Sets of integers that describe planes and directions in a crystal. For a plane, the reciprocals of its intercepts on the axes are reduced to the smallest integer ratio and written (hkl); a direction is written [uvw].
Japanese name: ミラー指数
Miller indices appears in 1 practice question across 1 topic, including Crystal structures and Miller indices.
The Miller indices of a plane are found by the following steps. First, read the points where the plane meets the crystal axes x, y and z (the intercepts) in units of the lattice parameters a, b and c. Next, take the reciprocal of each, clear fractions to get the smallest integer ratio, and write the result in parentheses as (hkl). For example, if the intercepts are 1, 2 and ∞, the reciprocals are 1, 1/2 and 0, giving (210). A plane parallel to an axis has an infinite intercept on it, so that index is 0. Negative intercepts are shown with a bar over the number. If the plane passes through the origin, move the origin to a neighboring lattice point before reading the intercepts.
Direction indices are the coordinates of a point reached by moving from the origin in that direction, reduced to the smallest integer ratio and written in square brackets as [uvw]. In cubic crystals many planes are equivalent by symmetry; (100), (010), (001) and so on are written together in braces as {100}, and equivalent directions together in angle brackets as <100>. Cubic crystals have the convenient property that the [hkl] direction is exactly perpendicular to the (hkl) plane, so angles between planes can also be found from dot products of directions. Note that this property does not hold in general for tetragonal or hexagonal crystals.
The interplanar spacing of a cubic crystal is calculated as d = a/√(h² + k² + l²). Planes with larger indices have smaller spacings, and their XRD peaks appear at higher angles. Depending on the arrangement of atoms in the unit cell, however, diffraction from some planes cancels out and does not appear. In BCC only planes with h + k + l even diffract, and in FCC only planes with h, k and l all even or all odd (the extinction rules). Correctly indexing the peaks is the starting point for determining lattice parameters and identifying phases.
For hexagonal crystals such as HCP, Miller–Bravais indices (hkil) are widely used, with three axes a₁, a₂ and a₃ at 120° to each other in the basal plane plus the c axis. The relation i = −(h + k) holds, and the advantage is that equivalent planes have the same pattern of numbers. For example, the basal plane is (0001) and prism planes take the form {10-10}. These indices appear constantly in materials and device specifications, for wafer orientations, the orientation of epitaxial films and the notation of slip systems, so it pays to be able to picture a plane's orientation from its indices.
Answer: C. (210)
Miller indices are found by expressing the intercepts on each axis in units of the lattice parameter, taking their reciprocals and reducing to the smallest integer ratio. The intercepts are 1, 2 and ∞, so the reciprocals are 1, 1/2 and 0; multiplying by 2 gives (210). (120) is the mistake of listing the intercepts 1 and 2 without taking reciprocals. A plane parallel to the z axis has an infinite z intercept, so l = 0 always, and (102), whose third index is not zero, is also wrong. If the plane passes through the origin, move the origin to a neighboring lattice point before reading the intercepts.
(hkl) denotes a single plane, {hkl} a family of planes equivalent by symmetry, [uvw] a single direction, and <uvw> a family of equivalent directions. The slip system of FCC, for example, is written {111}<110>.
A negative index obtained after taking the reciprocal is shown with a bar over the number. In plain text a minus sign is sometimes used instead, as in (1-10).
They have the same orientation, but (200) denotes a set of planes with half the spacing of (100). In XRD, the second-order reflection from (100) is treated as a first-order reflection from the (200) planes with half the spacing.
Miller indices are found by expressing the intercepts on each axis in units of the lattice parameter, taking their reciprocals and reducing to the smallest integer ratio.
In a cubic crystal, a plane intercepts the x axis at 1a and…Atomic packing factor Body-centered cubic Face-centered cubic Hexagonal close-packed