Crystallographic Axes
The crystallographic axes are imaginary lines that we can draw within the crystal lattice. These will define a coordinate system within the crystal. For 3-dimensional space lattices we need 3 or in some cases 4 crystallographic axes that define directions within the crystal lattices. Depending on the symmetry of the lattice, the directions may or may not be perpendicular to one another, and the divisions along the coordinate axes may or may not be equal along the axes. As we will see later, the lengths of the axes are in some way proportional to the lattice spacing along an axis and this is defined by the smallest group of points necessary to allow for translational symmetry to reproduce the lattice.
We here discuss the basic concepts of the crystallographic axes. As we will see, the axes are defined based on the symmetry of the lattice and the crystal. Each crystal system has different conventions that define the orientation of the axes, and the relative lengths of the axes.
Unit CellsThe "lengths" of the various crystallographic axes are defined on the basis of the unit cell. When arrays of atoms or molecules are laid out in a space lattice we define a group of such atoms as the unit cell. This unit cell contains all the necessary points on the lattice that can be translated to repeat itself in an infinite array. In other words, the unit cell defines the basic building blocks of the crystal, and the entire crystal is made up of repeatedly translated unit cells.
For example, in the 2-dimensional lattice shown here there are 6 possible choices to define the unit cell, labeled a through f. The lattice has 2-fold rotational symmetry about an axis perpendicular to the page. Since the lattice itself does not have 3-fold or 6-fold rotational symmetry, choices a and b would not be wise choices for the unit cell. Choice f can be eliminated because it is really just half of cell b. The edges of c and e are not coincident or parallel to any 2-fold axes that lie in the plane of the page. Thus our best choice would cell d.
In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices (an infinite array of discrete points). Space groups (symmetry groups of a configuration in space) are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.
The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).
Further information: Space group § Classification systemsCrystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".
A lattice system is a group of lattices with the same set of lattice point groups. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 crystallographic point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system.
A crystal family is determined by lattices and point groups. It is formed by combining crystal systems that have space groups assigned to a common lattice system. In three dimensions, the hexagonal and trigonal crystal systems are combined into one hexagonal crystal family.
Five of the crystal systems are essentially the same as five of the lattice systems. The hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. These are combined into the hexagonal crystal family.
The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table:
Crystal family Crystal system Required symmetries of the point group Point groups Space groups Bravais lattices Lattice system Triclinic Triclinic None 2 2 1 Triclinic Monoclinic Monoclinic 1 twofold axis of rotation or 1 mirror plane 3 13 2 Monoclinic Orthorhombic Orthorhombic 3 twofold axes of rotation or 1 twofold axis of rotation and 2 mirror planes 3 59 4 Orthorhombic Tetragonal Tetragonal 1 fourfold axis of rotation 7 68 2 Tetragonal Hexagonal Trigonal 1 threefold axis of rotation 5 7 1 Rhombohedral 18 1 Hexagonal Hexagonal 1 sixfold axis of rotation 7 27 Cubic Cubic 4 threefold axes of rotation 5 36 3 Cubic 6 7 Total 32 230 14 7The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below:
