Trigonal crystal system
In crystallography, the hexagonal crystal family is one of the 6 crystal families. In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes (a by a), an included angle of 120° (γ) and a height (c, which can be different from a) perpendicular to the two base axes.
The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system. There are 52 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral.
The hexagonal crystal family consists of two lattice systems: hexagonal and rhombohedral. Each lattice system consists of one Bravais lattice.
The conventional cell for the rhombohedral Bravais lattice is the rhombohedrally-centered (R-centered) hexagonal cell, consisting of two additional lattice points which occupy the longest body diagonal of the unit cell with coordinates ( 2⁄3, 1⁄3, 1⁄3) and ( 1⁄3, 2⁄3, 2⁄3). Hence, there are 3 lattice points per unit cell in total and the lattice is non-primitive.
The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes. The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice system). This is a unit cell with parameters a = b = c; α = β = γ ≠ 90°. In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. However, the rhombohedral axes are often shown (for the rhombohedral lattice system) in textbooks because this cell reveals 3m symmetry of crystal lattice.
The unit cell for the hexagonal Bravais lattice in rhombohedral axes, consisting of two additional lattice points which occupy the longest body diagonal of the unit cell with coordinates ( 1⁄3, 1⁄3, 1⁄3) and ( 2⁄3, 2⁄3, 2⁄3), is called the hexagonally-centered (D-centered) rhombohedral cell. However, such a description is rarely used.
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