Fractional coordinates

In crystallography , fractional coordinates form a coordinate system in which the edges of a unit cell are used as basis vectors to represent the position of atoms . A unit cell is a parallelepiped that creates the crystal lattice and can be described using the lengths of the three edges a, b, c and the angles α, β, γ between two edges (see picture).

Definition of the unit cell over the length of the edges a , b , c and the angles α, β, γ

Conversion of fractional to Cartesian coordinates

For the conversion from fractional to Cartesian coordinates , it is assumed that the Cartesian coordinate system is positioned as follows with regard to the unit cell, or the unit cell with regard to the Cartesian coordinate system: The coordinate origins match. The vector is arranged parallel to the x-axis, the vector lies in the xy-plane. The position of the vector then results from the two angles α and β, cf. Image. If the fractional coordinates of a point denote , the Cartesian coordinates are calculated as follows: ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {c}}}$ ${\displaystyle (x',y',z')}$ ${\displaystyle (x,y,z)}$

${\displaystyle {\begin{pmatrix}x\\y\\z\\\end{pmatrix}}={\begin{pmatrix}a&b\cos(\gamma )&c\cos(\beta )\\0&b\sin(\gamma )&c{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}\\0&0&c{\frac {\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}{\sin(\gamma )}}\\\end{pmatrix}}{\begin{pmatrix}x'\\y'\\z'\\\end{pmatrix}}}$

The elements of the matrix can be derived as follows:

The first column corresponds to the definition of the vector . Since this is aligned parallel to the x-axis, its length corresponds to the value of the first element, the other two are zero. ${\displaystyle {\vec {a}}}$

${\displaystyle {\vec {a}}={\begin{pmatrix}|{\vec {a}}|\\0\\0\\\end{pmatrix}}={\begin{pmatrix}a\\0\\0\\\end{pmatrix}}}$

For the second column, the scalar product between the vectors and gives : ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {b}}}$

${\displaystyle {\begin{aligned}|{\vec {a}}|\,|{\vec {b}}|\,\cos \sphericalangle ({\vec {a}},{\vec {b}})&={\vec {a}}\cdot {\vec {b}}\\&=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\\&=a_{1}|{\vec {b}}|\cos(\gamma )\\\end{aligned}}}$

and thus:

${\displaystyle {\vec {b}}={\begin{pmatrix}|{\vec {b}}|\cos(\gamma )\\|{\vec {b}}|\sin(\gamma )\\0\\\end{pmatrix}}}$

The first two elements of the third column result from the scalar products between the vectors and respectively and , the third from the length of the vector using Pythagoras : ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {c}}}$ ${\displaystyle {\vec {b}}}$ ${\displaystyle {\vec {c}}}$

${\displaystyle c_{1}=|{\vec {c}}|\cos(\beta )}$

${\displaystyle c_{2}=|{\vec {c}}|{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}}$

${\displaystyle c_{3}=|{\vec {c}}|{\frac {\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}{\sin(\gamma )}}}$

and thus:

${\displaystyle {\vec {c}}={\begin{pmatrix}|{\vec {c}}|\cos(\beta )\\|{\vec {c}}|{\frac {\cos(\alpha )-\cos(\beta )\cos(\gamma )}{\sin(\gamma )}}\\|{\vec {c}}|{\frac {\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}{\sin(\gamma )}}\\\end{pmatrix}}}$

Individual evidence

1. ^ Donald E. Sands: Introduction to Crystallography . Dover Publications, New York 1993, ISBN 0-486-67839-3 , pp. 7–9 ( limited preview in Google Book search).
2. ↑ Unit cell definition using parallelepiped with lengths a , b , c and angles between the edges given by α, β, γ archive link ( Memento from October 4, 2008 in the Internet Archive )
3. ↑ Michael Polyakov: Fractional to Orthonormal Coordinates Conversion Tutorial. Retrieved May 16, 2015 . 
4. ^ Bernhard Rupp: Coordinate system transformation. Retrieved May 16, 2015 .