Miller Indices Calculator
Calculate the interplanar spacing of a cubic crystal from its lattice constant and plane indices.
Crystal inputs
Interplanar spacing
Calculation breakdown
| Term | Entered index | Squared contribution |
|---|---|---|
| h | 2 | 4 |
| k | 0 | 0 |
| l | 1 | 1 |
| Total | – | 5 |
For a cubic lattice, only the sum h² + k² + l² enters the spacing formula. Signs affect plane orientation but not this distance because the indices are squared.
How to use this Miller indices calculator
What this calculator does
This calculator finds the separation between neighboring parallel crystal planes in a cubic unit cell. It uses the cubic-spacing identity dhkl = a / √(h² + k² + l²), where a is the cubic lattice constant and h, k, and l identify the plane family. The result is an exact geometric consequence of the entered model, but it does not identify an unknown crystal, determine whether a diffraction reflection is allowed, or replace a full crystallographic refinement. The International Union of Crystallography definition of Miller indices explains how the notation represents lattice planes.
When to use it
Use the calculator when checking a powder-diffraction assignment, comparing the spacing of low-index planes, preparing a crystallography exercise, or estimating a surface-plane separation before a materials simulation. It is specifically for cubic cells, where all three cell edges are equal and mutually perpendicular. For tetragonal, orthorhombic, monoclinic, or other systems, the spacing equation needs additional lattice parameters and sometimes angular terms.
How to calculate
- The calculator opens with a complete demonstration: a = 2 Å and plane (201). The displayed spacing and the example XLSX are ready immediately.
- Choose a Material preset or leave “Custom lattice constant.” A preset inserts a representative lattice constant; edit the value when your specimen or source gives a different parameter.
- Enter the positive Lattice constant, a in ångströms, then enter integer values for Miller index h, Miller index k, and Miller index l. At least one index must be nonzero.
- Read Interplanar distance, dhkl, the substituted equation, Index square sum, Index norm, and the calculation table. Results update as you type.
- Select Download Excel to export the current validated inputs and outputs as a real workbook. Reset clears the demonstration and all calculated data; export stays unavailable until a complete valid state is entered again.
Input guide
Material preset is an optional selector. Choose Custom, Aluminum, Copper, Silicon, or Sodium chloride. The preset writes a lattice constant in ångströms, but it is not a certificate value for every temperature, composition, or strain state. A common mistake is treating a representative preset as more authoritative than the lattice parameter reported for the actual sample.
Lattice constant, a is required and accepts an ordinary positive decimal such as 2, 3.615, or 5.431. Use a decimal point, not a decimal comma, and do not type the Å symbol inside the field. Increasing a while holding the indices fixed increases d in direct proportion. Zero, negative values, scientific notation, and mixed text are rejected.
Miller index h, Miller index k, and Miller index l are required signed integers, for example 2, 0, and 1. Negative indices are permitted because a barred crystallographic index may be typed as a negative integer; the distance is unchanged by sign. Fractions and decimals are rejected because Miller indices are conventionally reduced to integers. The triple (0,0,0) is invalid because it does not define a plane family.
Output guide
Plane displays the entered (hkl) triplet. a repeats the validated lattice constant. √Σ is the square root of h² + k² + l². The main Interplanar distance, dhkl is shown in ångströms to four decimals; higher-index combinations generally produce smaller spacing for the same a. Index square sum is the exact integer h² + k² + l², while Index norm is its square root. The calculation table lists each index and its squared contribution, allowing the denominator to be checked term by term. These outputs are identities for the ideal cubic geometry, not a recommendation or a measurement uncertainty statement.
Worked example
For the startup plane (201), h² + k² + l² = 2² + 0² + 1² = 5, so the index norm is √5 = 2.2361. With a = 2.0000 Å, the spacing is d201 = 2 / √5 = 0.894427... Å, displayed as 0.8944 Å. The result panel, breakdown table, and downloaded workbook all use the same unrounded canonical value. For broader context on how lattice-plane spacings connect to diffraction, see the LibreTexts treatment of Miller indices and interplanar spacing.
Formula, interpretation, and common mistakes
The cubic formula follows from reciprocal-space geometry. Because the cell axes have the same length and are orthogonal, the reciprocal-lattice vector associated with (hkl) has magnitude √(h² + k² + l²) / a, and the corresponding real-space plane spacing is its reciprocal. This is why changing the sign of an index does not change d, while increasing the magnitude of any index decreases d.
Keep the unit consistent. One ångström equals 10 – 10 metre, and lattice constants are often reported in ångströms or picometres. This calculator expects ångströms throughout. The BIPM SI Brochure provides the underlying SI length framework, while crystallographic practice commonly retains the ångström for atomic-scale distances.
Three errors are especially common: entering axis intercepts instead of their reciprocal integer indices, using the (000) triplet, and applying the cubic formula to a noncubic cell. Another subtle issue is confusing a plane, conventionally written (hkl), with a direction, conventionally written [uvw]. In cubic crystals the normal to (hkl) is parallel to [hkl], but that simple relationship does not generalize to every crystal system. The IUCr introduction to crystal planes and diffraction gives a fuller geometric discussion.