What are space groups and crystal systems?

Space groups classify the complete symmetry of a crystal structure, including translations, rotations, reflections, glide planes, and screw axes. There are 230 unique space groups.

Crystal Systems

The 230 space groups are organized into 7 crystal systems based on unit cell geometry:

| System | Cell Shape | Example | |————|—————-|————-| | Cubic | a=b=c, all right angles | NaCl (Fm-3m), Si (Fd-3m) | | Hexagonal | a=b, gamma=120 | ZnO (P6_3mc) | | Tetragonal | a=b, all right angles | TiO2 rutile (P4_2/mnm) | | Orthorhombic | All right angles | Many perovskites | | Monoclinic | One non-right angle | Common in minerals | | Triclinic | No angle constraints | Least symmetric | | Trigonal | a=b=c, equal angles | Quartz (P3_121) |

Why It Matters

Symmetry determines many material properties:

• Optical: Cubic materials are optically isotropic
• Piezoelectric: Requires non-centrosymmetric space group
• Elastic: Higher symmetry means fewer independent elastic constants
• Electronic: Band structure topology is constrained by symmetry