Section: 12 | Symmetry of Crystals |

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SYMMETRY OF CRYSTALS

L. I. Berger

The ability of a body to coincide with itself in its different positions regarding a coordinate system is called its symmetry. This property reveals itself in iteration of the parts of the body in space. The iteration may be done by reflection in mirror planes, rotation about certain axes, inversions and translations. These actions are called the symmetry operations. The planes, axes, points, etc., are known as symmetry elements. Essentially, mirror reflection is the only truly primitive symmetry operation. All other operations may be done by a sequence of reflections in certain mirror planes. Hence, the mirror plane is the only true basic symmetry element. But for clarity, it is convenient to use the other symmetry operations, and accordingly, the other aforementioned symmetry elements. The symmetry elements and operations are presented in Table 1.

The entire set of symmetry elements of a body is called its symmetry class. There are thirty-two symmetry classes that describe all crystals that have ever been noted in mineralogy or been synthesized (more than 150,000). The denominations and symbols of the symmetry classes are presented in Table 2.

There are several known approaches to classification of individual crystals in accordance with their symmetry and crystallochemistry. The particles that form a crystal are distributed in certain points in space. These points are separated by certain distances (translations) equal to each other in any chosen direction in the crystal. Crystal lattice is a diagram that describes the location of particles (individual or groups) in a crystal. The lattice parameters are three non-coplanar translations that form the crystal lattice. Three basic translations form the unit cell of a crystal. August Bravais (1848) has shown that all possible crystal lattice structures belong to one or another of fourteen lattice types (Bravais lattices). The Bravais lattices, both primitive and non-primitive, are the contents of Table 3.

Among the three-dimensional figures, there is a group of polyhedrons that are called regular, which have all faces of the same shape and all edges of the same size (regular polygons). It has been shown that there are only five regular polyhedrons. Because of their importance in crystallography and solid state physics, a brief description of these polyhedrons is included in Table 4.

The systematic description of crystal structures is presented primarily in the well-known Strukturbericht. The classification of crystals by the Strukturbericht does not reflect their crystal class, the Bravais lattice, but is based on the crystallochemical type. This makes it inconvenient to use the Strukturbericht categories for comparison of some individual crystals. Thus, there have been several attempts to provide a more convenient classification of crystals. Table 5 presents a compilation of different classifications which allows the reader to correlate the Strukturbericht type with the international and Schoenflies point and space groups and with Pearson’s symbols, based on the Bravais lattice and chemical composition of the class prototype. The information included in Table 5 has been chosen as an introduction to a more detailed crystallophysical and crystallochemical description of solids.

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TABLE 1. Symmetry Operations, Symmetry Elements, and Stereographic Projections

Symmetry operation	Name	International (Hermann-Mauguin)	Schoenflies	Parallel projection	Perpendicular projection
Reflection in a plane	Plane	m	C_s
Rotation by angle α = 360°/n about an axis	Axis	n = 1, 2, 3, 4, or 6	C_n
		n = 2	C₂
		n = 3	C₃
		n = 4	C₄
		n = 6	C₆
Rotation about an axis and inversion in a symmetry center lying on the axis	Inversion (improper) axis	n̄ = 3̄, 4̄, 6̄	C_ni
		n̄ = 3̄	C_3i
		n̄ = 4̄	C_4i
		n̄ = 6̄	C_6i
Inversion in a point	Center	1̄	C_i
Parallel translation	Translation vector a⃗, b⃗, c⃗
Reflection in a plane and translation parallel to the plane	Glide–plane	a, b, c, n, d
Rotation about an axis and translation parallel to the axis	Screw axis	n_m (m = 1, 2, .., n – 1)
Rotation about an axis and reflection in a plane perpendicular to the axis	Rotatory-reflection axis	ñ = 1̃, 2̃, 3̃, 4̃, 6̃	S_n

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TABLE 2. The Thirty-Two Symmetry Classes

Triclinic	Int	Sch	Int	Sch	Int	Sch	Int	Sch	Int	Sch	Int	Sch	Int	Sch
Triclinic	1	C₁	1	C_i
Monoclinic					m	C_s	2	C₂	2/m	C_2h
Orthorhombic					mm2	C_2v	222	D₂	mmm	D_2h
Trigonal	3	C₃	3	C_3i	3m	C_3v	32	D₃	3̄m	C_3d
Tetragonal	4	C₄	4/m	C_4h	4mm	C_4v	422	D₄	4/mmm	D_4h	4̄	S₄	4̄2m	D_2d
Hexagonal	6	C₆	6/m	C_6h	6mm	C_6v	622	D₆	6/mmm	D_6h	6̄	C_3h	6̄m2	D_3h
Cubic	23	T	m3	T_h	4̄3m	T_h	432	O	m3m	O_h

^a Per Fedorov Institute of Crystallography, Russian Academy of Sciences, nomenclature.