Crystal family Crystal system Point group / Crystal class Schönflies Hermann–Mauguin Orbifold Coxeter Point symmetry Order Abstract group triclinic pedial C1 1 11 [ ]+ enantiomorphic polar 1 trivial Z 1 {\displaystyle \mathbb {Z} _{1}} pinacoidal Ci (S2) 1 1x [2,1+] centrosymmetric 2 cyclic Z 2 {\displaystyle \mathbb {Z} _{2}} monoclinic sphenoidal C2 2 22 [2,2]+ enantiomorphic polar 2 cyclic Z 2 {\displaystyle \mathbb {Z} _{2}} domatic Cs (C1h) m *11 [ ] polar 2 cyclic Z 2 {\displaystyle \mathbb {Z} _{2}} prismatic C2h 2/m 2* [2,2+] centrosymmetric 4 Klein four V = Z 2 × Z 2 {\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}} orthorhombic rhombic-disphenoidal D2 (V) 222 222 [2,2]+ enantiomorphic 4 Klein four V = Z 2 × Z 2 {\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}} rhombic-pyramidal C2v mm2 *22 [2] polar 4 Klein four V = Z 2 × Z 2 {\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}} rhombic-dipyramidal D2h (Vh) mmm *222 [2,2] centrosymmetric 8 V × Z 2 {\displaystyle \mathbb {V} \times \mathbb {Z} _{2}} tetragonal tetragonal-pyramidal C4 4 44 [4]+ enantiomorphic polar 4 cyclic Z 4 {\displaystyle \mathbb {Z} _{4}} tetragonal-disphenoidal S4 4 2x [2+,2] non-centrosymmetric 4 cyclic Z 4 {\displaystyle \mathbb {Z} _{4}} tetragonal-dipyramidal C4h 4/m 4* [2,4+] centrosymmetric 8 Z 4 × Z 2 {\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}} tetragonal-trapezohedral D4 422 422 [2,4]+ enantiomorphic 8 dihedral D 8 = Z 4 ⋊ Z 2 {\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}} ditetragonal-pyramidal C4v 4mm *44 [4] polar 8 dihedral D 8 = Z 4 ⋊ Z 2 {\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}} tetragonal-scalenohedral D2d (Vd) 42m or 4m2 2*2 [2+,4] non-centrosymmetric 8 dihedral D 8 = Z 4 ⋊ Z 2 {\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}} ditetragonal-dipyramidal D4h 4/mmm *422 [2,4] centrosymmetric 16 D 8 × Z 2 {\displaystyle \mathbb {D} _{8}\times \mathbb {Z} _{2}} hexagonal trigonal trigonal-pyramidal C3 3 33 [3]+ enantiomorphic polar 3 cyclic Z 3 {\displaystyle \mathbb {Z} _{3}} rhombohedral C3i (S6) 3 3x [2+,3+] centrosymmetric 6 cyclic Z 6 = Z 3 × Z 2 {\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}} trigonal-trapezohedral D3 32 or 321 or 312 322 [3,2]+ enantiomorphic 6 dihedral D 6 = Z 3 ⋊ Z 2 {\displaystyle \mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}} ditrigonal-pyramidal C3v 3m or 3m1 or 31m *33 [3] polar 6 dihedral D 6 = Z 3 ⋊ Z 2 {\displaystyle \mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}} ditrigonal-scalenohedral D3d 3m or 3m1 or 31m 2*3 [2+,6] centrosymmetric 12 dihedral D 12 = Z 6 ⋊ Z 2 {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}} hexagonal hexagonal-pyramidal C6 6 66 [6]+ enantiomorphic polar 6 cyclic Z 6 = Z 3 × Z 2 {\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}} trigonal-dipyramidal C3h 6 3* [2,3+] non-centrosymmetric 6 cyclic Z 6 = Z 3 × Z 2 {\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}} hexagonal-dipyramidal C6h 6/m 6* [2,6+] centrosymmetric 12 Z 6 × Z 2 {\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2}} hexagonal-trapezohedral D6 622 622 [2,6]+ enantiomorphic 12 dihedral D 12 = Z 6 ⋊ Z 2 {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}} dihexagonal-pyramidal C6v 6mm *66 [6] polar 12 dihedral D 12 = Z 6 ⋊ Z 2 {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}} ditrigonal-dipyramidal D3h 6m2 or 62m *322 [2,3] non-centrosymmetric 12 dihedral D 12 = Z 6 ⋊ Z 2 {\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}} dihexagonal-dipyramidal D6h 6/mmm *622 [2,6] centrosymmetric 24 D 12 × Z 2 {\displaystyle \mathbb {D} _{12}\times \mathbb {Z} _{2}} cubic tetartoidal T 23 332 [3,3]+ enantiomorphic 12 alternating A 4 {\displaystyle \mathbb {A} _{4}} diploidal Th m3 3*2 [3+,4] centrosymmetric 24 A 4 × Z 2 {\displaystyle \mathbb {A} _{4}\times \mathbb {Z} _{2}} gyroidal O 432 432 [4,3]+ enantiomorphic 24 symmetric S 4 {\displaystyle \mathbb {S} _{4}} hextetrahedral Td 43m *332 [3,3] non-centrosymmetric 24 symmetric S 4 {\displaystyle \mathbb {S} _{4}} hexoctahedral Oh m3m *432 [4,3] centrosymmetric 48 S 4 × Z 2 {\displaystyle \mathbb {S} _{4}\times \mathbb {Z} _{2}}The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect them all through a single point, so that (x,y,z) becomes (−x,−y,−z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is centrosymmetric. Otherwise it is non-centrosymmetric. Still, even in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral or enantiomorphic and its symmetry group is enantiomorphic.[1]
A direction (meaning a line without an arrow) is called polar if its two-directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called a polar axis. Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.
