w]]
    [uuw] [[\bar{u}0w\hbox{] [}0\bar{v}w]]
6mm [001] [uv0]
[\bar{6}m2] [\hbox{[}1\bar{1}0\hbox{]}], [120], [\hbox{[}\bar{2}\bar{1}0\hbox{]}] [uuw] [[\bar{u}0w\hbox{] [}0\bar{v}w]]
[\bar{6}2m] [100], [010], [\hbox{[}\bar{1}\bar{1}0\hbox{]}] [u2uw] [\hbox{[}\overline{2u}\bar{u}w\hbox{]}\ \hbox{[}u\bar{u}w\hbox{]}]
Cubic [\!\left.\matrix{23\hfill\cr \bar{4}3m\hfill\cr}\right\}] [\!\matrix{\hbox{Four threefold}\hfill\cr\quad \hbox{axes along }\langle 111\rangle\hfill\cr}] [\!\matrix{\hbox{[}0vw\hbox{]}\ \hbox{[}u0w\hbox{]}\ \hbox{[}uv0\hbox{]}\cr \hbox{[}0vw\hbox{]}\ \hbox{[}u0w\hbox{]}\ \hbox{[}uv0\hbox{]}\cr}]
432 [{\hbox{None}}{\hbox to 3.9pc{}}\left\{\matrix{\cr\cr\cr}\right.] [\!\openup0.5pt\matrix{\hbox{[}0vw\hbox{]}\ \hbox{[}u0w\hbox{]}\ \hbox{[}uv0\hbox{]}\semi \hfill\cr \hbox{[}uuw\hbox{]}\ \hbox{[}uvv\hbox{]}\ \hbox{[}uvu\hbox{]}\semi \hfill\cr \hbox{[}u\bar{u}w\hbox{]}\ \hbox{[}uv\bar{v}\hbox{]}\ \hbox{[}\bar{u}vu\hbox{]}\hfill\cr}]
In class 1 any direction is polar; in class m all directions except [010] (or [001]) are polar.

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[link]). 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 3.2.1.4[link] and 3.2.2.4[link]). 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[link]), Nye (1957[link]) and Wooster (1973[link]).

References

Bhagavantam, S. (1966). Crystal Symmetry and Physical Properties. London: Academic Press.
Nye, J. F. (1957). Physical Properties of Crystals. Oxford: Clarendon Press. [Revised edition (1985).]
Wooster, W. A. (1973). Tensors and Group Theory for the Physical Properties of Crystals. Oxford: Clarendon Press.








































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