Fractional crystallographic coordinate ($(x,y,z)$, $(u,v,w)$)

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.

The crystallographic metric matrix is the Gram matrix of the three unit cell basis vectors. If $A$ has Cartesian columns $\mathbf{a},\mathbf{b},\mathbf{c}$, thenHere $\alpha ,\beta ,\gamma$ are the angles between $\mathbf{b},\mathbf{c}$; $\mathbf{a},\mathbf{c}$; and $\mathbf{a},\mathbf{b}$, respectively. The matrix converts fractional crystallographic coordinate differences into physical lengths: a fractional displacement $\mathbf{u}$ has squared length ${\mathbf{u}}^{T}G\mathbf{u}$.^[ref]

For an original oblique cell, choose $a=2$, $b=3$, $c=4$ angstroms, $\gamma =60^{\circ }$ and $\alpha =\beta =90^{\circ }$. ThenFor $\mathbf{u}=(1/2,-1/2,0{)}^T$,Ignoring the off-diagonal entries would instead give $13/4$, overestimating this distance. The cross terms encode the cell angles.

The reciprocal crystallographic metric matrix is $G^{-1}$ when the reciprocal basis has dot products ${\mathbf{a}}_i^{\ast }\cdot {\mathbf{a}}_j={\delta }_{ij}$, with no factor of $2\pi$. Thus the plane spacing for a nonzero Miller index column $\mathbf{h}$ satisfiesA basis change $A^{\prime }=AP$ gives $G^{\prime }=P^{T}GP$ and fractional coordinates ${\mathbf{u}}^{\prime }=P^{-1}\mathbf{u}$, preserving ${\mathbf{u}}^{\prime T}{G}^{\prime }{\mathbf{u}}^{\prime }={\mathbf{u}}^{T}G\mathbf{u}$. Distances do not change when the same crystal structure is described in another basis.^[ref]

The reciprocal crystallographic metric matrix is the Gram matrix of the reciprocal lattice basis. With direct-basis matrix $A=(\mathbf{a}\mathbf{b}\mathbf{c})$ and the convention ${\mathbf{a}}_i^{\ast }\cdot {\mathbf{a}}_j={\delta }_{ij}$, the reciprocal-basis matrix is $A^{\ast }=A^{-T}$. Consequently,where $G$ is the direct crystallographic metric matrix. If direct lengths are in angstroms, the reciprocal metric entries have units of inverse squared angstroms.^[ref]

For an original calculation, takeFor a Miller index column $\mathbf{h}$, the corresponding reciprocal vector is ${\mathbf{g}}_{\mathbf{h}}=A^{\ast }\mathbf{h}$. Its squared magnitude determines the interplanar spacing:For $(110)$ this gives $d^{-2}=7/27{\mathring{\mathrm{A}}}^{-2}$, so $d\simeq 1.964\mathring{\mathrm{A}}$.

The same matrix also gives angles between plane normals. In this example,The reciprocal angle is120degrees while the corresponding direct-cell angle is60degrees. In an oblique cell, reciprocal directions cannot be recovered by simply inverting the three edge lengths separately.

A convention using reciprocal vectors with dot products $2\pi {\delta }_{ij}$ multiplies this reciprocal metric by $(2\pi {)}^2$. In that convention the spacing formula includes the matching factor $(2\pi {)}^2$ in its numerator. Stating the convention prevents an otherwise systematic length error.