The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral).
Main article: Bravais latticeThere are seven different kinds of lattice systems, and each kind of lattice system has four different kinds of centerings (primitive, base-centered, body-centered, face-centered). However, not all of the combinations are unique; some of the combinations are equivalent while other combinations are not possible due to symmetry reasons. This reduces the number of unique lattices to the 14 Bravais lattices.
The distribution of the 14 Bravais lattices into 7 lattice systems is given in the following table.
Crystal family Lattice system Point groupaP
Monoclinic (m) C2hmP
mS
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Orthorhombic (o) D2hoP
oS
oI
oF
Tetragonal (t) D4htP
tI
Hexagonal (h) Rhombohedral D3dhR
Hexagonal D6hhP
Cubic (c) OhcP
cI
cF
In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups (also known as lattices) in three directions.
Such symmetry groups consist of translations by vectors of the form
where n1, n2, and n3 are integers and a1, a2, and a3 are three non-coplanar vectors, called primitive vectors.
These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They[clarification needed] represent the maximum symmetry a structure with the given translational symmetry can have.
All crystalline materials (not including quasicrystals) must, by definition, fit into one of these arrangements.
For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3, or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.
The Bravais lattices were studied by Moritz Ludwig Frankenheim in , who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in .
In two-dimensional space, there are four crystal systems (oblique, rectangular, square, hexagonal), four crystal families (oblique, rectanguar, square, hexagonal), and four lattice systems (oblique, rectangular, square, and hexagonal).[3][4]
Crystal family Crystal system Crystallographic point groups No. of plane groups Bravais lattices Oblique (monoclinic) Oblique 1, 2 2 mp Rectangular (orthorhombic) Rectangular m, 2mm 7 op, oc Square (tetragonal) Square 4, 4mm 3 tp Hexagonal Hexagonal 3, 6, 3m, 6mm 5 hp Total 4 10 17 5The four-dimensional unit cell is defined by four edge lengths (a, b, c, d) and six interaxial angles (α, β, γ, δ, ε, ζ). The following conditions for the lattice parameters define 23 crystal families
Crystal families in 4D space No. Family Edge lengths Interaxial angles 1 Hexaclinic a ≠ b ≠ c ≠ d α ≠ β ≠ γ ≠ δ ≠ ε ≠ ζ ≠ 90° 2 Triclinic a ≠ b ≠ c ≠ d α ≠ β ≠ γ ≠ 90°The names here are given according to Whittaker.[5] They are almost the same as in Brown et al.,[6] with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown et al. are given in parentheses.
The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.[5][6] Enantiomorphic systems are marked with an asterisk. The number of enantiomorphic pairs is given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group itself (considered as a geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P31 and P32, P and P. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.
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Crystal systems in 4D space No. of