Page 72 - Materials Chemistry, Second Edition

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2.3. The Crystalline State

Figure 2.36. The translational incompatibility of a fivefold rotation axis, which leaves voidspaces (red)
among the lattice objects.

Table 2.5. The 32 Crystallographic Point Groups
a
Crystal system (Bravais lattices) {defining symmetry Crystallographic point groups (molecular point
b
elements} groups )
Cubic (P, I, F) 23, m 3, 432, 43m, m 3m (T, T h , O, T d , O h )
{Four ⊥ threefold rotation axes}
Tetragonal (P, I) 4, 4, 4/m, 422, 4mm, 42m,4/mmm
{One fourfold rotation axis} (C 4 , S 4 , C 4h , D 4 , C 4u , D 2d , D 4h )
Orthorhombic (P, C, I, F) 222, mm2, mmm (D 2 , C 2u , D 2h )
{Three ⊥ twofold rotation axes or three ⊥ m’s}
Trigonal/Rhombohedral (P) 3, 3, 32, 3m, 3m ðC 3 ; C 3i ; D 3 ; C 3u ; D 3d Þ
{One threefold rotation axis}
Hexagonal (P) 6, 6, 6/m, 622, 6mm, 6m2, 6/mmm
{One sixfold rotation axis} (C 6 , C 3h , C 6h , D 6 , C 6u , D 3h , D 6h )
Monoclinic (P, C) 2, m,2/m (C 2 , C s , C 2h )
{One twofold rotation axis}
Triclinic (P) 1, 1 C 1 ; C i Þ
ð
{N/A}
a
The H–M symbolism derived from crystal symmetry operations. For image and movie representations of
each point group, see the website: http://cst-www.nrl.navy.mil/lattice/spcgrp/index.html
b
The analogous Schoenflies symbolism derived from molecular symmetry operations.

axes (i.e.,5 or 5) are not relevant (Figure 2.36). [36] Furthermore, since there is no
such thing as a linear 3-D crystal, the linear point groups are also irrelevant.
The remaining point group symmetry operations yield a total of 32 crystallographic
point groups, designated by Hermann-Mauguin symbols (Table 2.5). As they can
be deduced from the macroscopic crystal symmetry, they are also referred to as
the 32 crystal classes. For the same reason, the symmetry elements that give rise to
the crystal classes are sometimes referred to as external symmetry elements.