Bravais lattice in the context of "Hexagonal crystal family"

Cinnabar (/ˈsɪnəˌbɑːr/; from Ancient Greek κιννάβαρι (kinnábari)), also called cinnabarite (/ˌsɪnəˈbɑːraɪt/) or mercurblende, is the bright scarlet to brick-red form of mercury(II) sulfide (HgS). It is the most common source ore for refining elemental mercury and is the historic source for the brilliant red or scarlet pigment termed vermilion and associated red mercury pigments.

Cinnabar generally occurs as a vein-filling mineral associated with volcanic activity and alkaline hot springs. The mineral resembles quartz in symmetry and it exhibits birefringence. Cinnabar has a mean refractive index near 3.2, a hardness between 2.0 and 2.5, and a specific gravity of approximately 8.1. The color and properties derive from a structure that is a hexagonal crystalline lattice belonging to the trigonal crystal system, crystals that sometimes exhibit twinning.

In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems (hexagonal and trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent (see section crystal systems below). In particular, there are crystals that have trigonal symmetry but belong to the hexagonal lattice (such as α-quartz).

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.

In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of constituent particles to form symmetric patterns that repeat along the principal directions of three-dimensional space in matter.

The smallest group of particles in a material that constitutes this repeating pattern is the unit cell of the structure. The unit cell completely reflects the symmetry and structure of the entire crystal, which is built up by repetitive translation of the unit cell along its principal axes. The translation vectors define the nodes of the Bravais lattice.

In condensed-matter physics, a collision cascade (also known as a displacement cascade or a displacement spike) is a set of nearby adjacent energetic (much higher than ordinary thermal energies) collisions of atoms induced by an energetic particle in a solid or liquid.

If the maximum atom or ion energies in a collision cascade are higher than the threshold displacement energy of the material (tens of eVs or more), the collisions can permanently displace atoms from their lattice sites and produce defects. The initial energetic atom can be, e.g., an ion from a particle accelerator, an atomic recoil produced by a passing high-energy neutron, electron or photon, or be produced when a radioactive nucleus decays and gives the atom a recoil energy.

Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices.

In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers h, k, and ℓ, the Miller indices. They are written (hkℓ), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal to ${\displaystyle \mathbf {g} _{hk\ell }=h\mathbf {b} _{1}+k\mathbf {b} _{2}+\ell \mathbf {b} _{3}}$, where ${\displaystyle \mathbf {b} _{i}}$ are the basis or primitive translation vectors of the reciprocal lattice for the given Bravais lattice. (Note that the plane is not always orthogonal to the linear combination of direct or original lattice vectors ${\displaystyle h\mathbf {a} _{1}+k\mathbf {a} _{2}+\ell \mathbf {a} _{3}}$ because the direct lattice vectors need not be mutually orthogonal.) This is based on the fact that a reciprocal lattice vector ${\displaystyle \mathbf {g} }$ (the vector indicating a reciprocal lattice point from the reciprocal lattice origin) is the wavevector of a plane wave in the Fourier series of a spatial function (e.g., electronic density function) which periodicity follows the original Bravais lattice, so wavefronts of the plane wave are coincident with parallel lattice planes of the original lattice. Since a measured scattering vector in X-ray crystallography, ${\displaystyle \Delta \mathbf {k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }}$ with ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ as the outgoing (scattered from a crystal lattice) X-ray wavevector and ${\displaystyle \mathbf {k} _{\mathrm {in} }}$ as the incoming (toward the crystal lattice) X-ray wavevector, is equal to a reciprocal lattice vector ${\displaystyle \mathbf {g} }$ as stated by the Laue equations, the measured scattered X-ray peak at each measured scattering vector ${\displaystyle \Delta \mathbf {k} }$ is marked by Miller indices.

The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths,

The reciprocal lattice of the hexagonal lattice is a hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length