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Unit cell

In geometry, biology, mineralogy, and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice.[1] The geometry of the unit cell is defined as a parallelotope in n dimensions.

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 unit cell. The unit cell is a section of the tiling (a parallelogram or parallelepiped) that generates the whole tiling using only translations, and is as small as possible.

There are two special cases of the unit cell: the primitive cell and the conventional cell. The primitive cell is a unit cell corresponding to a single lattice point. In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is the smallest unit cell with the full symmetry of the lattice and may include more than one lattice point.

Primitive cell

A primitive cell is a unit cell that contains only one and 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.[2] An alternative conceptualization is to consistently pick only one of the n lattice points to belong to the given unit cell (so the other 1-n lattice points belong to adjacent unit cells).

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

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

where u1, u2, u3 are integers, translation by which leaves the lattice invariant.[note 1] 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

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

Conventional cell

A conventional cell is the smallest unit cell whose axes follow the symmetry axes of the crystal structure. Such conventional cells may have additional lattice points located in the middle of the faces or body of the unit cell. The number of lattice points, as well as the volume, of the conventional cell is an integer multiple (typically 1, 2, 3, or 4) of that of the primitive cell.[4]

Two dimensions

The parallelogram is the general primitive cell for the plane.

For any 2-dimensional lattice, the unit cells are parallelograms, which in special cases may have orthogonal angles, or equal lengths, or both. Some of the five two-dimensional Bravais lattices are represented using conventional primitive cells, as shown below.

Conventional primitive cell Shape name Bravais lattice Parallelogram Rectangle Square Primitive Oblique Primitive Rectangular Primitive Square

The centered rectangular lattice also has a primitive cell in the shape of a rhombus, but in order to allow easy discrimination on the basis of symmetry, it is represented by a conventional cell which contains two lattice points.

Primitive cell Rhombus Centered Rectangular

Three dimensions

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

For any 3-dimensional lattice, the unit cells are parallelepipeds, which in special cases may have orthogonal angles, or equal lengths, or both. Some of the fourteen three-dimensional Bravais lattices are represented using conventional primitive cells, as shown below.

 Conventional primitive cell Shape name Bravais lattice Parallelepiped Oblique rectangular prism Rectangular cuboid Square cuboid Trigonal trapezohedron Cube Primitive Triclinic Primitive Monoclinic Primitive Orthorhombic Primitive Tetragonal Primitive Rhombohedral Primitive Cubic

The other Bravais lattices also have primitive cells in the shape of a parallelepiped, but in order to allow easy discrimination on the basis of symmetry, they are represented by conventional cells which contain more than one lattice point.

Primitive cell Shape name Oblique rhombic prism Right rhombic prism Base-centered Monoclinic Base-centered Orthorhombic

Wigner–Seitz cell

An alternative to the unit cell, for every Bravais lattice there is another kind of cell called the Wigner–Seitz cell. In the Wigner–Seitz cell, the lattice point is at the center of the cell, and for most Bravais lattices, the shape is not a parallelogram or parallelepiped. This is a type of Voronoi cell. The Wigner–Seitz cell of the reciprocal lattice in momentum space is called the Brillouin zone.

Notes

1. ^ In n dimensions the crystal translation vector would be
${\displaystyle {\vec {T}}=\sum _{i=1}^{n}u_{i}{\vec {a}}_{i},\quad {\mbox{where }}u_{i}\in \mathbb {Z} \quad \forall i.}$
That is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T as from r.

References

1. ^ Ashcroft, Neil W. (1976). "Chapter 4". Solid State Physics. W. B. Saunders Company. p. 72. ISBN 0-03-083993-9.
2. ^ "DoITPoMS – TLP Library Crystallography – Unit Cell". Online Materials Science Learning Resources: DoITPoMS. University of Cambridge. Retrieved 21 February 2015.
3. ^ Kittel, Charles. Introduction to Solid State Physics (8 ed.). Wiley. p. 4. ISBN 978-0-471-41526-8.
4. ^ Ashcroft, Neil W. (1976). Solid State Physics. W. B. Saunders Company. p. 73. ISBN 0-03-083993-9.