International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 2.2, p. 18

Table 2.2.4.1 

Th. Hahna* and A. Looijenga-Vosb

aInstitut für Kristallographie, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, and bLaboratorium voor Chemische Fysica, Rijksuniversiteit Groningen, The Netherlands
Correspondence e-mail:  hahn@xtl.rwth-aachen.de

Table 2.2.4.1| top | pdf |
Lattice symmetry directions for two and three dimensions

Directions that belong to the same set of equivalent symmetry directions are collected between braces. The first entry in each set is taken as the representative of that set.

LatticeSymmetry direction (position in Hermann–Mauguin symbol)
PrimarySecondaryTertiary
Two dimensions
Oblique [\matrix{\hbox{Rotation}\hfill\cr \hbox{point}\hfill\cr\hbox{in plane}\hfill\cr}]    
Rectangular [10] [01]
Square [\left\{\matrix{[10]\cr [01]\cr}\right\}] [\left\{\matrix{[1\bar{1}]\cr [11]\cr}\right\}]
Hexagonal [\left\{\matrix{[10]\cr [01]\cr [\bar{1}\bar{1}]\cr}\right\}] [\left\{\matrix{[1\bar{1}]\cr [12]\cr [\bar{2}\bar{1}]\cr}\right\}]
Three dimensions
Triclinic None    
Monoclinic [010] (`unique axis b')  
  [001] (`unique axis c')  
Orthorhombic [100] [010] [001]
Tetragonal [001] [\left\{\matrix{[100]\cr [010]\cr}\right\}] [\left\{\matrix{[1\bar{1}0]\cr [110]\cr}\right\}]
Hexagonal [001] [\left\{\matrix{[100]\cr [010]\cr [\bar{1}\bar{1}0]\cr}\right\}] [\left\{\matrix{[1\bar{1}0]\cr [120]\cr [\bar{2}\bar{1}0]\cr}\right\}]
Rhombohedral (hexagonal axes) [001] [\left\{\matrix{[100]\cr [010]\cr [\bar{1}\bar{1}0]\cr}\right\}]  
Rhombohedral (rhombohedral axes) [111] [\left\{\matrix{[1\bar{1}0]\cr [01\bar{1}]\cr [\bar{1}01]\cr}\right\}]  
Cubic [\left\{\matrix{[100]\cr [010]\cr [001]\cr}\right\}] [\left\{\matrix{[111]\cr [1\bar{1}\bar{1}]\cr [\bar{1}1\bar{1}]\cr [\bar{1}\bar{1}1]\cr}\right\}] [\left\{\matrix{[1\bar{1}0]\ [110]\cr [01\bar{1}]\ [011]\cr [\bar{1}01]\ [101]\cr}\right\}]
For the full Hermann–Mauguin symbols see Section 2.2.4.1[link].