Chapter 10: Plane Groups and Space Groups

Plane Groups and Space Groups positioning generates new symmetry elements including glide planes, which involve a reflection operation followed by translation along the mirror plane, and screw axes, which combine rotation with translation parallel to the rotation axis. By systematically applying these combined symmetry operations in two dimensions, the chapter demonstrates how seventeen distinct plane groups arise from ten two-dimensional point groups and five two-dimensional Bravais lattices. This methodology extends to three-dimensional crystals, revealing the existence of exactly 230 space groups that represent all possible periodic symmetries in three-dimensional structures. The chapter distinguishes between symmorphic space groups, where point group operations maintain a common intersection point without requiring translational components, and non-symmorphic space groups, which necessarily incorporate translational elements such as screw rotations and glide reflections. A central portion of the chapter addresses practical interpretation of the International Tables for Crystallography, equipping readers with skills to decode space group symbols using Hermann-Mauguin notation, identify appropriate unit cell parameters, and determine asymmetric units. Key concepts including Wyckoff positions, which describe equivalent atomic sites within a unit cell, site symmetry, which characterizes the local symmetry environment of specific atomic positions, and multiplicity, which indicates the number of equivalent positions, are thoroughly explained. The chapter also presents matrix representations of symmetry operators as tools for determining space group generators and acknowledging the independent historical discoveries of these groups by Fedorov, Barlow, and Schonflies.