In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, a fractional coordinate system (crystal coordinate system) is a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
in which the basis vectors used to the describe the space are the lattice vectors of a crystal (periodic) pattern. The selection of an origin and a basis define a unit cell, a parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped (3D) in higher dimensions) defined by the lattice basis vectors
where
is the dimension of the space. These basis vectors are described by lattice parameters (lattice constants) consisting of the lengths of the lattice basis vectors
and the angles between them
.
Most cases in crystallography involve two- or three-dimensional space in which the basis vectors
are commonly displayed as
with their lengths and angles denoted by
and
respectively.
:
Crystal Structure
A
crystal structure
In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...
is defined as the spatial distribution of the atoms within a crystal, usually modeled by the idea of an infinite crystal pattern. An infinite crystal pattern refers to the infinite 3D periodic array which corresponds to a crystal, in which the lengths of the periodicities of the array may not be made arbitrarily small. The geometrical shift which takes a crystal structure coincident with itself is termed a symmetry translation (translation) of the crystal structure. The vector which is related to this shift is called a translation vector
. Since a crystal pattern is periodic, all integer linear combinations of translation vectors are also themselves translation vectors,
Lattice
The vector
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
(lattice)
is defined as the infinite set consisting of all of the translation vectors of a crystal pattern. Each of the vectors in the vector lattice are called lattice vectors. From the vector lattice it is possible to construct a point lattice. This is done by selecting an origin
with position vector
. The endpoints
of each of the vectors
make up the point lattice of
and
. Each point in a point lattice has periodicity i.e., each point is identical and has the same surroundings. There exist an infinite number of point lattices for a given vector lattice as any arbitrary origin
can be chosen and paired with the lattice vectors of the vector lattice. The points or particles that are made coincident with one another through a translation are called translation equivalent.
Coordinate systems
General coordinate systems
Usually when describing a space geometrically, a
coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
is used which consists of a choice of origin and a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of
linearly independent, non-coplanar basis vectors
, where
is the dimension of the space being described. With reference to this coordinate system, each point in the space can be specified by
coordinates (a coordinate
-tuple). The origin has coordinates
and an arbitrary point has coordinates
. The position vector
is then,
In
-dimensions, the lengths of the basis vectors are denoted
and the angles between them
. However, most cases in crystallography involve two- or three-dimensional space in which the basis vectors
are commonly displayed as
with their lengths and angles denoted by
and
respectively.
Cartesian coordinate system
A widely used coordinate system is the
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
, which consists of
orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...
basis vectors. This means that,
and
However, when describing objects with crystalline or periodic structure a Cartesian coordinate system is often not the most useful as it does not often reflect the symmetry of the lattice in the simplest manner.
Fractional (crystal) coordinate system
In
crystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...
, a fractional coordinate system is used in order to better reflect the symmetry of the underlying lattice of a crystal pattern (or any other periodic pattern in space). In a fractional coordinate system the basis vectors of the coordinate system are chosen to be lattice vectors and the basis is then termed a crystallographic basis (or lattice basis).
In a lattice basis, any lattice vector
can be represented as,
There are an infinite number of lattice bases for a crystal pattern. However, these can be chosen in such a way that the simplest description of the pattern can be obtained. These bases are used in the International Tables of Crystallography Volume A and are termed conventional bases. A lattice basis
is called primitive if the basis vectors are lattice vectors and all lattice vectors
can be expressed as,
However, the conventional basis for a crystal pattern is not always chosen to be primitive. Instead, it is chosen so the number of orthogonal basis vectors is maximized. This results in some of the coefficients of the equations above being fractional. A lattice in which the conventional basis is primitive is called a primitive lattice, while a lattice with a non-primitive conventional basis is called a centered lattice.
The choice of an origin and a basis implies the choice of a unit cell which can further be used to describe a crystal pattern. The unit cell is defined as the parallelotope (i.e., generalization of a parallelogram (2D) or parallelepiped (3D) in higher dimensions) in which the coordinates of all points are such that,
.
Furthermore, points outside of the unit cell can be transformed inside of the unit cell through standardization, the addition or subtraction of integers to the coordinates of points to ensure
. In a fractional coordinate system, the lengths of the basis vectors
and the angles between them
are called the lattice parameters (lattice constants) of the lattice. In two- and three-dimensions, these correspond to the lengths and angles between the edges of the unit cell.
The fractional coordinates of a point in space
in terms of the lattice basis vectors is defined as,
Calculations involving the unit cell
General transformations between fractional and Cartesian coordinates
Three Dimensions
The relationship between fractional and
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be described by the matrix transformation
:
Similarly, the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be converted back to fractional coordinates using the matrix transformation
:
Transformations using the cell tensor
Another common method of converting between fractional and Cartesian coordinates involves the use of a cell tensor
which contains each of the basis vectors of the space expressed in
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
.
Two Dimensions
= Cell tensor
=
In
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
the 2 basis vectors are represented by a
cell tensor
:
The area of the
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. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
,
, is given by the determinant of the cell matrix:
For the special case of a square or rectangular
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. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
, the matrix is diagonal, and we have that:
= Relationship between fractional and Cartesian coordinates
=
The relationship between fractional and
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be described by the matrix transformation
:
Similarly, the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be converted back to fractional coordinates using the matrix transformation
:
Three Dimensions
= Cell tensor
=
In
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
the 3 basis vectors are represented by a
cell tensor
:
The volume of the
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. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
,
, is given by the determinant of the cell tensor:
For the special case of a cubic, tetragonal, or orthorhombic cell, the matrix is diagonal, and we have that:
= Relationship between fractional and Cartesian coordinates
=
The relationship between fractional and
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be described by the matrix transformation
:
Similarly, the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be converted back to fractional coordinates using the matrix transformation
:
Arbitrary number of dimensions
= Cell tensor
=
In
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
the
basis vectors are represented by a
cell tensor
:
The hypervolume of the
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. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
,
, is given by the determinant of the cell tensor:
= Relationship between fractional and Cartesian coordinates
=
The relationship between fractional and
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be described by the matrix transformation
:
Similarly, the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
can be converted back to fractional coordinates using the transformation
:
Determination of cell properties in two and three dimensions using the metric tensor
The metric tensor
is sometimes used for calculations involving the unit cell and is defined (in matrix form) as:
In two dimensions,
In three dimensions,
The distance between two points
and
in the unit cell can be determined from the relation:
The distance from the origin of the unit cell to a point
within the unit cell can be determined from the relation:
The angle formed from three points
,
(apex), and
within the unit cell can determined from the relation:
The volume of the unit cell,
can be determined from the relation:
References
{{DEFAULTSORT:Fractional Coordinates
Crystallography
Coordinate systems