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The unique arrangements of lattice points are socalled Bravais lattice, named after Auguste Bravais. Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be selfconsistently arranged in space. In twodimensional (2D) lattices, there are five distinct Bravais lattices.
In threedimensional (3D) lattices, there are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Figure 4546 schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups.
Figure 4546. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.
Table 4546 also lists the relation between threedimensional crystal families, crystal systems, and lattice systems. The lattices are classified in 6 crystal families and are symbolized by 6 lower case letters a, m, o, t, h, and c. The classifications of crystal families and crystal systems are the same except the hexagonal family. The first letter of the Bravais lattice symbols is denoted with the symbol of the crystal family, and the second letter is a capital letter (P, S, I, F, R) that represents the Bravais lattice centering. The symbol S denotes an oneface centered lattice (See page3021 for detailed descriptions on the other letters).
Table 4546. The relation between threedimensional crystal families, crystal systems, and lattice systems.
Crystal
system 
Crystal
family 
Required symmetries of point group 
Point
group 
Space
group 
Bravais
lattices 
Pearson symbol 
Coordinate description 

Symbol 
No. 
Symbol 
Name 
Triclinic (anorthic) 
a 
None 
2 
2 
1 
aP 
Primitive triclinic (P) 
aP* 

Monoclinic

m 
1 twofold axis of rotation, rotary inversion axis along b, or 1 mirror plane 
3

13

2

mP 
Simple monoclinic (P) 
mP* 
a≠b≠c,
α=γ= 90°, β≠90° 
mS (mA, mB, mC) 
Baseface centered monoclinic (C) 
mS* 
Orthorhombic 
o 
Three mutually perpendicular 2fold axes of rotation, rotatoryinversion axes along a, b, and c, or one 2fold axis of rotation and two mirror planes. 
3 
59 
4 
oP 
Primitive orthorhombic (P) 
oP* 
a≠b≠c,
α=β=γ=90° 
oF 
Face centered orthorhombic (C, F) 
oF* 
oS (oA, oB, oC) 
Basecentered orthorhombic 
oS* 
oI 
Body centered orthorhombic (I) 
oI* 
Tetragonal 
t 
A single 4fold rotation or rotatoryinversion axis along c 
7 
68 
2 
tP 
Primative tetragonal (P) 
tP* 
a=b≠c,
α=β=γ=90° 
tI 
Body centered tetragonal (I) 
tI* 
Trigonal 
Hexagonal 
h 
Identity or inversion in any direction. 
5 
7 
1 
hR 
Trigonal (P) 

a≠b≠c,
α≠β≠γ≠90° 
One 3fold rotation axis along c 
18 
1 
Primative rhombohedral (P) 
hR* 
a=b=c,
α=β=γ≠90° 
Hexagonal 
A single 6fold rotation or rotatoryinversion axis along c 
7 
27 
1 
hP 
Hexagonal (P) 
hP* 
a=b≠c,
α=β=90°, γ=120° 
Cubic

c 
Four 3fold rotation axes along a+b+c, a+b+c, ab+c, ab+c. 
5 
36 
3 
cP 
Primitive cubic (P) 
cP* 
a=b=c,
α=β=γ=90°

cF 
Face centered cubic (F) 
cF* 
cI 
Body centered cubic (I) 
cI* 
Total: 7 
6 


32 
230 
14 




* = sum of multiplicities of all atom sites in the structure. 
Bravais lattices can be classified in terms of the number of lattice points in the unit cell. Figure 4546 shows the schematic illustrations of the Bravais Lattices. The array of points for each Bravais Lattice repeat periodically in threedimensional space.
Figure 4546. Schematic illustrations of the Bravais Lattices.
Pearson symbol represents crystal system, type of Bravais lattice and number of atoms in an unit cell. For instance, a facecentered cubic (FCC) structure is a facecentered Bravais lattice having 4 atoms in the unit cell, and thus its Pearson symbol is cF4. In this notation, the crystal system is simplified by a = triclinic, m = monoclinic, o = orthorhombic, t = tetragonal, h = hexagonal and trigonal and c = cubic. The Bravais lattice is symbolized by P, S (for A, B and C), R, F and I.
