Space Groups

Note that the 32 point groups, also called crystal classes are derived from only 13 operations: Congruent operations 1, 2, 3, 4, 6 Incongruent operations _ _ _ _ _ 1 = i, 2 = m, 3, 4, 6 = 3/m, 2/m, 4/m, 6/m

3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System based on the minimum symmetry present

These symmetry operations are all that is needed to catagorize the symmetry present in the external form of a crystal, such as the one below. However, in order to consider the positions of atoms in a crystal, the symmetry elements must work in conjunction with the Bravais lattices

Monoclinic (2nd setting) a=b=c, α = β = γ = 90° Cubic = Isometric a= b≠ c, α = β = γ = 90° Tetragonal a≠ b≠ c, α = β = γ = 90° Orthorhombic a=b=c, α = β = γ ≠ 90° Hexagonal a=b≠ c, α = β = 90°, γ = 120° a≠ b≠ c, α ≠ β ≠ γ Triclinic a≠ b≠ c, α = γ = 90° β > 90° Monoclinic (2nd setting) The 14 Bravais lattices

Combining the symmetry elements with the Bravais lattices requires the introduction of three additional symmetry operations: translation, glide and screw. Of these translation is simple. This merely repeating the unit cell of the crystal. t By repetition of this operation in three dimensions, the crystal is assembled. t

However, in order to generate the lattice points (or atoms, as determined), it is necessary to introduce two additional symmetry elements: the glide plane and the screw axis.

Glide Plane

Example of a screw operation screw axis Rotate 180° while translating ½ of the length of the axis Illustration of the operation of the screw 21

The operation of 31 and 32 screw axes are shown below, looking down the axis and showing the sense of rotation and the 1/3 2/3 axis axis 2/3 1/3 31 32 translation distance along the axis. The result is opposite senses of screw rotation; i.e. left-handed and right-handed threads as on a wood screw.

The presence of the elements 31 or 32 in α-quartz produces left-handed and right-handed quartz, as shown below. The two crystals are mirror images (enantiomorphic pairs); this can be seen in face s, which points upward to theleft or right.

Screw axes combine rotation with translation, but are similar to rotation axes because only 1-, 2-, 3- 4- and 6-fold rotations are possible; the translation possibilities are more numerous. Thus, a 21 screw is a translation of ½ along the axis, followed by a rotation of 180°. But also possible are 31, (t=1/3), 41, (t=1/4), and 61, (t=1/6). But because of the repeats of translations total possibilities are: 21 31, 32 41, 42,, 43 61, 62, 63, 64, 65

The combination of the symmetry elements with the 14 Bravais lattices results in 230 possible space groups. These represent the only possible arrangements of lattice nodes (or atoms) in space. The space groups were derived independently in the 19th Century by Barlowe in England and Federov in Russia. This was done by trial and error on an abstract consideration of lattice nodes, as atomic theory did not exist at that time. The space groups can be derived by application of group theory in mathematics. As an example, consider the triclinic system, which can only be the primitive lattice, designated P. The only symmetry elements present in the triclinic system are 1 and 1. Thus, the only space groups in the triclinic system are: P1 and P1 Only 228 more to go!

The 230 space groups are listed in Table 11.9, p. 248 in the text. One can see that the number of space groups in a crystal system increases as the symmetry content increases. Thus, there are a large number of possible space groups in the cubic (isometric) system. The space groups are designated by abbreviated Hermann- Mauguin symbols. They are constructed by first stating one of the 14 lattices, then listing the symmetries present. The space groups must each contain the symmetry present in one of the 32 point groups or crystal classes. A space group is thus isogonal with a point group, as shown in the table.

Although some of the Hermann-Mauguin symbols can be confusing, it is possible to easily extract the point group from the space group symbol. One must remember that every glide operation contains a mirror, and every screw operation contains a rotation. So, one converts glides to mirrors and screw axes to rotation axes, as below: For the glides a, b, c, d and n → m For the screw axes 21, 31, 32 . . . → 2, 3 . . . For example, consider α-quartz (low quartz): P3121 → 32 So space group P3121 is isogonal with point group 32

One can see that α-quartz contains a single 3-fold rotation axis and thus belongs to the trigonal subsystem of the hexagonal system. Now, consider the space group of wurtzite (α-ZnS): R3m → 3m Wurtzite also belongs to the trigonal subsystem of the hexagonal system but has a rhombohedral lattice. The foregoing illustrates the point that it is logical to define the hexagonal system in terms of the lattice type rather than the symmetry, 3-fold or 6-fold rotation axis. We would then have: Hexagonal lattice division hexagonal (6-fold) or trigonal (3-fold) symmetry e.g. β-quartz e.g. α-quartz Rhombohedral lattice division trigonal (3-fold symmetry) e.g. wurtzite