The Seven Crystal Systems: Overview

The seven crystal systems are cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, and hexagonal. They differ based on the lengths and angles of their lattice vectors. Cubic systems have equal vectors at 90° angles, while triclinic systems have unequal vectors at unequal angles. Each system exhibits distinct symmetry properties that determine its geometric constraints.

A cubic crystal system is the simplest system where all three lattice vectors (a, b, c) are equal in length and orthogonal to each other, with all angles at 90°. Sodium chloride (NaCl) exemplifies this system. Cubic crystals display high symmetry with multiple perpendicular axes and planes of symmetry.

A tetragonal lattice has two vectors of equal length (a = b) while the third differs (c ≠ a), with all angles at 90°. Unlike cubic systems, tetragonal lattices introduce fourfold axes parallel to the c direction. This reduced symmetry compared to cubic systems creates a more elongated or compressed unit cell geometry.

A triclinic lattice has no specific constraints on its defining vectors: they are unequal in length (a ≠ b ≠ c) and form unequal angles (α ≠ β ≠ γ). This system has the lowest symmetry, with only inversion centers as symmetry elements. Triclinic lattices are necessarily primitive and conventionally use the three shortest vectors.

A hexagonal lattice has two equal-length vectors (a = b) that subtend a 120° angle in the basal plane, while the third vector is perpendicular to this plane. This configuration creates a six-fold rotational symmetry. The hexagonal system represents an intermediate level of symmetry between highly symmetric cubic and less symmetric triclinic systems.

An orthorhombic lattice has three mutually perpendicular vectors of different lengths (a ≠ b ≠ c), with all angles at 90°. This system exhibits three mutually perpendicular sets of twofold axes and reflection planes. Orthorhombic lattices can be centered or primitive and represent an intermediate symmetry level between monoclinic and tetragonal systems.

Although crystals belong to the same point group or symmetry class, they can adopt different external shapes like cubes and octahedra because external morphology depends on growth conditions and environmental factors beyond internal symmetry. The 32 three-dimensional point groups are categorized into seven crystal systems based on unit-cell geometry, but symmetry elements alone do not determine final crystal shape.