Crystal Planes and Miller Indices

Introduction

Crystal planes are labeled by Miller indices such as (100), (110) and (111). The labels arise from reciprocals of axis intercepts and describe plane orientation rather than a chemical formula. In cubic crystals, they also connect simply to interplanar spacing and diffraction peak positions.

Core explanation

To find indices, express a plane's intercepts on crystal axes in cell-edge units, take reciprocals and clear fractions to obtain the smallest integer triple. A plane meeting x at a, y at b and running parallel to z has intercepts (1,1,∞). Reciprocals are (1,1,0), giving (110). An infinite intercept means the plane does not cut that axis, so its index is zero.

The labels (100), (010) and (001) correspond to planes perpendicular to one cubic axis and parallel to the other two. For a cubic cell with edge a, the (100) plane spacing is a; (110) spacing is a/√2 and (111) spacing is a/√3. The general cubic relation is dₕₖₗ=a/√(h²+k²+l²). It does not apply unchanged to noncubic cells, where edges and angles require a different metric.

Negative intercepts give negative indices, conventionally written with a bar over the relevant digit, such as (1̄10) for a negative first-axis index in a suitable notation. The chosen plane can be translated parallel to itself to avoid an origin intercept that is zero and inconvenient for reciprocal construction. Miller indices label orientation and repeating plane family, not one unique sheet passing through one chosen origin.

Parentheses (hkl) refer to a particular plane orientation, while braces {hkl} denote a symmetry-related family in common crystallographic notation. Brackets [uvw] refer to a direction, not a plane. In a cubic crystal, direction [hkl] is perpendicular to plane (hkl), but that simple perpendicular relation is not universal in noncubic metrics. Keeping brackets and parentheses distinct prevents conceptual confusion.

Diffraction connects Miller indices with measured spacing. With known X-ray λ and θ, Bragg's law gives d. If a cubic cell is assumed and an index assigned, a=d√(h²+k²+l²). Several peaks should give a consistent a; one peak alone may fit several possible index assignments. Reflection intensities and systematic absences help identify actual lattice and basis.

The indices do not tell where atoms sit on a plane or what bonding occurs there. They describe geometric orientation. Surface chemistry and slip systems often depend on crystal planes, but such properties require atom occupancy and interactions in addition to hkl labels.

Step-by-step reasoning

1. Express axis intercepts in units of a, b and c. 2. Replace parallel-axis infinity with reciprocal zero. 3. Take reciprocals of all intercepts. 4. Clear fractions to get the smallest integer triple (hkl). 5. For a cubic cell only, use dₕₖₗ=a/√(h²+k²+l²).

Visual explanation

Draw a cubic cell with a (100) face, a diagonal (110) cut and a triangular (111) cut. Label their x, y and z intercepts and show the reciprocal step beneath each. Add a bracketed arrow [100] to distinguish a direction from plane (100).

Real-world analogy

A map can label the direction a wall faces without specifying which particular parallel wall is meant. Miller indices identify orientation and spacing families, not one unique physical plane in an infinite lattice.

Real-world example

Crystal diffraction peaks may be indexed as (111) or (200). In a cubic structure, their measured d values help calculate the unit-cell edge and test whether the assigned phase is consistent.

Why?

Why does a plane parallel to one axis have zero for that Miller index? Its intercept on that axis is effectively infinite, and the reciprocal of infinity is zero in the indexing construction.

Common misconception

“(110) means the plane contains one atom of type 1 and one of type 0.” The digits are reciprocal-intercept orientation labels, not atom counts or chemical formulas.

Worked example

A plane intercepts x at a, y at 2b and is parallel to z. Intercepts in edge units are (1,2,∞); reciprocals are (1,1/2,0). Multiply all by two to obtain smallest integers (2,1,0), so the plane is (210). For a cubic cell, its spacing would be a/√(2²+1²+0²)=a/√5.

Quick check

1. What index corresponds to an axis parallel to a plane? Answer: Zero for that axis.

Exam focus

Show intercept, reciprocal and integer steps, distinguish planes from directions and use the simple spacing equation only when cubic geometry is stated.

Advanced insight

Systematic reflection absences arise from lattice centering and basis symmetry, so not every mathematically possible (hkl) plane family produces an observed diffraction peak in a given crystal.

Summary

Miller indices label crystal-plane orientation through reciprocal intercepts. In cubic cells, dₕₖₗ=a/√(h²+k²+l²). Indices describe geometry, while diffraction intensities and material properties need additional structural information.

Practice questions

1. What are indices for intercepts (1,1,∞)? Answer: (110). 2. What is cubic d₁₁₁ if a=0.300 nm? Answer: 0.300/√3≈0.173 nm. 3. Why is [100] not the same notation as (100)? Answer: Square brackets name a direction; parentheses name a plane orientation.