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TABLE 3. The Fourteen Possible Space Lattices (Bravais Lattices) (P, C, I, F, R Indicate Lattice Type; a, b, c, α, β, γ Indicate Characteristic Parameters)

Crystal system	Metric category of the system	No. of different lattices in the system	P	C	I	F	R	No. of identipoints per unit cell	a	b	c	α	β	γ	Description of characteristic parameters a⊂X, b⊂Y, c⊂Z α≡(b,c), β≡(a,c), γ≡(b,c)	Lattice symmetry (Int)	Lattice symmetry (Sch)
Triclinic	Trimetric	1	+					1	+	+	+	+	+	+	a ≠ b ≠ c, α ≠ β ≠ γ	1	C
Monoclinic	Trimetric	2	+	+				1 or 2	+	+	+		+		a ≠ b ≠ c, α = γ = 90° ≠ β	2/m	C_2h
Orthorhombic	Trimetric	4	+	+	+	+		1, 2, or 4	+	+	+				a ≠ b ≠ c, α = β = γ = 90°	mmm	D_2h
Trigonal (rhombohedral)	Dimetric	1					+	1	+			+			a = b = c, 120° > α = β = γ ≠ 90°	3m	D_3d
Tetragonal	Dimetric	2	+		+			1 or 2	+		+				a = b ≠ c, α = β = γ = 90°	4/mmm	D_4h
Hexagonal	Dimetric	1	+					1	+		+				a = b ≠ c, α = β = 90°, γ = 120°	6/mmm	D_6h
Isometric (cubic)	Monometric	3	+		+	+		1, 2, or 4	+						a = b = c, α = β = γ = 90°	m3m	O_h

^a Designations of the space-lattice types: P – primitive, C – side-centered (base-centered), I – body-centered, F – face-centered, R – rhombohedral.

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TABLE 4. The Five Possible Regular Polyhedrons

Polyhedron	Symmetry class	Symmetry elements	Form of faces	No. of faces (F)	No. of edges (E)	No. of vertices (V)
Tetrahedron	T	4C₃3C₂	Equilateral triangle	4	6	4
Cube (hexahedron)	O	3C₄4C₃6C₂	Square	6	12	8
Octahedron	O	3C₄4C₃6C₂	Equilateral triangle	8	12	6
Pentagonal dodecahedron	J	6C₅10C₃15C₂	Regular pentagon	12	30	20
Icosahedron	J	6C₅10C₃15C₂	Equilateral triangle	20	30	12

^a Per formula by Leonhard Euler: F + V – E = 2.

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TABLE 5. Classification of Crystals

Strukturbericht symbol	Structure name	Symmetry group (International)	Symmetry group (Schoenflies)	Pearson symbol^a	Standard ASTM E157-82a symbol^b
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A1	Cu	Fm3m	O⁴_h	cF4	F
A2	W	Im3m	O⁹_h	cI2	B
A3	Mg	P6₃/mmc	D⁴_6h	hP2	H
A4	C	Fd3m	O⁷_h	cF8	F
A5	Sn	If₁/amd	D¹⁹_4h	tI4	U
A6	In	I4/mmm	D¹⁷_4h	tI2	U
A7	As	R3̄m	D⁵_3d	hR2	R
A8	Se	P3₁21 or P3₂21	D⁴₃ (D⁶₃)	hP3	H
A10	Hg	R3̄m	D⁵_3d	hR1	R
A11	Ga	Cmca	D¹⁸_2h	oC8	Q
A12	α-Mn	I43̄m	T³_d	cI58	B
A13	β-Mn	P4₁32	O⁷	cP20	C
A15	OW₃	Pm3n	O³_h	cP8	C
A20	α-U	Cmcm	D¹⁷_2h	oC4	Q
B1	ClNa	Fm3m	O⁵_h	cF8	F
B2	ClCs	Pm3m	O¹_h	cP2	C
B3	SZn	F4̄3m	T²_d	cF8	F
B4	SZn	P6₃mc	C⁴_6v	hP4	H
B8₁	AsNi	P6₃/mmc	D⁴_6h	hP4	H

^a The first letter denotes the crystal system: triclinic (a), monoclinic (m), orthorhombic (o), tetragonal (t), hexagonal (h), and cubic (c). Trigonal (rhombohedral) system is denoted by combination hR. The second letter of Pearson’s symbol denotes lattice type: primitive (P), edge-(base-) centered (C), body-centered (I), or face-centered (F). The following number denotes number of atoms in the crystal unit cell.
^b Standard ASTM E157-82a has the Bravais lattice designations as follows: C – primitive cubic; B – body-centered cubic; F – face-centered cubic; T – primitive tetragonal; U – body-centered tetragonal; R – rhombohedral; H – hexagonal; O – primitive orthorhombic; P – body-centered orthorhombic; Q – base-centered orthorhombic; S – face-centered orthorhombic; M – primitive monoclinic; N – centered monoclinic; A – triclinic.

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SYMMETRY OF CRYSTALS

TABLE 1. Symmetry Operations, Symmetry Elements, and Stereographic Projections

TABLE 2. The Thirty-Two Symmetry Classes

TABLE 3. The Fourteen Possible Space Lattices (Bravais Lattices) (P, C, I, F, R Indicate Lattice Type; a, b, c, α, β, γ Indicate Characteristic Parameters)

TABLE 4. The Five Possible Regular Polyhedrons

TABLE 5. Classification of Crystals

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