Three Miller indices specify a lattice plane orientation in chosen unit cell axes. Choose a parallel plane away from the origin, express its intercepts in axis units, take reciprocals, and clear fractions. An infinite intercept gives index zero; a negative intercept gives a negative index, written with an overbar.[ref]
Miller indices. Online Dictionary of Crystallography by International Union of Crystallography; Definition, Direct space: reciprocal intercepts; Definition, Reciprocal space: normal to the plane family; primitive versus centered-cell qualifications
For example, intercepts have reciprocal ratios , so the orientation is after multiplying by six. Relative to the same axes, intercepts give . Choosing a different parallel member changes the common intercept scale, not the orientation.
Primitive axes require relatively prime Miller indices for complete families. Centered conventional unit cells need not, as the FCC lattice example shows. Reflection labels can also retain higher diffraction orders. Use the reciprocal lattice for the normal vector with oblique axes; the direct vector need not be perpendicular.
If new direct-cell vectors are columns of in old fractional coordinates, an unchanged reciprocal vector has the displayed new coordinates. Noninteger coordinates must not be rescaled when tracking the same diffraction order. Under the Bain path cell change, . The old FCC reflection becomes and remains absent; doubling that triple gives the different, allowed reflection that originated as FCC . Orientation indices and order-resolved reciprocal vectors carry different information.
A lattice direction lies parallel to a plane exactly when . Indeed, their direct and reciprocal lattice vectors have dot product proportional to this expression. This statement holds for noncubic cells as well; angles between directions still require the direct-lattice metric tensor.
