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.
