In crystallography, a fractional coordinate system (crystal coordinate system) is a coordinate system in which 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
a1,a2,...,ad
d
a1,a2,...,ad
\alpha1,\alpha2,...,
\alpha | ||||
|
Most cases in crystallography involve two- or three-dimensional space. In the three-dimensional case, the basis vectors
a1,a2,a3
a,b,c
a,b,c
\alpha,\beta,\gamma
\alpha
b
c
\beta
c
a
\gamma
a
b
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
t
t=c1t1+c2t2wherec1,c2\inZ
The vector lattice (lattice)
T
X0
x0
Xi
xi=x0+ti
X0
T
X0
Usually when describing a space geometrically, a coordinate system is used which consists of a choice of origin and a basis of
d
a1,a2,...,ad
d
d
d
(0,0,...,0)
(x1,x2,...,xd)
\vec{OP}
\vec{OP}=x=
d | |
\sum | |
i=1 |
xiai
In
d
a1,a2,...,ad
\alpha1,\alpha2,...,
\alpha | ||||
|
a1,a2,a3
a,b,c
a,b,c
\alpha,\beta,\gamma
A widely used coordinate system is the Cartesian coordinate system, which consists of orthonormal basis vectors. This means that,
a1=|a1|=a2=|a2|=...=ad=|ad|=1
\alpha1=\alpha2=...=
\alpha | ||||
|
=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.
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
t
t=
d | |
\sum | |
i=1 |
ciaiwhereci\inQ
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
a1,a2,...,ad
t
t=
d | |
\sum | |
i=1 |
ciaiwhereci\inZ
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\leqx1,x2,...,xd<1
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\leqx1,x2,...,xd<1
a1,a2,...,ad
\alpha1,\alpha2,...,
\alpha | ||||
|
The fractional coordinates of a point in space
\rho=
(\rho | |
x1 |
,
\rho | |
x2 |
,...,
\rho | |
xd |
)
\rho=
\rho | |
x1 |
a1+
\rho | |
x2 |
a2+...+
\rho | |
xd |
adwhere\rho\in[0,1)
The relationship between fractional and Cartesian coordinates can be described by the matrix transformation
r=A\boldsymbol\rho
\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\ r | |
x3 |
\end{pmatrix}=\begin{pmatrix} a1\sin(\alpha2)\sqrt{1-(\cot(\alpha1)\cot(\alpha2)-\csc(\alpha1)\csc(\alpha2)\cos(\alpha3))2}&0&0\\ a1\csc(\alpha1)\cos(\alpha3)-a1\cot(\alpha1)\cos(\alpha2)&a2\sin(\alpha1)&0\\ a1\cos(\alpha2)&a2\cos(\alpha1)&a3\\ \end{pmatrix} \begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\ \rho | |
x3 |
\end{pmatrix}
Similarly, the Cartesian coordinates can be converted back to fractional coordinates using the matrix transformation
\boldsymbol\rho=A-1r
\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\ \rho | |
x3 |
\end{pmatrix}=\begin{pmatrix}
\csc(\alpha2) | |
a1\sqrt{1-(\cot(\alpha1)\cot(\alpha2)-\csc(\alpha1)\csc(\alpha2)\cos(\alpha3))2 |
Another common method of converting between fractional and Cartesian coordinates involves the use of a cell tensor
h
In Cartesian coordinates the 2 basis vectors are represented by a
2 x 2
h
h=\begin{pmatrix}a1&a2\end{pmatrix}\operatorname{T}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
\ a | |
2,x1 |
&
a | |
2,x2 |
\end{pmatrix}
The area of the unit cell,
A
A=\det(h)=
a | |
1,x1 |
a | |
2,x2 |
-
a | |
1,x2 |
a | |
2,x2 |
For the special case of a square or rectangular unit cell, the matrix is diagonal, and we have that:
A=\det(h)=
a | |
1,x1 |
a | |
2,x2 |
The relationship between fractional and Cartesian coordinates can be described by the matrix transformation
r=h\boldsymbol\rho
\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\end{pmatrix}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
\ a | |
2,x1 |
&
a | |
2,x2 |
\end{pmatrix}\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\end{pmatrix}
Similarly, the Cartesian coordinates can be converted back to fractional coordinates using the matrix transformation
\boldsymbol\rho=h-1r
\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\end{pmatrix}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
\ a | |
2,x1 |
&
a | |
2,x2 |
\end{pmatrix}-1\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\end{pmatrix}
In Cartesian coordinates the 3 basis vectors are represented by a
