32 Point Groups
The 32 point groups - also called the 32 crystal classes - represent every possible combination of point symmetry operations that a crystal can possess. [1] In two dimensions, only mirrors and rotations are possible symmetry elements, producing 10 distinct two-dimensional point groups. Adding the third dimension makes inversion possible, and the number of valid combinations rises to 32. [1] These 32 classes are not arbitrary - the combinations are limited because symmetry elements must be mutually compatible and must also be compatible with the translational symmetry of the crystal lattice. [1]
Organisation by Crystal System
The 32 point groups divide among the six crystal systems based on common symmetry elements. [1] Each crystal system has a characteristic symmetry - the element that defines membership in that system:
| Crystal System | Characteristic Symmetry | [1] |
|---|---|---|
| Triclinic | One-fold rotation with or without inversion | [1] |
| Monoclinic | A single 2-fold axis and/or a single mirror | [1] |
| Orthorhombic | Three 2-fold axes or two mirrors plus a 2-fold axis | [1] |
| Tetragonal | A single 4-fold rotation or rotoinversion axis | [1] |
| Hexagonal | A single 3-fold or 6-fold rotation axis | [1] |
| Isometric | Four 3-fold rotation axes through the body diagonals of the unit cell | [1] |
Crystal Axes and Symmetry
The positions of the crystal axes for each crystal system are not chosen arbitrarily - they must be in rational orientations relative to the symmetry elements of the class. [1] In the orthorhombic system, for example, the three crystal axes coincide with the three 2-fold rotation axes. In the tetragonal system, the c axis is parallel to the 4-fold axis, and the a and b axes are perpendicular to it. These conventions ensure that the Hermann-Mauguin symbols describe the symmetry in a consistent and unambiguous way across all minerals in the same system.
Identifying Crystal Classes
Point group symmetry can be determined from a well-formed hand specimen by identifying the symmetry elements present - mirrors, rotation axes, and inversion - using the external faces of the crystal. [1] However, distinguishing between different Bravais lattices within the same crystal system requires X-ray diffraction - the point group and crystal class are visible in the external form, but the internal lattice type is not. [1] The 32 point groups are used in all mineral descriptions to characterise the symmetry of the mineral species, and each class has distinctive physical and optical properties that follow from its symmetry.
References
- Nesse, W. D. (2017). Introduction to Mineralogy, 3rd ed. Oxford University Press.
Related Topics
Mineralogy
Mineralogy is the study of minerals.{/* SRC: Nesse p.4: "ing and exciting field of study called geomicrobiology. Numerous different minerals and mineraloids are now" */} {/* EDITORIAL */} It is...
Translational Symmetry
Translational symmetry is the simplest and most fundamental type of symmetry in crystals. It is the repetition of a motif - which in a mineral represents a collection of atoms - by translation:...
Rotoinversion
Rotoinversion axes are compound symmetry operations that combine rotation with inversion through the centre of the crystal in a single step.{/* SRC: Nesse p.20: "Rotation axes may be combined with...
Point Symmetry
Point symmetry deals with how a motif can be repeated about a fixed central point. In minerals, the motifs being repeated are either crystal faces on the outside of a hand specimen or particular...
References & Citations
- 1.Introduction to Mineralogy Nesse
Master UPSC Geology Optional
Ex-ONGC Geologist & Rank Holder
Learn the exact analytical answer-writing patterns needed for UPSC Optional from an AIR 2 & AIR 25 holder.
Offline in Delhi