# Miller index ( planar spacing distance in fcc system)

## Description

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 each index denotes a plane orthogonal to a direction (h, k, ℓ) in the basis of the reciprocal lattice vectors. For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similar to the reciprocal lattice. Thus, in this common case, the Miller indices (hkℓ) simply denote normals/directions in Cartesian coordinates. For cubic crystals (in face-centered cubic system) with lattice constant a, the spacing d between adjacent (hkℓ) lattice planes can be calculated by the integers h,k,l and the atomic radius.

(For face-centered cubic and body-centered cubic lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell. For face-centered (fcc) cubic crystals, the primitive cell is a parallelepiped or rhombohedron, respectively, but the conventional unit cell used to describe these structures is a simple-cubic supercell.)

## Variables

d_{fcc} | The spacing distance between adjacent (hkℓ) lattice planes (Å) |

r | The atomic radius (Å) |

h | Integer (dimensionless) |

k | Integer (dimensionless) |

l | Integer (dimensionless) |