Crystallographic metric matrix

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.