Fractional coordinates

A crystal structure 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 \mathbf{t} . Since a crystal pattern is periodic, all integer linear combinations of translation vectors are also themselves translation vectors, \mathbf{t} = c_1\mathbf{t}_1+c_2\mathbf{t}_2 \text { where } c_1, c_2 \in \mathbb{Z} == Lattice ==

The vector lattice (lattice) \mathbf{T} 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 X_0 with position vector \mathbf{x}_0. The endpoints X_i of each of the vectors \mathbf{x}_i = \mathbf{x}_0 + \mathbf{t}_i make up the point lattice of X_0 and \mathbf{T}. 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 X_0 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 is used which consists of a choice of origin and a basis of d linearly independent, non-coplanar basis vectors \mathbf{a}_1, \mathbf{a}_2, \dots, \mathbf{a}_d , where d is the dimension of the space being described. With reference to this coordinate system, each point in the space can be specified by d coordinates (a coordinate d-tuple). The origin has coordinates (0, 0,\dots,0) and an arbitrary point has coordinates (x_1,x_2,...,x_d). The position vector \vec{OP} is then, \vec{OP} = \mathbf{x} = \sum_{i=1}^{d} x_i\mathbf{a}_i In d-dimensions, the lengths of the basis vectors are denoted a_1, a_2, \dots, a_d and the angles between them \alpha_1, \alpha_2, \dots, \alpha_{\frac{d(d-1)}{2}}. However, most cases in crystallography involve two- or three-dimensional space in which the basis vectors \mathbf {a}_1, \mathbf {a}_2, \mathbf {a}_3 are commonly displayed as \mathbf{a}, \mathbf{b}, \mathbf{c} with their lengths and angles denoted by a, b, c and \alpha, \beta, \gamma respectively. Cartesian coordinate system A widely used coordinate system is the Cartesian coordinate system, which consists of orthonormal basis vectors. This means that, a_1 = |\mathbf{a}_1| = a_2 = |\mathbf{a}_2| = \dots = a_d = |\mathbf{a}_d| = 1 and \alpha_1 = \alpha_2 = \dots = \alpha_{\frac{d(d-1)}{2}} = 90^\circ 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, 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 \mathbf{t} can be represented as, \mathbf{t} = \sum_{i=1}^{d} c_i\mathbf{a}_i \text{ where } c_i \in \mathbb{Q} 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 \mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_d is called primitive if the basis vectors are lattice vectors and all lattice vectors \mathbf{t} can be expressed as, \mathbf{t} = \sum_{i=1}^{d} c_i \mathbf{a}_i \text{ where } c_i \in \mathbb{Z} 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, 0 \leq x_1,x_2,\dots,x_d . 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 0 \leq x_1,x_2,\dots,x_d . In a fractional coordinate system, the lengths of the basis vectors \mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_d and the angles between them \alpha_1, \alpha_2, \dots, \alpha_{\frac{d(d-1)}{2}} 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 \rho = (\rho_{x_1}, \rho_{x_2}, \dots, \rho_{x_d}) in terms of the lattice basis vectors is defined as, \rho = \rho_{x_1}\mathbf{a}_1 + \rho_{x_2}\mathbf{a}_2 + \dots + \rho_{x_d}\mathbf{a}_d \text{ where } \rho \in [0,1) == Calculations involving the unit cell ==

