Reciprocal crystallographic metric matrix ($G^{\ast }=G^{-1}$)

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.