3 x 3
h
h=\begin{pmatrix}a1&a2&a3\end{pmatrix}\operatorname{T}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
&
a | |
1,x3 |
\ a | |
2,x1 |
&
a | |
2,x2 |
&
a | |
2,x3 |
\ a | |
3,x1 |
&
a | |
3,x2 |
&
a | |
3,x3 |
\end{pmatrix}
The volume of the unit cell,
V
V=\det(h)=
a | |
1,x1 |
(a | |
2,x2 |
a | |
3,x3 |
-a | |
2,x3 |
a | |
3,x2 |
)-
a | |
1,x2 |
(a | |
2,x1 |
a | |
3,x3 |
-
a | |
2,x3 |
a | |
3,x1 |
)-
a | |
1,x3 |
(a | |
2,x1 |
a | |
3,x2 |
-
a | |
2,x2 |
a | |
3,x1 |
)
For the special case of a cubic, tetragonal, or orthorhombic cell, the matrix is diagonal, and we have that:
V=\det(h)=
a | |
1,x1 |
a | |
2,x2 |
a | |
3,x3 |
The relationship between fractional and Cartesian coordinates can be described by the matrix transformation
r=h\boldsymbol\rho
\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\ r | |
x3 |
\end{pmatrix}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
&
a | |
1,x3 |
\ a | |
2,x1 |
&
a | |
2,x2 |
&
a | |
2,x3 |
\ a | |
d,x1 |
&
a | |
d,x2 |
&
a | |
d,xd |
\end{pmatrix}\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\ \rho | |
x3 |
\end{pmatrix}
Similarly, the Cartesian coordinates can be converted back to fractional coordinates using the matrix transformation
\boldsymbol\rho=h-1r
\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\ \rho | |
x3 |
\end{pmatrix}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
&
a | |
1,x3 |
\ a | |
2,x1 |
&
a | |
2,x2 |
&
a | |
2,x3 |
\ a | |
d,x1 |
&
a | |
d,x2 |
&
a | |
d,xd |
\end{pmatrix}-1\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\ r | |
x3 |
\end{pmatrix}
In Cartesian coordinates the
d
d x d
h
h=\begin{pmatrix}a1&a2&...&ad\end{pmatrix}\operatorname{T}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
&...&
a | |
1,xd |
\ a | |
2,x1 |
&
a | |
2,x2 |
&...&
a | |
2,xd |
\ \vdots&\vdots&\ddots&\vdots
\ a | |
d,x1 |
&
a | |
d,x2 |
&...&
a | |
d,xd |
\end{pmatrix}
The hypervolume of the unit cell,
V
V=\det(h)
The relationship between fractional and Cartesian coordinates can be described by the matrix transformation
r=h\boldsymbol\rho
\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\ \vdots
\ r | |
xd |
\end{pmatrix}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
&...&
a | |
1,xd |
\ a | |
2,x1 |
&
a | |
2,x2 |
&...&
a | |
2,xd |
\ \vdots&\vdots&\ddots&\vdots
\ a | |
d,x1 |
&
a | |
d,x2 |
&...&
a | |
d,xd |
\end{pmatrix}\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\ \vdots
\ \rho | |
xd |
\end{pmatrix}
Similarly, the Cartesian coordinates can be converted back to fractional coordinates using the transformation
\boldsymbol\rho=h-1r
\begin{pmatrix}
\rho | |
x1 |
\ \rho | |
x2 |
\ \vdots
\ \rho | |
xd |
\end{pmatrix}=\begin{pmatrix}
a | |
1,x1 |
&
a | |
1,x2 |
&...&
a | |
1,xd |
\ a | |
2,x1 |
&
a | |
2,x2 |
&...&
a | |
2,xd |
\ \vdots&\vdots&\ddots&\vdots
\ a | |
d,x1 |
&
a | |
d,x2 |
&...&
a | |
d,xd |
\end{pmatrix}-1\begin{pmatrix}
r | |
x1 |
\ r | |
x2 |
\ \vdots
\ r | |
xd |
\end{pmatrix}
The metric tensor
G
In two dimensions,
G=\begin{pmatrix}g11&g12\ g21&g22\end{pmatrix}=\begin{pmatrix}a1 ⋅ a1&a1 ⋅ a2\ a2 ⋅ a1&a2 ⋅ a2\end{pmatrix}=\begin{pmatrix}
2 | |
a | |
1 |
&a1a2\cos(\alpha1)\ a1a2\cos(\alpha1)&
2 | |
a | |
2 |
\end{pmatrix}
In three dimensions,
G=\begin{pmatrix}g11&g12&g13\ g21&g22&g23\ g31&g32&g33\end{pmatrix}=\begin{pmatrix}a1 ⋅ a1&a1 ⋅ a2&a1 ⋅ a3\ a2 ⋅ a1&a2 ⋅ a2&a2 ⋅ a3\ a3 ⋅ a1&a3 ⋅ a2&a3 ⋅ a3\end{pmatrix}=\begin{pmatrix}
2 | |
a | |
1 |
&a1a2\cos(\alpha3)&a1a3\cos(\alpha2)\ a1a2\cos(\alpha3)&
2 | |
a | |
2 |
&a2a3\cos(\alpha1)\ a1a3\cos(\alpha2)&a2a3\cos(\alpha1)&
2 | |
a | |
3 |
\end{pmatrix}
The distance between two points
Q
R
2 | |
d | |
qr |
=\sumi,gij(ri-qi)(rj-qj)
The distance from the origin of the unit cell to a point
Q
OQ=rq;
2 | |
r | |
q |
=\sumi,gijqiqj
The angle formed from three points
Q
P
R
\cos(QPR)=(rpq)-1(rpr)-1\sumi,gij(qi-pi)(rj-pj)
The volume of the unit cell,
V
V2=\det(G)