Chemistry:Fractional coordinates

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In crystallography, a fractional coordinate system is a coordinate system in which the edges of the unit cell are used as the basic vectors to describe the positions of atomic nuclei. The unit cell is a parallelepiped defined by the lengths of its edges [math]\displaystyle{ a, b, c }[/math] and angles between them [math]\displaystyle{ \alpha, \beta, \gamma }[/math].

General case

Consider a system of periodic structure in space and use [math]\displaystyle{ {\mathbf a} }[/math] , [math]\displaystyle{ \mathbf b }[/math], and [math]\displaystyle{ \mathbf c }[/math] as the three independent period vectors, forming a right-handed triad, which are also the edge vectors of a cell of the system. Then any vector [math]\displaystyle{ \mathbf r }[/math] in Cartesian coordinates can be written as a linear combination of the period vectors

[math]\displaystyle{ {\mathbf r} = u {\mathbf a} + v {\mathbf b} + w {\mathbf c}. }[/math]

Our task is to calculate the scalar coefficients known as fractional coordinates [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], and [math]\displaystyle{ w }[/math], assuming [math]\displaystyle{ \mathbf r }[/math], [math]\displaystyle{ \mathbf a }[/math], [math]\displaystyle{ \mathbf b }[/math], and [math]\displaystyle{ \mathbf c }[/math] are known.

For this purpose, let us calculate the following cell surface area vector

[math]\displaystyle{ \mathbf\sigma_{\mathbf a} = {\mathbf b} \times {\mathbf c} , }[/math]

then

[math]\displaystyle{ {\mathbf b} \cdot \mathbf\sigma_{\mathbf a} = 0 , {\mathbf c} \cdot \mathbf\sigma_{\mathbf a} = 0 , }[/math]

and the volume of the cell is

[math]\displaystyle{ \Omega = {\mathbf a} \cdot \mathbf\sigma_{\mathbf a}. }[/math]

If we do a vector inner (dot) product as follows

[math]\displaystyle{ \begin{align} {\mathbf r} \cdot \mathbf\sigma_{\mathbf a} &= u {\mathbf a} \cdot \mathbf\sigma_{\mathbf a} + v {\mathbf b} \cdot \mathbf\sigma_{\mathbf a} + w {\mathbf c} \cdot \mathbf\sigma_{\mathbf a} \\ &= u {\mathbf a} \cdot \mathbf\sigma_{\mathbf a} \\ &= u \Omega, \end{align} }[/math]

then we get

[math]\displaystyle{ u = \frac 1 \Omega {{\mathbf r} \cdot \mathbf\sigma_{\mathbf a}}. }[/math]

Similarly,

[math]\displaystyle{ \mathbf\sigma_{\mathbf b} = {\mathbf c} \times {\mathbf a} , {\mathbf c} \cdot \mathbf\sigma_{\mathbf b} = 0 , {\mathbf a} \cdot \mathbf\sigma_{\mathbf b} = 0 , {\mathbf b} \cdot \mathbf\sigma_{\mathbf b} = \Omega , }[/math]

[math]\displaystyle{ {\mathbf r} \cdot \mathbf\sigma_{\mathbf b} = u {\mathbf a} \cdot \mathbf\sigma_{\mathbf b} + v {\mathbf b} \cdot \mathbf\sigma_{\mathbf b} + w {\mathbf c} \cdot \mathbf\sigma_{\mathbf b} = v {\mathbf b} \cdot \mathbf\sigma_{\mathbf b} = v \Omega, }[/math]

we arrive at

[math]\displaystyle{ v = \frac 1 \Omega {{\mathbf r} \cdot \mathbf\sigma_{\mathbf b}} , }[/math]

and

[math]\displaystyle{ \mathbf\sigma_{\mathbf c} = {\mathbf a} \times {\mathbf b} , {\mathbf a} \cdot \mathbf\sigma_{\mathbf c} = 0 , {\mathbf b} \cdot \mathbf\sigma_{\mathbf c} = 0 , {\mathbf c} \cdot \mathbf\sigma_{\mathbf c} = \Omega , }[/math]

[math]\displaystyle{ {\mathbf r} \cdot \mathbf\sigma_{\mathbf c} = u {\mathbf a} \cdot \mathbf\sigma_{\mathbf c} + v {\mathbf b} \cdot \mathbf\sigma_{\mathbf c} + w {\mathbf c} \cdot \mathbf\sigma_{\mathbf c} = w {\mathbf c} \cdot \mathbf\sigma_{\mathbf c} = w \Omega, }[/math]

[math]\displaystyle{ w = \frac 1 \Omega {{\mathbf r} \cdot \mathbf\sigma_{\mathbf c}}. }[/math]

If there are many [math]\displaystyle{ \mathbf r }[/math]s to be converted with respect to the same period vectors, to speed up, we can have

