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In
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state phy ...
and
crystallography Crystallography is the branch of science devoted to the study of molecular and crystalline structure and properties. The word ''crystallography'' is derived from the Ancient Greek word (; "clear ice, rock-crystal"), and (; "to write"). In J ...
, a
crystal structure In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...
is described by a unit cell repeating periodically over space. There are an infinite number of choices for unit cells, with different shapes and sizes, which can describe the same crystal, and different choices can be useful for different purposes. Say that a crystal structure is described by a unit cell U. Another unit cell S is a supercell of unit cell U, if S is a cell which describes the same crystal, but has a 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.


Unit cell transformation

The basis vectors of unit cell U (\vec,\vec,\vec) can be transformed to basis vectors of supercell S (\vec',\vec',\vec') by linear transformation \begin \vec' & \vec' & \vec' \\ \end = \begin \vec & \vec & \vec \\ \end \hat= \begin \vec & \vec & \vec \\ \end \begin P_ & P_ & P_ \\ P_ & P_ & P_ \\ P_ & P_ & P_ \\ \end where \hat is a
transformation matrix In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping \mathbb^n to \mathbb^m and \mathbf x is a column vector with n entries, then there exists an m \times n matrix A, called the transfo ...
. All elements P_ should be integers with \det(\hat) > 1 (with \det(\hat) = 1 the transformation preserves volume). For example, the matrix P_= \begin 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end transforms a primitive cell to body-centered. Another particular case of the transformation is a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagon ...
(i.e., P_=0). This called diagonal supercell expansion and can be represented as repeating of the initial cell over crystallographic axes of the initial cell.


Application

Supercells are also commonly used in computational models of
crystal defect A crystallographic defect is an interruption of the regular patterns of arrangement of atoms or molecules in crystalline solids. The positions and orientations of particles, which are repeating at fixed distances determined by the unit cell par ...
s to allow the use of periodic
boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
s.


See also

*
Crystal structure In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...
*
Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 ...
* Primitive cell *
Space group In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...


References


External links


IUCR online dictionary of crystallography
Crystallography {{Crystallography-stub