How to Read and Use Miller Indices in CrystallographyCrystallography studies the orderly, repeating arrangement of atoms in solids. A central language in that field is Miller indices — a compact notation that identifies crystal planes and directions. This article explains what Miller indices are, how to read and write them, how they relate to crystal geometry, and how to use them in common crystallographic tasks. Examples, step-by-step procedures, and common pitfalls are included to help beginners become confident with Miller indices.
What are Miller indices?
Miller indices are a set of three (or sometimes four) integers, typically written as (h k l), that denote the orientation of a crystallographic plane in a lattice. They are reciprocal intercepts of the plane with the crystallographic axes, scaled to the smallest set of integers. For directions, square brackets u v w denote a vector in the lattice.
- Planes: (h k l)
- Directions: [u v w]
- Families of planes/directions: {h k l} and
Why Miller indices matter
Miller indices let you:
- Identify and compare crystal planes and directions unambiguously.
- Relate real-space lattice geometry to diffraction patterns (X-ray, electron).
- Describe slip systems and cleavage planes important for materials properties.
- Work with symmetry-related sets of planes and directions.
Conventions and notations
- Parentheses (h k l) denote a single plane.
- Curly braces {h k l} denote all symmetrically equivalent planes.
- Square brackets [u v w] denote a specific direction (a vector).
- Angle brackets denote a family of equivalent directions.
- A bar over a number indicates a negative integer, e.g., (1 1 -1) is often shown as (1 1 ar{1}).
- For hexagonal crystals with four-index Miller–Bravais indices, use (h k i l) with i = −(h + k).
How to determine Miller indices for a plane (step-by-step)
- Identify the intercepts of the plane with the crystallographic axes in terms of lattice constants a, b, c. If a plane intercepts at x = pa, y = qb, z = rc, then the intercepts are p, q, r in units of the lattice vectors.
- Express the intercepts as fractions of the unit cell axes: p : q : r.
- Take the reciprocals: 1/p : 1/q : 1/r.
- Clear fractions by multiplying by the smallest common factor to get integers (h k l).
- Enclose in parentheses: (h k l).
Special cases:
- If the plane is parallel to an axis, its intercept is at infinity; the reciprocal (1/∞) is zero. So a plane parallel to the z-axis has l = 0.
- If a plane passes through the origin, translate to a different origin (choose an equivalent lattice point) to find finite intercepts.
Example:
- A plane that cuts the axes at a/2, b, ∞ (i.e., parallel to z) has intercepts ⁄2 : 1 : ∞ → reciprocals 2 : 1 : 0 → Miller indices (2 1 0).
From Miller indices to plane geometry
To sketch a plane given (h k l):
- Write intercepts as a/h, b/k, c/l (with 0 where h, k, or l = 0).
- Plot these intercepts in a unit cell and join them to visualize the plane.
Note: For non-orthogonal unit cells (e.g., monoclinic, triclinic), the geometric interpretation requires using the basis vectors and not just orthogonal coordinates.
Miller–Bravais indices (hexagonal systems)
Hexagonal lattices use four-index Miller–Bravais notation (h k i l) to symmetrically represent three equivalent a-axes and the c-axis. The relation is: i = −(h + k)
Conversion:
- From three-index (h k l) in hexagonal: use i = −(h + k) to get four indices.
- From four-index (h k i l) to three-index: drop i, keeping (h k l) with the constraint above.
Families of planes are often written with four indices for clarity in hexagonal crystals.
Miller indices for directions
To find the direction [u v w] that connects two lattice points:
- Express the vector between the points in terms of the lattice vectors: Δ = u*a + v*b + w*c.
- If the vector components are fractional, multiply by the smallest factor to obtain integers.
- Enclose in square brackets: [u v w].
Relation between planes and directions:
- The direction normal to the plane (h k l) in an orthogonal system is [h k l]. In non-orthogonal systems, use reciprocal lattice vectors: the plane (h k l) has normal vector proportional to ha* + kb* + lc, where a, b, c are reciprocal-lattice vectors.
Reciprocal lattice and diffraction
Miller indices are directly tied to the reciprocal lattice, which is central to diffraction. A plane (h k l) corresponds to a reciprocal-lattice vector G = ha* + kb* + lc*. Diffraction occurs when the scattering vector matches a reciprocal-lattice vector (Laue/Bragg conditions).
Bragg’s law in terms of d-spacing: 2d_{hkl} sin θ = nλ
where the interplanar spacing d{hkl} depends on lattice type and lattice constants. For a cubic lattice: d{hkl} = a / sqrt(h^2 + k^2 + l^2)
Calculating d-spacing (examples)
- Cubic: d_{hkl} = a / sqrt(h^2 + k^2 + l^2)
- Tetragonal: d_{hkl} = 1 / sqrt( (h^2 + k^2)/a^2 + l^2/c^2 )
- Orthorhombic: d_{hkl} = 1 / sqrt( h^2/a^2 + k^2/b^2 + l^2/c^2 )
(For monoclinic and triclinic systems the formula involves angles between axes; use lattice metric tensor.)
Common applications
- X-ray and electron diffraction indexing.
- Determining slip systems in metallurgy: slip often occurs on close-packed planes like {111} in FCC and {110} in BCC.
- Surface science: low-index surfaces like (100), (110), (111) have distinct atomic arrangements and properties.
- Structure determination and unit cell characterization.
Examples and practice problems
-
Find Miller indices for a plane cutting the axes at (⁄2)a, (⁄3)b, and c.
- Intercepts: ⁄2 : ⁄3 : 1 → reciprocals 2 : 3 : 1 → (2 3 1)
-
Convert the hexagonal plane with three-index (1 0 -1) to four-index format:
- Compute i = −(h + k) = −(1 + 0) = −1 → (1 0 ar{1} l) — include l as appropriate for the c-axis component.
-
What is d for (2 1 0) in a cubic lattice with a = 4 Å?
- d = a / sqrt(2^2 + 1^2 + 0^2) = 4 / sqrt(5) ≈ 1.789 Å
Common mistakes and tips
- Forgetting that parallel to an axis gives a zero index.
- Not clearing fractions to smallest integers.
- Confusing plane indices (parentheses) and direction indices (brackets).
- For hexagonal crystals, forgetting the third index i in four-index notation.
- In non-orthogonal cells, assuming [h k l] is perpendicular to (h k l) — only true in orthogonal systems.
Practical tip: always sketch the unit cell and intercepts the first few times you work with Miller indices; spatial visualization builds intuition.
Summary
Miller indices are a concise, powerful notation to describe crystal planes and directions. They connect real-space lattice geometry with diffraction, materials behavior, and surface properties. With the steps and examples above you should be able to read, write, convert, and apply Miller indices across common crystal systems.
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