[math]\displaystyle{ \begin{align} u &= {{\mathbf r} \cdot \mathbf\sigma^\prime_{\mathbf a}}, \\ v &= {{\mathbf r} \cdot \mathbf\sigma^\prime_{\mathbf b}}, \\ \end{align} }[/math]

where

[math]\displaystyle{ \begin{align} \mathbf\sigma^\prime_{\mathbf a} = \frac 1 \Omega {\mathbf\sigma_{\mathbf a}}, \\ \mathbf\sigma^\prime_{\mathbf b} = \frac 1 \Omega {\mathbf\sigma_{\mathbf b}}, \\ \mathbf\sigma^\prime_{\mathbf c} = \frac 1 \Omega {\mathbf\sigma_{\mathbf c}}. \end{align} }[/math]

In crystallography

In crystallography, the lengths ([math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], [math]\displaystyle{ c }[/math]) of and angles ([math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math], [math]\displaystyle{ \gamma }[/math]) between the edge (period) vectors ([math]\displaystyle{ \mathbf a }[/math], [math]\displaystyle{ \mathbf b }[/math], [math]\displaystyle{ \mathbf c }[/math]) of the parallelepiped unit cell are known. For simplicity, it is chosen so that edge vector [math]\displaystyle{ \mathbf a }[/math] in the positive [math]\displaystyle{ x }[/math]-axis direction, edge vector [math]\displaystyle{ \mathbf b }[/math] in the [math]\displaystyle{ x-y }[/math] plane with positive [math]\displaystyle{ y }[/math]-axis component, edge vector [math]\displaystyle{ \mathbf c }[/math] with positive [math]\displaystyle{ z }[/math]-axis component in the Cartesian-system, as shown in the figure below.

Unit cell definition using parallelepiped with lengths [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], [math]\displaystyle{ c }[/math] and angles between the sides given by [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math], and [math]\displaystyle{ \gamma }[/math]

Then the edge vectors can be written as

[math]\displaystyle{ \begin{align} {\mathbf a} &= (a, 0, 0), \\ {\mathbf b} &= (b \cos (\gamma), b \sin (\gamma), 0 ), \\ {\mathbf c} &= ( c_x, c_y, c_z), \end{align} }[/math]

where all [math]\displaystyle{ a }[/math], [math]\displaystyle{ b }[/math], [math]\displaystyle{ c }[/math], [math]\displaystyle{ \sin (\gamma) }[/math], [math]\displaystyle{ c_z }[/math] are positive. Next, let us express all [math]\displaystyle{ \mathbf c }[/math] components with known variables. This can be done with

[math]\displaystyle{ \begin{align} {\mathbf c} \cdot {\mathbf a} &= a c \cos (\beta) = c_x a , \\ {\mathbf c} \cdot {\mathbf b} &= b c \cos (\alpha) =c_x b \cos (\gamma) + c_y b \sin (\gamma), \\ {\mathbf c} \cdot {\mathbf c} &= c^2 = c^2_x + c^2_y + c^2_z. \end{align} }[/math]

Then

[math]\displaystyle{ \begin{align} c_x &= c \cos (\beta) , \\ c_y &= c \frac {\cos (\alpha) - \cos (\gamma) \cos (\beta)} { \sin (\gamma)}, \\ c^2_z &= c^2 - c^2_x - c^2_y = c^2 \left \{ 1 - \cos^2 (\beta) - \frac {[\cos (\alpha) - \cos (\gamma) \cos (\beta) ]^2} { \sin^2 (\gamma)} \right \}. \end{align} }[/math]

The last one continues

[math]\displaystyle{ \begin{align} c^2_z &= c^2 \frac { \sin^2 (\gamma) - \sin^2 (\gamma) \cos^2 (\beta) - [\cos (\alpha) - \cos (\gamma) \cos (\beta) ]^2 } { \sin^2 (\gamma)} \\ &= \frac {c^2} { \sin^2 (\gamma)} \left \{ \sin^2 (\gamma) - \sin^2 (\gamma) \cos^2 (\beta) - [\cos (\alpha) - \cos (\gamma) \cos (\beta) ]^2 \right \} \end{align} }[/math]

where

[math]\displaystyle{ \begin{align} & \sin^2 (\gamma) - \sin^2 (\gamma) \cos^2 (\beta) - [\cos (\alpha) - \cos (\gamma) \cos (\beta) ]^2 \\ &= \sin^2 (\gamma) - \sin^2 (\gamma) \cos^2 (\beta) - \cos^2 (\alpha) - \cos^2 (\gamma) \cos^2 (\beta) + 2 \cos (\alpha) \cos (\gamma) \cos (\beta) \\ &= \sin^2 (\gamma) - \cos^2 (\alpha) - \sin^2 (\gamma) \cos^2 (\beta) - \cos^2 (\gamma) \cos^2 (\beta) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) \\ &= \sin^2 (\gamma) - \cos^2 (\alpha) - [ \sin^2 (\gamma) + \cos^2 (\gamma)] \cos^2 (\beta) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) \\ &= \sin^2 (\gamma) - \cos^2 (\alpha) - \cos^2 (\beta) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) \\ &= 1 - \cos^2 (\alpha) - \cos^2 (\beta) - \cos^2 (\gamma) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) . \end{align} }[/math]

