 Fractional crystallographic coordinate

A fractional crystallographic coordinate is one of the dimensionless coefficients expressing a position in the basis of specified unit cell vectors:The three coefficients are also called fractional crystal coordinates. Reporting them requires specifying the cell vectors and origin. The core CIF coordinate items store these fractions rather than Cartesian lengths.^[ref]

For an original orthorhombic example, let the mutually perpendicular cell edges have lengths4,6 and8 angstroms. The fractional position $(1/4,1/2,1/8)$ has Cartesian coordinates $(1,3,1)$ angstroms. Adding an integer to one fractional coordinate selects the corresponding translated position in another cell; it does not change the motif's location relative to the repeating cell.

For an oblique cell, multiplying the fractions by edge lengths is insufficient to obtain Cartesian components. In a two-dimensional example, take $\mathbf{a}=(2,0)$ and $\mathbf{b}=(1,\sqrt{3})$ angstroms. The fractional position $(1/2,1/2)$ isThe crystallographic metric matrix $G_{ij}={\mathbf{a}}_i\cdot {\mathbf{a}}_j$ records the necessary lengths and angles. A displacement with fractional column vector $\mathbf{u}$ has squared length ${\mathbf{u}}^{T}G\mathbf{u}$. Applying the ordinary Cartesian distance formula directly to the fractions would discard the geometry of the crystal structure.

Fractional crystal coordinates are the dimensionless coefficients in $r=u{a}_1+v{a}_2+w{a}_3$ for specified unit cell vectors. They are not Cartesian lengths unless the vectors themselves are included. With primitive vectors, replacing $u,v,w$ by values differing by integers selects an equivalent position in another unit cell.

In the two-dimensional example under crystal structure, B has position $(0.65,0.55)$ in ordinary length coordinates, but fractional crystal coordinates $(0.65/2,0.55/1.5)=(13/40,11/30)$. Multiplying these coefficients by the two translation vectors recovers the drawn position. Reporting the axes is therefore part of reporting a structure.