Primitive cell

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In geometry, biology, mineralogy, and solid state physics, a primitive cell is 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.

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In some cases, the full symmetry of a crystal structure is not obvious from the primitive unit 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 whose axes follow the symmetry axes of the crystal structure. The volume of the conventional cell is always an integer multiple (typically 1, 2, 3, or 4) of the primitive cell volume. [1]

The primitive cell is a primitive place. 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

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. [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).

Two dimensions

The parallelogram is the general primitive cell for the plane. Fundamental parallelogram.png
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.

Conventional primitive cell 2d mp.svg 2d op rectangular.svg 2d tp.svg
Shape name Parallelogram Rectangle Square
Bravais latticePrimitive ObliquePrimitive RectangularPrimitive 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 2d oc rhombic.svg
Shape name Rhombus
Conventional cell 2d oc rectangular.svg
Bravais latticeCentered Rectangular

Three dimensions

A parallelepiped is a general primitive cell for 3-dimensional space. Parallelepiped 2013-11-29.svg
A parallelepiped is a general primitive cell for 3-dimensional space.

The primitive translation vectorsa1, 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. [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

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. When the lattice points are on the corner, each lattice point is shared by eight different primitive cells, so each lattice point will contribute only 1/8 of a lattice point to each of those cells. However, there are eight corners, so there is still a total of one lattice point per cell, as required by definition. Some of the fourteen three-dimensional Bravais lattices are represented using such parallelepiped primitive cells, as shown below.

Conventional primitive cell 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 latticePrimitive 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 Clinorhombic prism.svg Rhombic prism.svg
Shape nameOblique rhombic prism Right rhombic prism
Conventional cell Monoclinic-base-centered.svg Orthorhombic-base-centered.svg
Bravais latticeBase-centered Monoclinic Base-centered Orthorhombic

Wigner–Seitz cell

An alternative to the unit cell, 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.

See also

Notes

  1. In n dimensions the crystal translation vector would be
    That is, for a point in the lattice r, the arrangement of points appears the same from r′ = r + T as from r.

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Lattice (group)

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Monoclinic crystal system one of the 7 crystal systems in crystallography

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Brillouin zone Primitive cell in the reciprocal space lattice of crystals

In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone.

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Reciprocal lattice Fourier transform of real-space lattices, important in solid-state physics

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Translational symmetry

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Supercell (crystal)

In solid-state physics and crystallography, a crystal structure is described by a unit cell. There are an infinite number of unit cells with different shapes and sizes which can describe the same crystal. Say that a crystal structure is described by a unit cell U. The supercell S of unit cell U is a cell which describes the same crystal, but has larger volume than cell U. Many methods which use a supercell perturbate it somehow to determine properties which cannot be determined by the initial cell. For example, during phonon calculations by the small displacement method, phonon frequencies in crystals are calculated using force values on slightly displaced atoms in the supercell. Another very important example of a supercell is the conventional cell of body-centered (bcc) or face-centered (fcc) cubic crystals.

The Pearson symbol, or Pearson notation, is used in crystallography as a means of describing a crystal structure, and was originated by W.B. Pearson. The symbol is made up of two letters followed by a number. For example:

Hexagonal crystal family

In crystallography, the hexagonal crystal family is one of the six crystal families, which includes two crystal systems and two lattice systems. While commonly confused, the trigonal crystal system and the rhombohedral lattice system are not equivalent. In particular, there are crystals with trigonal symmetry but belong to the hexagonal lattice.

In geometry a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.

References

  1. Ashcroft, Neil W. (1976). Solid State Physics . W. B. Saunders Company. p. 73. 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.