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] Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessarily have unit size, or even a particular size at all. Rather, the primitive cell is the closest analogy to a unit vector, since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed.
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, which is a section of the tiling (a parallelogram or parallelepiped) that generates the whole tiling using only translations.
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, it is the smallest possible unit cell. 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 a unit cell with the full symmetry of the lattice and may include more than one lattice point. The conventional unit cells are parallelotopes in n dimensions.
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.[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 vectorsa→1, a→2, a→3 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.[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) a→1, a→2, a→3, the volume Vp of the primitive cell is given by the parallelepiped from the above axes as
Usually, primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds, with an atom at each corner of the cell. This choice of primitive cell is not unique, but volume of primitive cells will always be given by the expression above.[4]
In addition to the parallelepiped primitive cells, for every Bravais lattice there is another kind of primitive 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.
Conventional cell
For each particular lattice, a conventional cell has been chosen on a case-by-case basis by crystallographers based on convenience of calculation.[5] These 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 (1, 2, 3, or 4) of that of the primitive cell.[6]
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, equal lengths, or both. Four of the five two-dimensional Bravais lattices are represented using conventional primitive cells, as shown below.
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.
A parallelepiped is a general primitive cell for 3-dimensional space.
For any 3-dimensional lattice, the conventional unit cells are parallelepipeds, which in special cases may have orthogonal angles, or equal lengths, or both. Seven of the fourteen three-dimensional Bravais lattices are represented using conventional primitive cells, as shown below.
The other seven Bravais lattices (known as the centered 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.
^Mehl, Michael J.; Hicks, David; Toher, Cormac; Levy, Ohad; Hanson, Robert M.; Hart, Gus; Curtarolo, Stefano (2017). "The AFLOW Library of Crystallographic Prototypes: Part 1". Computational Materials Science. Elsevier BV. 136: S1–S828. arXiv:1806.07864. doi:10.1016/j.commatsci.2017.01.017. ISSN 0927-0256. S2CID 119490841.
^Aroyo, M. I., ed. (2016-12-31). International Tables for Crystallography. Chester, England: International Union of Crystallography. p. 25. doi:10.1107/97809553602060000114. ISBN978-0-470-97423-0.
^Ashcroft, Neil W. (1976). Solid State Physics. W. B. Saunders Company. p. 73. ISBN0-03-083993-9.
January 07, 2023
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This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Unit cell news newspapers books scholar JSTOR May 2021 Learn how and when to remove this template message 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 Despite its suggestive name the unit cell unlike a unit vector for example does not necessarily have unit size or even a particular size at all Rather the primitive cell is the closest analogy to a unit vector since it has a determined size for a given lattice and is the basic building block from which larger cells are constructed 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 which is a section of the tiling a parallelogram or parallelepiped that generates the whole tiling using only translations 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 it is the smallest possible unit cell 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 a unit cell with the full symmetry of the lattice and may include more than one lattice point The conventional unit cells are parallelotopes in n dimensions Contents 1 Primitive cell 1 1 Wigner Seitz cell 2 Conventional cell 3 Two dimensions 4 Three dimensions 5 See also 6 Notes 7 ReferencesPrimitive cell EditA 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 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 a 1 a 2 a 3 span a lattice cell of smallest volume for a particular three dimensional lattice and are used to define a crystal translation vector T u 1 a 1 u 2 a 2 u 3 a 3 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 a 1 a 2 a 3 the volume Vp of the primitive cell is given by the parallelepiped from the above axes as V p a 1 a 2 a 3 displaystyle V mathrm p left vec a 1 cdot vec a 2 times vec a 3 right Usually primitive cells in two and three dimensions are chosen to take the shape parallelograms and parallelepipeds with an atom at each corner of the cell This choice of primitive cell is not unique but volume of primitive cells will always be given by the expression above 4 Wigner Seitz cell Edit Main article Wigner Seitz cell In addition to the parallelepiped primitive cells for every Bravais lattice there is another kind of primitive 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 Conventional cell EditFor each particular lattice a conventional cell has been chosen on a case by case basis by crystallographers based on convenience of calculation 5 These 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 1 2 3 or 4 of that of the primitive cell 6 Two dimensions Edit 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 equal lengths or both Four of the five two dimensional Bravais lattices are represented using conventional primitive cells as shown below Conventional primitive cell Shape name Parallelogram Rectangle Square RhombusBravais lattice Primitive Oblique Primitive Rectangular Primitive Square Primitive HexagonalThe 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 Shape name RhombusConventional cell Bravais lattice Centered RectangularThree dimensions Edit A parallelepiped is a general primitive cell for 3 dimensional space For any 3 dimensional lattice the conventional unit cells are parallelepipeds which in special cases may have orthogonal angles or equal lengths or both Seven of the fourteen three dimensional Bravais lattices are represented using conventional primitive cells as shown below Conventional primitive cell Shape name Parallelepiped Oblique rectangular prism Rectangular cuboid Square cuboid Trigonal trapezohedron Cube Right rhombic prismBravais lattice Primitive Triclinic Primitive Monoclinic Primitive Orthorhombic Primitive Tetragonal Primitive Rhombohedral Primitive Cubic Primitive HexagonalThe other seven Bravais lattices known as the centered 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 prismConventional cell Bravais lattice Base centered Monoclinic Base centered Orthorhombic Body centered Orthorhombic Face centered Orthorhombic Body centered Tetragonal Body centered Cubic Face centered CubicSee also EditWigner Seitz cell Bravais lattice Wallpaper group Space groupNotes Edit In n dimensions the crystal translation vector would be T i 1 n u i a i where u i Z i 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 Edit Ashcroft Neil W 1976 Chapter 4 Solid State Physics W B Saunders Company p 72 ISBN 0 03 083993 9 DoITPoMS TLP Library Crystallography Unit Cell Online Materials Science Learning Resources DoITPoMS University of Cambridge Retrieved 21 February 2015 Kittel Charles 11 November 2004 Introduction to Solid State Physics 8 ed Wiley p 4 ISBN 978 0 471 41526 8 Mehl Michael J Hicks David Toher Cormac Levy Ohad Hanson Robert M Hart Gus Curtarolo Stefano 2017 The AFLOW Library of Crystallographic Prototypes Part 1 Computational Materials Science Elsevier BV 136 S1 S828 arXiv 1806 07864 doi 10 1016 j commatsci 2017 01 017 ISSN 0927 0256 S2CID 119490841 Aroyo M I ed 2016 12 31 International Tables for Crystallography Chester England International Union of Crystallography p 25 doi 10 1107 97809553602060000114 ISBN 978 0 470 97423 0 Ashcroft Neil W 1976 Solid State Physics W B Saunders Company p 73 ISBN 0 03 083993 9 Retrieved from https en wikipedia org w index php title Unit cell amp oldid 1129548772, wikipedia, wiki, book, books, library,