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

Table 3.2.2.2 

H. Klappera and Th. Hahna

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.