Space group

Space Group

A space group is a mathematical concept used in crystallography to describe the symmetries of a crystal structure. It encompasses the set of symmetry operations, including translations, rotations, reflections, and glide reflections, that map a crystal lattice onto itself. Understanding space groups is essential for the classification of crystal structures and the determination of their properties.

Symmetry Operations

Space groups are defined by a combination of symmetry operations:

• Translation: Moves every point of a crystal by the same distance in a given direction.
• Rotation: Rotates the crystal around an axis.
• Reflection: Reflects the crystal across a plane.
• Glide Reflection: Combines reflection with translation parallel to the reflecting plane.

Crystallographic Notation

Space groups are denoted using the Hermann-Mauguin notation, which succinctly describes the symmetry elements present in the crystal. For example, the space group P2₁/c indicates a primitive lattice with a two-fold screw axis and a glide plane.

Classification

There are 230 unique space groups in three-dimensional space, classified into:

• Triclinic: The least symmetric, with only one or two symmetry operations.
• Monoclinic: Includes a single two-fold axis or a mirror plane.
• Orthorhombic: Three mutually perpendicular two-fold axes or mirror planes.
• Tetragonal: A single four-fold axis.
• Trigonal: A single three-fold axis.
• Hexagonal: A single six-fold axis.
• Cubic: Multiple three-fold axes.

Importance in Crystallography

Space groups are crucial for:

• Determining Crystal Structures: They help in solving the phase problem in X-ray crystallography.
• Predicting Physical Properties: Symmetry can influence optical, electrical, and mechanical properties.
• Material Science: Understanding space groups aids in the design of new materials with desired properties.

History

The concept of space groups was developed in the late 19th and early 20th centuries, with significant contributions from mathematicians such as Evgraf Fedorov and Arthur Moritz Schoenflies. Their work laid the foundation for modern crystallography.

Also see

• Crystallography
• Symmetry in Crystals
• Bravais Lattice
• Hermann-Mauguin Notation
• X-ray Crystallography