In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes, an included angle of 120° and a height perpendicular to the two base axes. The hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell. There are two ways to do this, which can be thought of as two notations which represent the same structure. In the usual so-called obverse setting, the additional lattice points are at coordinates and, whereas in the alternative reverse setting they are at the coordinates and. In either case, 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. 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 in textbooks because this cell reveals m symmetry of crystal lattice. The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates and. However, such a description is rarely used.
Crystal systems
The hexagonal crystal family consists of two crystal systems: trigonal and hexagonal. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis, which includes space groups 143 to 167. These 5 point groups have 7 corresponding space groups assigned to the rhombohedral lattice system and 18 corresponding space groups assigned to the hexagonal lattice system. The hexagonal crystal system consists of the 7 point groups that have a single six-fold rotation axis. These 7 point groups have 27 space groups, all of which are assigned to the hexagonal lattice system. Graphite is an example of a crystal that crystallizes in the hexagonal crystal system.
Crystal classes
Trigonal crystal system
The trigonal crystal system is the only crystal system whose point groups have more than one lattice system associated with their space groups: the hexagonal and rhombohedral lattices both appear. The 5 point groups in this crystal system are listed below, with their international number and notation, their space groups in name and example crystals.
Hexagonal close packed is one of the two simple types of atomic packing with the highest density, the other being the face centered cubic. However, unlike the fcc, it is not a Bravais lattice as there are two nonequivalent sets of lattice points. Instead, it can be constructed from the hexagonal Bravais lattice by using a two atom motif associated with each lattice point.
Example: Quartz
is a crystal that belongs to the hexagonal lattice system but exists in two polymorphs that are in two different crystal systems. The crystal structures of α-quartz are described by two of the 18 space groups associated with the trigonal crystal system, while the crystal structures of β-quartz are described by two of the 27 space groups associated with the hexagonal crystal system.
Rhombohedral lattice angle
The lattice angles and the lengths of the lattice vectors are all the same for both the cubic and rhombohedral lattice systems. The lattice angles for simple cubic, face-centered cubic, and body-centered cubic lattices are /2 radians, /3 radians, and radians, respectively. A rhombohedral lattice will result from lattice angles other than these.
