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. 738

Table 3.2.2.1 

H. Klappera and Th. Hahna

Table 3.2.2.1| top | pdf |
The 11 Laue classes, the 21 noncentrosymmetric crystallographic point groups (crystal classes) and the occurrence (+) of specific crystal properties

The number of non-zero independent tensor components is given in parentheses. For comparison, the icosahedral and the sphere groups (isotropy) are added. For first-rank tensors the direction [uvw] of the property vector is given. There are 10 pyroelectric, 20 piezoelectric, 11 enantiomorphic and 15 optically active crystal classes.

Crystal systemLaue classNoncentrosymmetric point groupFirst-rank tensor (e.g. pyroelectricity, piezoelectricity under hydrostatic pressure)Third-rank tensor (e.g. piezoelectricity, second-harmonic generation)EnantiomorphismAxial second-rank tensor (optical activity = optical gyration)
Triclinic [\bar{1}] 1 + (3) [uvw] + (18) + + (6)
Monoclinic [2 /m] 2 + (1) [010] + (8) + + (4)
m + (2) [u0w] + (10) + (2)
Orthorhombic [{2 / m}\, {2 / m}\, {2 / m}] 222 + (3) + + (3)
mm2 + (1) [001] + (5) + (1)
Tetragonal [{4 /m}] 4 + (1) [001] + (4) + + (2)
[\overline{4}] + (4) + (2)
[{4/ m}\, {2/ m} \,{2/ m}] 422 + (1) + + (2)
4mm + (1) [001] + (3)
[\overline{4}2m] + (2) + (1)
Trigonal [\overline{3}] 3 + (1) [001] + (6) + + (2)
[\overline{3} \,{2 / m}] 32 + (2) + + (2)
3m + (1) [001] + (4)
Hexagonal [{6 / m}] 6 + (1) [001] + (4) + + (2)
[\overline 6] + (2)
[{6 / m}\, {2 / m} \,{2/ m}] 622 + (1) + + (2)
6mm + (1) [001] + (3)
[\overline{6}2m] + (1)
Cubic [{2/m} \,\overline{3}] 23 + (1) + + (1)
[{4/ m}\,{\overline 3}\,{2/ m}] 432 + + (1)
[{\overline 4}3m] + (1)
 
Icosahedral [m\,\overline3\,\overline5] ([\overline5\,\overline3\, m]) 235 (532) + + (1)
Spherical [m\overline\infty] ([\infty\infty m]) [2\infty] ([\infty\infty]) + + (1)
Unique axis b.
Hexagonal axes.