Section: 12 | Symmetry of Crystals |
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John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 102nd Edition (Internet Version 2021), CRC Press/Taylor & Francis, Boca Raton, FL.
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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.

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



Symmetry operationNameInternational (Hermann-Mauguin)SchoenfliesParallel projectionPerpendicular projection
Reflection in a planePlanemCsparallel.jpgperpendicular.jpg
Rotation by angle α = 360°/n about an axisAxisn = 1, 2, 3, 4, or 6Cn
n = 2C2axisc2parallel.jpgaxisc2perp.jpg
n = 3C3axis_c3_parallel.jpgaxis_c3_perp.jpg
n = 4C4axis_c4_parallel.jpgaxis_c4_perp.jpg
n = 6C6axis_c6_parallel.jpgaxis_c6_perp.jpg
Rotation about an axis and inversion in a symmetry center lying on the axisInversion (improper) axisn̄ = 3̄, 4̄, 6̄Cni
n̄ = 3̄C3iinv_c3i_parallel.jpginv_c3i_perp.jpg
n̄ = 4̄C4iinv_c4i_parallel.jpginv_c4i_perp.jpg
n̄ = 6̄C6iinv_c6i_parallel.jpginv_c6i_perp.jpg
Inversion in a pointCenterCicentre_parallel.jpgcentre_perp.jpg
Parallel translationTranslation vector a⃗, b⃗, c⃗
Reflection in a plane and translation parallel to the planeGlide–planea, b, c, n, d
Rotation about an axis and translation parallel to the axisScrew axisnm (m = 1, 2, .., n – 1)
Rotation about an axis and reflection in a plane perpendicular to the axisRotatory-reflection axisñ = 1̃, 2̃, 3̃, 4̃, 6̃Sn


TABLE 2. The Thirty-Two Symmetry Classes



TriclinicIntSchIntSchIntSchIntSchIntSchIntSchIntSch
Triclinic1C11Ci
MonoclinicmCs2C22/mC2h
Orthorhombicmm2C2v222D2mmmD2h
Trigonal3C33C3i3mC3v32D33̄mC3d
Tetragonal4C44/mC4h4mmC4v422D44/mmmD4hS44̄2mD2d
Hexagonal6C66/mC6h6mmC6v622D66/mmmD6hC3h6̄m2D3h
Cubic23Tm3Th4̄3mTh432Om3mOh
  • a Per Fedorov Institute of Crystallography, Russian Academy of Sciences, nomenclature.


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 systemMetric category of the systemNo. of different lattices in the systemPCIFRNo. of identipoints per unit cellabcαβγDescription of characteristic parameters a⊂X, b⊂Y, c⊂Z α≡(b,c), β≡(a,c), γ≡(b,c)Lattice symmetry (Int)Lattice symmetry (Sch)
TriclinicTrimetric1+1++++++a ≠ b ≠ c, α ≠ β ≠ γ1C
MonoclinicTrimetric2++1 or 2++++a ≠ b ≠ c, α = γ = 90° ≠ β2/mC2h
OrthorhombicTrimetric4++++1, 2, or 4+++a ≠ b ≠ c, α = β = γ = 90°mmmD2h
Trigonal (rhombohedral)Dimetric1+1++a = b = c, 120° > α = β = γ ≠ 90°3mD3d
TetragonalDimetric2++1 or 2++a = b ≠ c, α = β = γ = 90°4/mmmD4h
HexagonalDimetric1+1++a = b ≠ c, α = β = 90°, γ = 120°6/mmmD6h
Isometric (cubic)Monometric3+++1, 2, or 4+a = b = c, α = β = γ = 90°m3mOh
  • a Designations of the space-lattice types: P – primitive, C – side-centered (base-centered), I – body-centered, F – face-centered, R – rhombohedral.


TABLE 4. The Five Possible Regular Polyhedrons



PolyhedronSymmetry classSymmetry elementsForm of facesNo. of faces (F)No. of edges (E)No. of vertices (V)
TetrahedronT4C33C2Equilateral triangle464
Cube (hexahedron)O3C44C36C2Square6128
OctahedronO3C44C36C2Equilateral triangle8126
Pentagonal dodecahedronJ6C510C315C2Regular pentagon123020
IcosahedronJ6C510C315C2Equilateral triangle203012

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


TABLE 5. Classification of Crystals



Strukturbericht symbolStructure nameSymmetry group (International)Symmetry group (Schoenflies)Pearson symbolaStandard ASTM E157-82a symbolb
Continued on next page...
A1CuFm3mO4hcF4F
A2WIm3mO9hcI2B
A3MgP63/mmcD46hhP2H
A4CFd3mO7hcF8F
A5SnIf1/amdD194htI4U
A6InI4/mmmD174htI2U
A7AsR3̄mD53dhR2R
A8SeP3121 or P3221D43 (D63)hP3H
A10HgR3̄mD53dhR1R
A11GaCmcaD182hoC8Q
A12α-MnI43̄mT3dcI58B
A13β-MnP4132O7cP20C
A15OW3Pm3nO3hcP8C
A20α-UCmcmD172hoC4Q
B1ClNaFm3mO5hcF8F
B2ClCsPm3mO1hcP2C
B3SZnF4̄3mT2dcF8F
B4SZnP63mcC46vhP4H
B81AsNiP63/mmcD46hhP4H

  • 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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