Miller index

Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices. Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices. In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written (hkℓ), and denote the family of planes orthogonal to h b 1 + k b 2 + ℓ b 3 {displaystyle hmathbf {b_{1}} +kmathbf {b_{2}} +ell mathbf {b_{3}} } , where b i {displaystyle mathbf {b_{i}} } are the basis of the reciprocal lattice vectors. (Note that the plane is not always orthogonal to the linear combination of direct lattice vectors h a 1 + k a 2 + ℓ a 3 {displaystyle hmathbf {a_{1}} +kmathbf {a_{2}} +ell mathbf {a_{3}} } because the reciprocal lattice vectors need not be mutually orthogonal.) By convention, negative integers are written with a bar, as in 3 for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1. There are also several related notations: In the context of crystal directions (not planes), the corresponding notations are: Miller indices were introduced in 1839 by the British mineralogist William Hallowes Miller, although an almost identical system (Weiss parameters) had already been used by German mineralogist Christian Samuel Weiss since 1817. The method was also historically known as the Millerian system, and the indices as Millerian, although this is now rare.