Remembering [math]\displaystyle{ c_z }[/math], [math]\displaystyle{ c }[/math], and [math]\displaystyle{ \sin (\gamma) }[/math] being positive, one gets

[math]\displaystyle{ c_z = \frac {c} { \sin (\gamma)} \sqrt { 1 - \cos^2 (\alpha) - \cos^2 (\beta) - \cos^2 (\gamma) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) } . }[/math]

Since the absolute value of the bottom surface area of the cell is

[math]\displaystyle{ \left |\mathbf\sigma_{\mathbf c} \right |= ab \sin (\gamma) , }[/math]

the volume of the parallelepiped cell can also be expressed as

[math]\displaystyle{ \Omega = c_z \left |\mathbf\sigma_{\mathbf c} \right | = abc \sqrt { 1 - \cos^2 (\alpha) - \cos^2 (\beta) - \cos^2 (\gamma) + 2 \cos (\alpha) \cos (\beta) \cos (\gamma) } }[/math].

Once the volume is calculated as above, one has

[math]\displaystyle{ c_z =\frac {\Omega} {ab\sin (\gamma)}. }[/math]

Now let us summarize the expression of the edge (period) vectors

[math]\displaystyle{ \begin{align} {\mathbf a} &= ({ a}_x, { a}_y, { a}_z) = (a, 0, 0), \\ {\mathbf b} &= ({ b}_x, { b}_y, { b}_z) = (b \cos (\gamma), b \sin (\gamma), 0 ), \\ {\mathbf c} &= ({ c}_x, { c}_y, { c}_z) = (c \cos (\beta) , c \frac {\cos (\alpha) - \cos (\beta) \cos (\gamma)} { \sin (\gamma)}, \frac {\Omega} {ab\sin (\gamma)} ). \end{align} }[/math]

Conversion from Cartesian coordinates

Let us calculate the following surface area vector of the cell first

[math]\displaystyle{ \mathbf\sigma_{\mathbf a} = (\mathbf\sigma_{\mathbf a,x}, \mathbf\sigma_{\mathbf a,y}, \mathbf\sigma_{\mathbf a,z}) = {\mathbf b} \times {\mathbf c} , }[/math]

where

[math]\displaystyle{ \begin{align} \mathbf\sigma_{\mathbf a,x} &= { b}_y { c}_z - { b}_z { c}_y = b \sin (\gamma) \frac {\Omega} {ab\sin (\gamma)} = \frac {\Omega} {a} , \\ \mathbf\sigma_{\mathbf a,y} &= { b}_z { c}_x - { b}_x { c}_z =- b \cos (\gamma) \frac {\Omega} {ab\sin (\gamma)} = - \frac {\Omega \cos (\gamma)} {a \sin (\gamma)} , \\ \mathbf\sigma_{\mathbf a,z} &= { b}_x { c}_y - { b}_y { c}_x = b \cos (\gamma) c \frac {\cos (\alpha) - \cos (\beta) \cos (\gamma)} { \sin (\gamma)} - b \sin (\gamma) c \cos (\beta) \\ &= bc \left \{\cos (\gamma) \frac {\cos (\alpha) - \cos (\beta) \cos (\gamma)} { \sin (\gamma)} - \sin (\gamma) \cos (\beta) \right \} \\ &= \frac {bc} { \sin (\gamma)} \left \{\cos (\gamma) [\cos (\alpha) - \cos (\beta) \cos (\gamma)] - \sin^2 (\gamma) \cos (\beta) \right \} \\ &= \frac {bc} { \sin (\gamma)} \left \{\cos (\gamma) \cos (\alpha) - \cos (\beta) \cos^2 (\gamma) - \sin^2 (\gamma) \cos (\beta) \right \} \\ &= \frac {bc} { \sin (\gamma)} \left \{\cos (\alpha) \cos ( \gamma) - \cos (\beta) \right \} . \\ \end{align} }[/math]

