What are Miller indices?

Miller indices (hkl) are a set of three integers that describe the orientation of a crystal plane relative to the unit cell axes. They are the reciprocals of the fractional intercepts of the plane with the axes, reduced to the smallest integers. A bar over an index (e.g., h̄) denotes a negative value. The notation (hkl) describes a specific plane, {hkl} describes the family of equivalent planes, [hkl] denotes a direction, and ⟨hkl⟩ the family of equivalent directions.

How do I find Miller indices from intercepts?

Take the reciprocal of each intercept (intercept of infinity for a parallel axis gives 0). Then multiply by the LCM (least common multiple) of the denominators to obtain the smallest integer set. For example, intercepts (1, 2, infinity) give reciprocals (1, 1/2, 0), which scale to (2, 1, 0).

What is the d-spacing formula for cubic crystals?

For a cubic crystal with lattice parameter a, the interplanar spacing is d = a / sqrt(h² + k² + l²). This is a simplification of the general formula valid only for cubic systems (a = b = c, all angles 90°). For non-cubic systems (tetragonal, hexagonal, orthorhombic, etc.), different formulas apply.

What is Bragg's law and how does d-spacing relate to XRD?

Bragg's law states that constructive interference (diffraction peaks) occurs when 2d·sin(θ) = nλ, where d is the interplanar d-spacing, θ is the diffraction angle, n is the reflection order, and λ is the X-ray wavelength. A larger d-spacing shifts diffraction peaks to smaller 2θ angles. This allows phase identification in XRD patterns by matching observed d-spacing values to known crystal structures.

What is a plane family {hkl}?

A plane family {hkl} includes all crystallographically equivalent planes related by the point group symmetry of the crystal. For a cubic crystal, {100} includes (100), (010), (001), (1̄00), (01̄0), and (001̄) — 6 planes total. All planes in a family share the same d-spacing. The number of equivalent planes is the multiplicity of the reflection.

What lattice parameter units should I use?

Enter the lattice parameter a in angstroms (Å) for most crystallography work — the result will be in angstroms as well. You can also use nanometers (nm); just be consistent. For example, silicon has a = 5.431 Å, iron (BCC) a = 2.866 Å, aluminum (FCC) a = 4.046 Å.

Miller Indices Calculator

Enter lattice intercepts (a, b, c) or direct h, k, l values to get Miller index notation and d-spacing for cubic crystal systems.

Crystal Plane Inputs

Input Mode

Lattice Intercepts — enter 0 for parallel (∞)

a-axis

b-axis

c-axis

Fractional unit-cell intercepts. Enter 0 if the plane is parallel to that axis.

Lattice Parameter a (Å)

Cubic crystal lattice parameter in angstroms (Å). Used for d-spacing calculation.

Common Cubic Crystals

Notation Reference

(hkl)Specific plane

{hkl}Plane family (all equivalents)

[hkl]Crystal direction

h̄ (bar notation)Negative index

d = a / √(h²+k²+l²)d-spacing (cubic)

Enter crystal plane intercepts or indices and click Calculate.

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Summary

The Miller Indices Calculator converts crystal plane intercepts into standard (hkl) notation and computes the interplanar d-spacing for cubic crystal systems. You can enter raw intercepts (a, b, c) — where the tool takes their reciprocals and reduces to the smallest integer set — or enter h, k, l integers directly. Negative indices are displayed using bar (overbar) notation, as is conventional in crystallography. The d-spacing is calculated from the cubic lattice spacing formula, making results directly applicable to Bragg's law and X-ray diffraction analysis.

How it works

Choose whether to enter lattice intercepts (a, b, c) or direct Miller indices (h, k, l).
For intercept mode: enter the fractional intercepts of the plane with the unit cell axes. Enter 0 for a parallel plane (infinity intercept).
For direct mode: enter h, k, l as integers. At least one must be nonzero.
Enter the lattice parameter a (in angstroms or nanometers) for the cubic crystal.
Click Calculate to get the (hkl) Miller index notation, bar notation for negatives, d-spacing, and plane family information.
Use the preset crystal buttons to auto-fill lattice parameters for common cubic materials.

Use cases

Determine Miller indices for crystal planes in cubic metals (FCC, BCC, simple cubic) during materials science coursework.
Calculate d-spacing for X-ray diffraction (XRD) peak identification and Bragg's law calculations.
Verify interplanar spacing for common cubic materials such as silicon, aluminum, iron, and NaCl.
Identify plane families and their multiplicity for texture analysis and pole figure interpretation.
Convert intercept data from crystallographic drawings into standard (hkl) notation for publications.
Understand systematic absences in XRD patterns by comparing d-spacing values across plane families.
Support materials characterization lab reports with precise crystallographic calculations.

Frequently Asked Questions

Last updated: 2026-09-30 · Reviewed by Nham Vu