Crystallographic Point Groups

Crystallographic point groups represent the various symmetry operations that can occur within crystals. They are unique in that at least one point will always remain unchanged during these actions. For instance, consider the triclinic system. This system, devoid of any axis or plane of symmetry, aligns with the C₁ and C_i point groups.where Cᵢ is characterized solely by a center of inversion.

Contrastingly, the monoclinic system introduces an element of symmetry. This system with one plane and one symmetry axis encompasses three different point groups: C_2h, C₂, and C_s. The Cₛ point group contains a single mirror plane (σ), while the C₂ point group contains a single two-fold rotation axis. The C₂h point group combines a two-fold rotation axis with a mirror plane perpendicular to it, resulting in the presence of a center of inversion located at their intersection. Thus, the monoclinic system includes both non-centrosymmetric (C₂, Cₛ) and centrosymmetric (C₂h) point groups. This is a clear step up from the triclinic system regarding complexity.

The orthorhombic system takes this complexity further. Equipped with three axes and three planes of symmetry, it falls under the D_2h, C_2v, and D₂ point groups. These belong to the dihedral family, characterized by multiple C₂ axes arranged perpendicular to a principal axis, often accompanied by mirror planes and a centre of inversion

However, the truly intricate systems are the tetragonal and hexagonal ones. Each of these contains seven distinct point groups. The tetragonal system includes the C_4h, C₄, S₄, D_4h, C_4v, D₄, and D_2d groups. Meanwhile, the hexagonal system encompasses the C_6h, C₆, C_3h, D_6h, C_6v, D₆, and D_3h groups.

The trigonal system, which includes the C_3i, C₃, D_3d, D₃, and C_3v groups, adds another level of intricacy. Finally, the cubic system, with nine planes of symmetry and thirteen axes of symmetry, encompasses the T_h, T, O_h, O, and T_d groups.

In summary, crystallographic point groups offer a fascinating glimpse into crystal symmetry. From systems with no symmetry to those with maximum symmetry elements, they comprehensively explain the geometric organization within crystals.