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If the lattice or crystal is 2-dimensional, the primitive cell has a minimum area; likewise in 3 dimensions the primitive cell has a minimum volume. Despite this rigid minimum-size requirement, there is not one unique choice of primitive unit cell. In fact, all cells whose borders are primitive translation vectors will be primitive unit cells.
A primitive cell is a unit cell that contains exactly one lattice point. For unit cells generally, lattice points that are shared by n cells are counted as 1 / n of the lattice points contained in each of those cells; so for example a primitive unit cell in three dimensions which has lattice points only at its eight vertices is considered to contain 1 / 8 of each of them. [3]
Instead, it is chosen so the number of orthogonal basis vectors is maximized. This results in some of the coefficients of the equations above being fractional. A lattice in which the conventional basis is primitive is called a primitive lattice, while a lattice with a non-primitive conventional basis is called a centered lattice.
Vectors and are primitive translation vectors. The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. [ 1 ] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
where n 1, n 2, and n 3 are integers and a 1, a 2, and a 3 are three non-coplanar vectors, called primitive vectors. These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only.
For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.
The primitive rectangular lattice can also be described by a centered rhombic unit cell, while the centered rectangular lattice can also be described by a primitive rhombic unit cell. Note that the length a {\displaystyle a} in the lower row is not the same as in the upper row.
P primitive; I body centered (from the German Innenzentriert) F face centered (from the German Flächenzentriert) A centered on A faces only; B centered on B faces only; C centered on C faces only; R rhombohedral; A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the ...