Practical Electron Microscopy and Database

Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be self-consistently arranged in space. There are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Depending on their geometry, crystals are commonly classified into seven systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic. Figure 4548a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. Table 4548a also lists the relation between three-dimensional crystal families, crystal systems, and lattice systems. Table 4548b also shows the seven crystal systems and the restrictions on their cell dimensions (cell edges and cell angles).

Figure 4548a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.

Table 4548a. The relation between three-dimensional crystal families, crystal systems, and lattice systems.

As indicated in Table 4548a, the 14 basic Bravais lattice types give rise to a total of 230 possible crystal types when they are combined with other symmetry operators such as mirrors planes, glide planes, and screw axes.

Table 4548b. The seven crystal systems (families) and the restrictions on their cell dimensions.

Figure 4548b shows the schematic illustration of the different classes of crystal systems and their properties.

Seven crystal systems
32 point groups
No center of symmetry (21 point groups)	Center of symmetry (11 point groups)
Piezoelectric (20 point groups)	Non-piezoelectric (1 point groups)
Polar or pyroelectric (10 point groups)	Non-pyroelectric
Ferroelectric (Only if polarization Is reversible)	Non-ferroelectric

Figure 4548b. Schematic illustration of the different classes of crystal systems and their properties.