Fractional coordinates

General case

Let us consider a system of periodic structure in space and use ${\mathbf {a} }$ , $\mathbf b$ , and $\mathbf {c}$ 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 $\mathbf {r}$ in Cartesian coordinates can be written as a linear combination of the period vectors

{\mathbf {r} }=u{\mathbf {a} }+v{\mathbf {b} }+w{\mathbf {c} }.

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

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

\mathbf {\sigma } _{\mathbf {a} }={\mathbf {b} }\times {\mathbf {c} },

then

{\mathbf {b} }\cdot \mathbf {\sigma } _{\mathbf {a} }=0,{\mathbf {c} }\cdot \mathbf {\sigma } _{\mathbf {a} }=0,

and the volume of the cell is

\Omega ={\mathbf {a} }\cdot \mathbf {\sigma } _{\mathbf {a} }.

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

{\begin{aligned}{\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{aligned}}

then we get

u={\frac {1}{\Omega }}{{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {a} }}.

Similarly,

\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 ,

{\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 ,

we arrive at

v={\frac {1}{\Omega }}{{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {b} }},

and

\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 ,

{\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 ,

w={\frac {1}{\Omega }}{{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {c} }}.

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

{\begin{aligned}u&={{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {a} }^{\prime }},\\v&={{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {b} }^{\prime }},\\w&={{\mathbf {r} }\cdot \mathbf {\sigma } _{\mathbf {c} }^{\prime }},\end{aligned}}

where

{\begin{aligned}\mathbf {\sigma } _{\mathbf {a} }^{\prime }={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {a} }},\\\mathbf {\sigma } _{\mathbf {b} }^{\prime }={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {b} }},\\\mathbf {\sigma } _{\mathbf {c} }^{\prime }={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {c} }}.\end{aligned}}

In Crystallography

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

Unit cell definition using parallelepiped with lengths

a

b

c

and angles between the sides given by

\alpha

\beta

, and

\gamma

^[1]

Then the edge vectors can be written as

{\begin{aligned}{\mathbf {a} }&=(a,0,0),\\{\mathbf {b} }&=(b\cos(\gamma ),b\sin(\gamma ),0),\\{\mathbf {c} }&=(c_{x},c_{y},c_{z}),\end{aligned}}

where all $a$ , $b$ , $c$ , $\sin(\gamma )$ , $c_{z}$ are positive. Next, let us express all $\mathbf {c}$ components with known variables. This can be done with

{\begin{aligned}{\mathbf {c} }\cdot {\mathbf {a} }&=ac\cos(\beta )=c_{x}a,\\{\mathbf {c} }\cdot {\mathbf {b} }&=bc\cos(\alpha )=c_{x}b\cos(\gamma )+c_{y}b\sin(\gamma ),\\{\mathbf {c} }\cdot {\mathbf {c} }&=c^{2}=c_{x}^{2}+c_{y}^{2}+c_{z}^{2}.\end{aligned}}

Then

{\begin{aligned}c_{x}&=c\cos(\beta ),\\c_{y}&=c{\frac {\cos(\alpha )-\cos(\gamma )\cos(\beta )}{\sin(\gamma )}},\\c_{z}^{2}&=c^{2}-c_{x}^{2}-c_{y}^{2}=c^{2}\left\{1-\cos ^{2}(\beta )-{\frac {[\cos(\alpha )-\cos(\gamma )\cos(\beta )]^{2}}{\sin ^{2}(\gamma )}}\right\}.\end{aligned}}

The last one continues

{\begin{aligned}c_{z}^{2}&=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{aligned}}

where

{\begin{aligned}&\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{aligned}}

Remembering $c_{z}$ , $c$ , and $\sin(\gamma )$ being positive, one gets

c_{z}={\frac {c}{\sin(\gamma )}}{\sqrt {1-\cos ^{2}(\alpha )-\cos ^{2}(\beta )-\cos ^{2}(\gamma )+2\cos(\alpha )\cos(\beta )\cos(\gamma )}}.

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

\left|\mathbf {\sigma } _{\mathbf {c} }\right|=ab\sin(\gamma ),

the volume of the parallelepiped cell can also be expressed as

\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 )}}

.^[2]

Once the volume is calculated as above, one has

c_{z}={\frac {\Omega }{ab\sin(\gamma )}}.

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

{\begin{aligned}{\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{aligned}}

Conversion from cartesian coordinates

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

\mathbf {\sigma } _{\mathbf {a} }=(\mathbf {\sigma } _{\mathbf {a} ,x},\mathbf {\sigma } _{\mathbf {a} ,y},\mathbf {\sigma } _{\mathbf {a} ,z})={\mathbf {b} }\times {\mathbf {c} },

where

{\begin{aligned}\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{aligned}}

Another surface area vector of the cell

\mathbf {\sigma } _{\mathbf {b} }=(\mathbf {\sigma } _{\mathbf {b} ,x},\mathbf {\sigma } _{\mathbf {b} ,y},\mathbf {\sigma } _{\mathbf {b} ,z})={\mathbf {c} }\times {\mathbf {a} },

where

{\begin{aligned}\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{aligned}}

The last surface area vector of the cell

\mathbf {\sigma } _{\mathbf {c} }=(\mathbf {\sigma } _{\mathbf {c} ,x},\mathbf {\sigma } _{\mathbf {c} ,y},\mathbf {\sigma } _{\mathbf {c} ,z})={\mathbf {a} }\times {\mathbf {b} },

where

{\begin{aligned}\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{aligned}}

Summarize

{\begin{aligned}\mathbf {\sigma } _{\mathbf {a} }^{\prime }&={\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 } _{\mathbf {b} }^{\prime }&={\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 } _{\mathbf {c} }^{\prime }&={\frac {1}{\Omega }}{\mathbf {\sigma } _{\mathbf {c} }}=\left(0,0,{\frac {ab\sin(\gamma )}{\Omega }}\right).\end{aligned}}

As a result^[3]

\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]

where $(a$ , $b$ , $c)$ are the components of the arbitrary vector $\mathbf {r}$ 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^[4]^[5]

\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].

For the special case of a monoclinic cell (a common case) where $\alpha =\gamma =90^{\circ }$ and $\beta >90^{\circ }$ , this gives:

{\begin{aligned}x&=au+cw\cos(\beta ),\\y&=bv,\\z&={\frac {\Omega }{ab}}w=cw\sin(\beta ).\end{aligned}}

Supporting file formats

CPMD input
CIF

References

↑ "Unit cell definition using parallelepiped with lengths a, b, c and angles between the edges given by α, β, γ". Ccdc.cam.ac.uk. Archived from the original on 2008-10-04. Retrieved 2016-08-17.
↑ "Coordinate system transformation". www.ruppweb.org. Retrieved 2016-10-19.
↑ "Coordinate system transformation". Ruppweb.org. Retrieved 2016-10-19.
↑ Sussman, J.; Holbrook, S.; Church, G.; Kim, S (1977). "A Structure-Factor Least-Squares Refinement Procedure For Macromolecular Structures Using Constrained And Restrained Parameters". Acta Crystallogr. A. 33: 800–804. doi:10.1107/S0567739477001958.
↑ Rossmann, M.; Blow, D. (1962). "The Detection Of Sub-Units Within The Crystallographic Asymmetric Unit". Acta Crystallogr. 15: 24–31. doi:10.1107/S0365110X62000067.

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