Primitive cell

In geometry, crystallography, mineralogy, and solid state physics, a primitive cell is a minimum volume cell (a unit cell) corresponding to a single lattice point of a structure with discrete translational symmetry. The concept is used particularly in describing crystal structure in two and three dimensions, though it makes sense in all dimensions. A lattice can be characterized by the geometry of its primitive cell.

The primitive cell is a primitive unit. A primitive unit is a section of the tiling (usually a parallelogram or a set of neighboring tiles) that generates the whole tiling using only translations, and is as small as possible.

The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.

The primitive translation vectors $a \to 1$ , $a \to 2$ , $a \to 3$ span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

{\vec {T}}=u_{1}{\vec {a}}_{1}+u_{2}{\vec {a}}_{2}+u_{3}{\vec {a}}_{3},

where $u 1$ , $u 2$ , $u 3$ are integers, translation by which leaves the lattice invariant.^[1] That is, for a point in the lattice $r$ , the arrangement of points appears the same from $r' = r + T \to$ as from $r$ .^[2]

Since the primitive cell is defined by the primitive axes (vectors) $a \to 1$ , $a \to 2$ , $a \to 3$ , the volume $V p$ of the primitive cell is given by the parallelepiped from the above axes as

V_{\mathrm {p} }=\left|{\vec {a}}_{1}\cdot ({\vec {a}}_{2}\times {\vec {a}}_{3})\right|.

A primitive cell is considered to contain 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]

A Wigner–Seitz cell is a primitive cell centered on the single lattice point it contains. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone.

2-dimensional primitive cell

A 2-dimensional primitive cell is a parallelogram, which in special cases may have orthogonal angles, or equal lengths, or both.

2-dimensional primitive cells
Parallelogram (Monoclinic)	Rhombus (Orthorhombic)	Rectangle (Orthorhombic)	Square (Tetragonal)

3-dimensional primitive cell

A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the basis). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.

A 3-dimensional primitive cell is a parallelepiped, which in special cases may have orthogonal angles, or equal lengths, or both.

3-dimensional primitive cells
Parallelepiped (Triclinic)	Clinorhombic prism (Monoclinic)	Right parallelogrammic prism (Monoclinic)	Right rhombic prism (Orthorhombic)	Rectangular cuboid (Orthorhombic)	Square cuboid (Tetragonal)	Trigonal trapezohedron (Rhombohedral)	Cube (Cubic)

Notes

^ In $n$ dimensions the crystal translation vector would be
${\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.$
^ Kittel, Charles (2004). Introduction to Solid State Physics (8th ed.). p. 4.^{[ISBN missing]}
^ "DoITPoMS - TLP Library Crystallography - Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015.

[n-1] In $n$ dimensions the crystal translation vector would be
${\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.$

[Kittel-2] Kittel, Charles (2004). Introduction to Solid State Physics (8th ed.). p. 4.^{[ISBN missing]}

[doitpoms-3] "DoITPoMS - TLP Library Crystallography - Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015.

[1]

[2]

[3]

2-dimensional primitive cell

3-dimensional primitive cell

See also

Notes