Visualizing Crystal Structures with Miller Indices

How to Calculate Miller Indices for Crystal PlanesUnderstanding Miller indices is essential for studying crystallography, materials science, and solid-state physics. Miller indices provide a concise notation to describe the orientation of crystal planes and directions in a lattice. This article explains the concept from first principles, walks through step-by-step calculations, gives examples for common lattices, and highlights practical tips and common pitfalls.

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What are Miller indices?

Miller indices are a set of three integers (h k l) that denote the orientation of a plane in a crystal lattice relative to the lattice vectors. They are defined such that the plane intercepts the crystal axes at positions that are the reciprocals of these integers (after clearing fractions). Miller indices are always written in parentheses for planes, e.g., (1 0 0), and in square brackets for directions, e.g., [1 0 0]. For families of equivalent planes or directions related by symmetry, curly braces { } and angle brackets < > are used, respectively.

Key fact: Miller indices are integers proportional to the reciprocals of the plane’s intercepts with the unit-cell axes.

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Why Miller indices matter

• They uniquely describe plane orientations in a lattice (up to a common factor).
• They are used to index X-ray diffraction peaks and interpret diffraction patterns.
• They help predict slip systems and mechanical behavior in crystals.
• They indicate surface orientations for thin-film growth and etching.

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Step-by-step method to calculate Miller indices

1. Identify the intercepts of the plane with the crystallographic axes (x, y, z) in terms of the unit cell dimensions a, b, c. Use coordinates along the unit cell axes; intercepts can be at infinity if the plane is parallel to an axis.

2. Express the intercepts as fractions of the unit cell lengths: x/a, y/b, z/c. If the plane intercepts the axes at positions p, q, r (in units of a, b, c respectively), write intercepts as p, q, r.

3. Take the reciprocals of these intercepts: 1/p, 1/q, 1/r. If an intercept is at infinity (plane parallel to that axis), its reciprocal is 0.

4. Clear fractions by multiplying by the least common multiple to obtain smallest integers (h, k, l). If all three values share a common factor, reduce to the smallest integer set.

5. Enclose the three integers in parentheses to denote the plane: (h k l). Negative indices are shown with a bar above the number (in plain text often written as a minus sign, e.g., (1 -1 0) or (1̅10)).

Example summary: plane intercepts (⁄₂ a, 1 b, ∞) → reciprocals (2, 1, 0) → Miller indices (2 1 0).

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Worked examples

Example 1 — Plane intercepts at 1a, 1b, 1c:

• Intercepts: 1, 1, 1
• Reciprocals: 1, 1, 1
• Miller indices: (1 1 1)

Example 2 — Plane intercepts at ⁄₂ a, 1 b, ∞:

• Intercepts: ⁄₂, 1, ∞
• Reciprocals: 2, 1, 0
• Miller indices: (2 1 0)

Example 3 — Plane intercepts at -1a, 2b, 1c (plane crosses negative x):

• Intercepts: -1, 2, 1
• Reciprocals: -1, ⁄₂, 1
• Clear fractions (multiply by 2): -2, 1, 2
• Miller indices: (−2 1 2) (write negative as a bar above the 2: (2̅ 1 2))

Example 4 — Cubic system: plane passing through lattice points (1,0,0), (0,1,0), (0,0,1) — that is the plane through three face centers:

• The intercepts with axes are 1, 1, 1 → (1 1 1) (this is a common close-packed plane in FCC crystals).

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Miller indices for directions vs planes

• Plane: (h k l)
• Direction: [u v w] — direction indices are determined from vector components along the unit cell axes, reduced to smallest integers.
• Important relation in cubic crystals: the direction [h k l] is perpendicular to the plane (h k l). This is generally true only for cubic lattices where axes are orthogonal and have equal scale. For non-cubic crystals, use reciprocal lattice vectors to relate planes and directions.

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Using reciprocal lattice vectors (general crystallography)

For non-orthogonal or non-cubic unit cells, Miller indices (h k l) correspond to a plane whose normal is given by the reciprocal-lattice vector: G = h a* + k b* + l c, where a, b, c are reciprocal basis vectors. This formalism ensures correct treatment of skewed unit cells and different axis lengths.

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Examples in common lattices

• Simple cubic (SC): (1 0 0) are faces, (1 1 0) are edge-centered planes, (1 1 1) are body-diagonal planes.
• Face-centered cubic (FCC): low-index close-packed plane is (1 1 1); common directions <1 1 0> are close-packed directions.
• Body-centered cubic (BCC): close-packed directions are <1 1 0>; common low-index planes include (1 1 0) and (1 1 1) (less close-packed than FCC).

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Negative indices notation

In printed text, a negative index is shown with a bar above the number, e.g., (1̅ 1 0). In plain ASCII you can write ( -1 1 0 ) or (1- 1 0) — prefer (−1 1 0) or (1̅10) where formatting permits.

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Common pitfalls

• Forgetting to take reciprocals before clearing fractions.
• Treating intercepts in angstroms instead of units of the lattice constants a, b, c.
• Not accounting for infinite intercepts (plane parallel to axis → index 0).
• Assuming direction [h k l] is perpendicular to (h k l) in non-cubic lattices.

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Quick reference procedure

1. Find intercepts in units of a, b, c.
2. Take reciprocals.
3. Clear fractions to obtain smallest integers.
4. Enclose as (h k l), use bar notation for negatives.

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Visual tips

• If a plane crosses the x-axis at ⁄₂ of a, put 2 as the first index.
• If a plane is parallel to an axis, its corresponding index is 0.
• Think of Miller indices as a compact code for how the plane “cuts” the unit cell.

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Final example (complete calculation)

Plane through points (1,0,0), (0,⁄₂,0), (0,0,⁄₃) in a cubic cell:

• Intercepts: 1, ⁄₂, ⁄₃
• Reciprocals: 1, 2, 3
• Miller indices: (1 2 3)

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If you want, I can provide diagrams, a step-by-step worksheet for practice problems, or worked examples for non-cubic cells using reciprocal lattice vectors.