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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Closepacking 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 6_{3} 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 6fold symmetry) and the trigonal (with 3fold 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 threefold axis of rotation 
5 
7 
1 
Rhombohedral 
18 
1 
Hexagonal 
Hexagonal 
1 sixfold 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 
Lattice 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

m3 
23, m3 
m3m 
432, 43m, m3m 
Table 3549c. Other characteristics of trigonal structures.
Contents 
Page 
Angles in unit cells 
page3555 
Volume of unit cells 
page3033 
Bravais lattices 
page4546 
Relationship between threedimensional crystal families, crystal systems, space group, point group, lattice systems and symmetries 
page4549 
