Primitive cell

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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.

Overview[edit]

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.

By definition, a primitive cell must contain exactly one and only 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.[1]

Two dimensions[edit]

The parallelogram is the general primitive cell for the plane.

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
Wallpaper group diagram p1.svg
Parallelogram
(Monoclinic)
Wallpaper group diagram p1 rhombic.svg
Rhombus
(Orthorhombic)
Wallpaper group diagram p1 rect.svg
Rectangle
(Orthorhombic)
Wallpaper group diagram p1 square.svg
Square
(Tetragonal)

Three dimensions[edit]

Parallelepiped[edit]

A parallelepiped is a general primitive cell for 3-dimensional space.

The primitive translation vectors a1, a2, a3 span a lattice cell of smallest volume for a particular three-dimensional lattice, and are used to define a crystal translation vector

where u1, u2, u3 are integers, translation by which leaves the lattice invariant.[2] That is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T as from r.[3]

Since the primitive cell is defined by the primitive axes (vectors) a1, a2, a3, the volume Vp of the primitive cell is given by the parallelepiped from the above axes as

For any 3-dimensional lattice, you can find primitive cells which are parallelepipeds, which in special cases may have orthogonal angles, or equal lengths, or both. While not mathematically required, by convention, one usually defines the parallelepiped primitive cell so that there is a lattice point on each corner. Some of the fourteen three-dimensional Bravais lattices are represented using such parallelepiped primitive cells, as shown below.

Conventional 3D primitive cells
Triclinic.svg Monoclinic.svg Orthorhombic.svg Tetragonal.svg Rhombohedral.svg Cubic.svg
Shape name Parallelepiped Oblique rectangular prism Rectangular cuboid Square cuboid Trigonal trapezohedron Cube
Bravais lattice Primitive Triclinic Primitive Monoclinic Primitive Orthorhombic Primitive Tetragonal Primitive Rhombohedral Primitive Cubic

The other Bravais lattices also have a primitive unit cell in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are conventionally represented by a non-primitive unit cell which contains more than one lattice point.

Non-conventional 3D primitive cells
Rhombic prism.svg Clinorhombic prism.svg
Shape name Right rhombic prism Oblique rhombic prism
Bravais lattice Base-centered Orthorhombic Base-centered Monoclinic

Wigner-Seitz cell[edit]

An alternative to the parallelepiped, there is another primitive cell called the Wigner-Seitz cell. Rather than placing that lattice points on the corners of the cell, a Wigner–Seitz cell is 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.

See also[edit]

Notes[edit]

  1. ^ "DoITPoMS - TLP Library Crystallography - Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015. 
  2. ^ In n dimensions the crystal translation vector would be
  3. ^ Kittel, Charles (2004). Introduction to Solid State Physics (8th ed.). p. 4. [ISBN missing]