= Fractional crystallographic coordinate
{title2=$(x,y,z)$}
{title2=$(u,v,w)$}
= Fractional crystal coordinate
{synonym}
A \b[fractional crystallographic coordinate] is one of the dimensionless coefficients expressing a position in the basis of specified <unit cell> vectors:
$$
\mathbf r=x\mathbf a+y\mathbf b+z\mathbf c.
$$
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.<Fractional coordinates in the IUCr core CIF dictionary>[`_atom_site_fract_x`, `_atom_site_fract_y`, `_atom_site_fract_z`, Definition]{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 $\mathbf b=(1,\sqrt3)$ angstroms. The fractional position $(1/2,1/2)$ is
$$
\mathbf r=\tfrac12\mathbf a+\tfrac12\mathbf b
=(3/2,\sqrt3/2)\,\mathrm{\mathring A},
\qquad |\mathbf r|=\sqrt3\,\mathrm{\mathring A}.
$$
The <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 $r=ua_1+va_2+wa_3$ 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.
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