Another surface area vector of the cell

[math]\displaystyle{ \mathbf\sigma_{\mathbf b} = (\mathbf\sigma_{\mathbf b,x}, \mathbf\sigma_{\mathbf b,y}, \mathbf\sigma_{\mathbf b,z})= {\mathbf c} \times {\mathbf a} , }[/math]

where

[math]\displaystyle{ \begin{align} \mathbf\sigma_{\mathbf b,x} &= { c}_y { a}_z - { c}_z { a}_y = 0 , \\ \mathbf\sigma_{\mathbf b,y} &= { c}_z { a}_x - { c}_x { a}_z = a \frac {\Omega} {ab\sin (\gamma)} = \frac {\Omega} {b\sin (\gamma)} , \\ \mathbf\sigma_{\mathbf b,z} &= { c}_x { a}_y - { c}_y { a}_x =- ac \frac {\cos (\alpha) - \cos (\beta) \cos (\gamma)} { \sin (\gamma)} \\ &= \frac {ac} { \sin (\gamma)} \left \{\cos (\beta) \cos (\gamma) - \cos (\alpha) \right \} . \end{align} }[/math]

The last surface area vector of the cell

[math]\displaystyle{ \mathbf\sigma_{\mathbf c} = (\mathbf\sigma_{\mathbf c,x}, \mathbf\sigma_{\mathbf c,y}, \mathbf\sigma_{\mathbf c,z})= {\mathbf a} \times {\mathbf b} , }[/math]

where

[math]\displaystyle{ \begin{align} \mathbf\sigma_{\mathbf c,x} &= { a}_y { b}_z - { a}_z { b}_y = 0 , \\ \mathbf\sigma_{\mathbf c,y} &= { a}_z { b}_x - { a}_x { b}_z = 0, \\ \mathbf\sigma_{\mathbf c,z} &= { a}_x { b}_y - { a}_y { b}_x = ab \sin (\gamma) . \end{align} }[/math]

Summarize

[math]\displaystyle{ \begin{align} \mathbf\sigma^\prime_{\mathbf a} &= \frac 1 \Omega {\mathbf\sigma_{\mathbf a}}= \left ( \frac 1 {a}, - \frac {\cos (\gamma)} {a \sin (\gamma)} , bc \frac {\cos (\alpha) \cos ( \gamma) - \cos (\beta) } {\Omega \sin (\gamma) } \right ), \\ \mathbf\sigma^\prime_{\mathbf b} &= \frac 1 \Omega {\mathbf\sigma_{\mathbf b}}= \left ( 0, \frac 1 {b\sin (\gamma)}, ac \frac {\cos (\beta) \cos (\gamma) - \cos (\alpha)} { \Omega \sin (\gamma)} \right ), \\ \mathbf\sigma^\prime_{\mathbf c} &= \frac 1 \Omega {\mathbf\sigma_{\mathbf c}} = \left ( 0, 0, \frac { ab \sin (\gamma)} {\Omega} \right ). \end{align} }[/math]

As a result

[math]\displaystyle{ \left [ \begin{matrix} u \\ v \\ w \end{matrix} \right ] = \left [ \begin{matrix} \frac 1 {a}& - \frac {\cos (\gamma)} {a \sin (\gamma)} & bc \frac {\cos (\alpha) \cos ( \gamma) - \cos (\beta) } {\Omega \sin (\gamma) } \\ 0 & \frac 1 {b\sin (\gamma)}& ac \frac {\cos (\beta) \cos (\gamma) - \cos (\alpha)} { \Omega \sin (\gamma)} \\ 0& 0& \frac { ab \sin (\gamma)} {\Omega} \end{matrix} \right ] \left [ \begin{matrix} x \\ y\\ z \end{matrix} \right ] }[/math]

where [math]\displaystyle{ (x }[/math], [math]\displaystyle{ y }[/math], [math]\displaystyle{ z) }[/math] are the components of the arbitrary vector [math]\displaystyle{ \mathbf r }[/math] in Cartesian coordinates.

Conversion to Cartesian coordinates

To return the orthogonal coordinates in ångströms from fractional coordinates, one can employ the first equation on top and the expression of the edge (period) vectors

[math]\displaystyle{ \left [ \begin{matrix} x \\ y \\ z \end{matrix} \right ] = \left [ \begin{matrix} a& b \cos (\gamma)& c \cos (\beta) \\ 0& b \sin (\gamma)& c \frac {\cos (\alpha) - \cos (\beta) \cos (\gamma)} { \sin (\gamma)} \\ 0& 0 & \frac {\Omega} {ab\sin (\gamma)} \end{matrix} \right ] \left [ \begin{matrix} u \\ v\\ w \end{matrix} \right ]. }[/math]

For the special case of a monoclinic cell (a common case) where [math]\displaystyle{ \alpha = \gamma = 90^{\circ} }[/math] and [math]\displaystyle{ \beta \gt 90^{\circ} }[/math], this gives:

[math]\displaystyle{ \begin{align} x&=a u + c w \cos(\beta), \\ y&=b v, \\ z&=\frac {\Omega} {a b} w = cw\sin(\beta). \end{align} }[/math]

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