General transformations between fractional and Cartesian coordinates Three Dimensions The relationship between fractional and Cartesian coordinates can be described by the matrix transformation \mathbf{r} = \mathbf{A}\boldsymbol\rho . Letting , , , , , and : \begin{pmatrix} r_{x_1} \\ r_{x_2} \\ r_{x_3} \end{pmatrix} = \begin{pmatrix} a_1s_1^{-1} \sqrt{(s_1s_2)^2-(c_3 - c_1c_2)^2} & 0 & 0 \\ a_1s_1^{-1}(c_3 - c_1c_2) & a_2 s_1 & 0 \\ a_1 c_2 & a_2 c_1 & a_3 \\ \end{pmatrix} \begin{pmatrix} \rho_{x_1} \\ \rho_{x_2} \\ \rho_{x_3} \end{pmatrix} Similarly, the Cartesian coordinates can be converted back to fractional coordinates using the matrix transformation \boldsymbol\rho = \mathbf{A}^{-1}\mathbf{r} : \mathbf{h} = \begin{pmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{pmatrix}^\operatorname{T} = \begin{pmatrix} a_{1,x_1} & a_{1,x_2} \\ a_{2,x_1} & a_{2,x_2} \end{pmatrix} The area of the unit cell, A, is given by the determinant of the cell matrix: A = \det(\mathbf{h}) = a_{1,x_1}a_{2,x_2} - a_{1,x_2}a_{2,x_2} For the special case of a square or rectangular unit cell, the matrix is diagonal, and we have that: A = \det(\mathbf{h}) = a_{1,x_1}a_{2,x_2} Relationship between fractional and Cartesian coordinates The relationship between fractional and Cartesian coordinates can be described by the matrix transformation \mathbf{r} = \mathbf{h}\boldsymbol\rho : In two dimensions, \mathbf{G} = \begin{pmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{pmatrix} = \begin{pmatrix} \mathbf{a}_1\cdot\mathbf{a}_1 & \mathbf{a}_1\cdot\mathbf{a}_2 \\ \mathbf{a}_2\cdot\mathbf{a}_1 & \mathbf{a}_2\cdot\mathbf{a}_2 \end{pmatrix} = \begin{pmatrix} a_1^2 & a_1a_2\cos(\alpha_1) \\ a_1a_2\cos(\alpha_1) & a_2^2 \end{pmatrix} In three dimensions, \mathbf{G} = \begin{pmatrix} g_{11} & g_{12} & g_{13} \\ g_{21} & g_{22} & g_{23} \\ g_{31} & g_{32} & g_{33} \end{pmatrix} = \begin{pmatrix} \mathbf{a}_1\cdot\mathbf{a}_1 & \mathbf{a}_1\cdot\mathbf{a}_2 & \mathbf{a}_1\cdot\mathbf{a}_3 \\ \mathbf{a}_2\cdot\mathbf{a}_1 & \mathbf{a}_2\cdot\mathbf{a}_2 & \mathbf{a}_2\cdot\mathbf{a}_3 \\ \mathbf{a}_3\cdot\mathbf{a}_1 & \mathbf{a}_3\cdot\mathbf{a}_2 & \mathbf{a}_3\cdot\mathbf{a}_3 \end{pmatrix} = \begin{pmatrix} a_1^2 & a_1a_2\cos(\alpha_3) & a_1a_3\cos(\alpha_2) \\ a_1a_2\cos(\alpha_3) & a_2^2 & a_2a_3\cos(\alpha_1) \\ a_1a_3\cos(\alpha_2) & a_2a_3\cos(\alpha_1) & a_3^2 \end{pmatrix} The distance between two points Q and R in the unit cell can be determined from the relation: can be used to calculate the distance d between two atoms from the lattice parameters and atomic coordinates: d = \sqrt{(a \Delta x)^2 + (b \Delta y)^2 + (c \Delta z)^2 + 2bc \Delta y \Delta z \cos \alpha + 2ac \Delta x \Delta z \cos \beta + 2ab \Delta x \Delta y \cos \gamma} where \Delta x, \Delta y and \Delta z are the differences between the coordinates of the two atoms. The angle \omega at atom 2 in a group of three atoms 1, 2 and 3 can be calculated from the three distances d_{12}, d_{23} and d_{13} between them according to the cosine formula: \cos \omega = -\sqrt{\frac{d_{13}^2 - d_{12}^2 - d_{23}^2}{2d_{12}d_{23}}} ==References==