= Miller index
{c}
{title2=$(hkl)$}
Three <Miller indices> $(hkl)$ 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.<Online Dictionary of Crystallography by International Union of Crystallography/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]{ref}
For example, intercepts $(2a,3b,\infty)$ have reciprocal ratios $(1/2,1/3,0)$, so the orientation is $(320)$ after multiplying by six. Relative to the same axes, intercepts $(a,-b,\infty)$ give $(1\bar10)$. 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> $(200)$ example shows. Reflection labels can also retain higher <diffraction orders>. Use the <reciprocal lattice> for the <normal vector> with oblique axes; the direct vector $ha_1+ka_2+la_3$ need not be perpendicular.
Back to article page
