Trigonal Crystal Systems
- Practical Electron Microscopy and Database -
- An Online Book -

http://www.globalsino.com/EM/

This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.

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Close-packing of equal spheres (e.g. the same atoms in a crystal) can form the trigonal, hexagonal or cubic crystal systems. The structure belongs to the trigonal system if the structure has the minimum symmetry, while it belongs to the hexagonal system if it has a 63 axis of symmetry.

Table 3549a and Figure 3549 show the trigonal crystal systems (in orange) and the schematic illustrations of the trigonal lattices, respectively. As indicated in Table 3549a, the structures belonging to the trigonal system can have either a rhombohedral or hexagonal lattice. Both the hexagonal (with 6-fold symmetry) and the trigonal (with 3-fold symmetry) systems require a hexagonal axial system. The trigonal system has the rhombohedral unit cell as an unique feature. In this case, the smallest primitive cell may be chosen with a = b = c, α = β = γ ≠ 90°. Furthermore, Table 3549c shows the cell edges and angles of trigonal crystals.

Table 3549a. Trigonal crystal systems.

Crystal
family
Crystal
system
Required
symmetries
of point group
Point
group
Space
group
Bravais
lattices
Lattice
system
Hexagonal Trigonal 1 three-fold axis of rotation 5 7 1 Rhombohedral
18 1 Hexagonal
Hexagonal 1 six-fold axis of rotation 7 27

Figure 3549. Schematic illustrations of the Bravais lattices of trigonal crystals: (a) and the grey cell in (b).

Table 3549b. Relationship between Laue classes and point groups.

System Essential symmetry  Laue class (Diffraction symmetry) Point Groups (Hermann–Mauguin notation)
Triclinic None 1, -1
Monoclinic 2/m 2/m 2, m, 2/m
Orthorhombic 222 or 2mm
mmm
mmm 222, mm2, mmm
Tetragonal 4/mmm
4/m
4, -4, 4/m
4/mmm 422, -42m, 4mm, 4/mmm
Trigonal 3 3, -3
-3m1 321, 3m1, -3m1
-31m 312, 31m, -31m
Hexagonal 6/mmm
6/m 6, -6, 6/m
6/mmm 622, -62m, 6mm, 6/mmm
Cubic 23 m3m
m-3 23, m-3
m-3m 432, -43m, m-3m

Table 3549c. Other characteristics of trigonal structures.

Contents
Page
Angles in unit cells page3555
Volume of unit cells page3033
Bravais lattices page4546
Relationship between three-dimensional crystal families, crystal systems, space group, point group, lattice systems and symmetries page4549

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