Last updated: July 3, 2026
Miller Indices Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
The Miller indices calculator reports cubic interplanar spacing from a lattice parameter and plane (hkl). Miller indices are obtained from fractional intercept reciprocals, then reduced to the smallest integers. For cubic crystals, dₕₖₗ = a / √(h² + k² + l²).
For a cubic crystal, enter Miller indices h, k, and l plus lattice parameter a. The interplanar spacing is a divided by the square root of h squared plus k squared plus l squared.
Key Takeaways
- Miller indices are found by taking reciprocals of fractional intercepts and clearing fractions to the smallest integers.
- A zero index means the plane is parallel to that crystallographic axis.
- For cubic crystals, interplanar spacing is d = a / √(h² + k² + l²).
- Higher h²+k²+l² values produce smaller plane spacings for the same lattice parameter.
- The cubic formula should not be used for non-cubic crystal systems.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Miller indices are reciprocals of fractional intercepts; for cubic crystals d = a / √(h² + k² + l²)
Where:
- h,k,l=Miller indices of the plane(dimensionless integers)
- a=Cubic lattice parameter(Å)
- dₕₖₗ=Interplanar spacing for plane (hkl)(Å)
- x/a,y/b,z/c=Fractional intercepts used to derive Miller indices(lattice parameters)
Worked Examples
Cubic (111) plane with a = 4.0 Å
A common close-packed-style plane in a cubic cell; all three indices contribute equally.
- 1Write the plane as (111), so h² + k² + l² = 1² + 1² + 1² = 3.
- 2Take the square root: √3 = 1.7320508.
- 3Compute d = a/√(h²+k²+l²) = 4.0/1.7320508 ≈ 2.309 Å.
Cubic (200) plane with a = 3.6 Å
Only the h index contributes because k and l are zero; the plane is parallel to two axes.
- 1Write the plane as (200), so h² + k² + l² = 2² + 0² + 0² = 4.
- 2Take the square root: √4 = 2.
- 3Compute d = 3.6/2 = 1.8 Å.
Cubic (110) plane with a = 5.0 Å
The l index is zero, as for planes obtained from intercepts (1, 1, ∞).
- 1For intercepts (1, 1, ∞), reciprocals are (1, 1, 0), giving plane (110).
- 2Compute h² + k² + l² = 1² + 1² + 0² = 2 and √2 = 1.4142136.
- 3Compute d = 5.0/1.4142136 ≈ 3.536 Å.
Introduction
Miller indices describe the orientation of crystal planes using the integer triplet (hkl). They are obtained by taking reciprocals of a plane's fractional intercepts with the unit-cell axes and clearing fractions to the smallest integers. This calculator focuses on cubic crystals: enter the plane indices and lattice parameter to get the interplanar spacing dₕₖₗ. It pairs naturally with the cubic cell calculator for lattice parameters and the lattice energy calculator for ionic-solid trends. The notation follows crystallography conventions summarized by IUCr teaching resources and standard solid-state chemistry texts.
What are Miller indices?
Miller indices are a compact way to name planes in a crystal lattice. The symbols h, k, and l are integers proportional to the reciprocals of the plane intercepts on the crystallographic a, b, and c axes. A zero index means the plane is parallel to that axis, because its intercept is at infinity. The notation (111), (200), or (110) identifies one plane family in a cubic lattice.
(100) cuts the a axis and is parallel to b and c.
(110) cuts a and b but is parallel to c.
(111) cuts all three cubic axes symmetrically.
Negative indices are written with a bar in crystallography, such as (1̄10).
How to find Miller indices from intercepts
To derive Miller indices, first express each intercept in units of the lattice parameters: x/a, y/b, and z/c. Next take reciprocals. Finally multiply by the least common denominator to make the smallest whole-number set. For example, intercepts (1, 1, ∞) give reciprocals (1, 1, 0), so the plane is (110). Intercepts (1/2, 1, 1/3) give reciprocals (2, 1, 3), so the plane is (213).
Infinity becomes a reciprocal of zero; it does not make the Miller index undefined.
Cubic interplanar spacing formula
For a cubic crystal, the unit cell has a = b = c and all angles are 90°. That symmetry reduces the general reciprocal-lattice expression to d = a / √(h² + k² + l²). Larger index sums give smaller spacings, so higher-index planes are more closely packed in reciprocal space. X-ray diffraction uses this spacing in Bragg's law to connect diffraction angle and wavelength.
Use the same length unit for a and d. With a in Å, the calculator returns d in Å.
Reference values for common cubic planes
The table shows the denominator √(h²+k²+l²) for common cubic planes. Multiply your lattice parameter by the reciprocal of that denominator to get d.
| Plane | h²+k²+l² | d/a |
|---|---|---|
| (100) | 1 | 1 |
| (110) | 2 | 0.7071 |
| (111) | 3 | 0.5774 |
| (200) | 4 | 0.5000 |
| (220) | 8 | 0.3536 |
How to use this Miller indices calculator
Enter integer h, k, and l values for the plane, then enter the cubic lattice parameter a in angstroms. The primary result is interplanar spacing. The secondary output formats the plane label as a Miller string, which is useful when comparing with diffraction tables or results from the atomic mass calculator and molar mass calculator for material identity checks.
Use zero for a parallel axis.
Use negative numbers for barred indices when needed.
Do not enter all three indices as zero; (000) is not a plane.
Use a measured lattice constant when comparing against diffraction data.
Limitations and best practices
The d = a/√(h²+k²+l²) formula is specific to cubic crystals. Tetragonal, orthorhombic, hexagonal, monoclinic, and triclinic systems require different metric relationships. For experimental diffraction work, account for wavelength calibration, sample strain, instrument broadening, and temperature. Authoritative background is available from LibreTexts solid-state chemistry and the Crystallography Open Database.
Confirm the crystal system before applying the cubic formula.
Keep h, k, and l as integers after reducing from intercepts.
Report lattice parameter and temperature with calculated spacings.
Compare calculated d values with diffraction peaks using Bragg's law.
Quick Reference Card
Miller Indices — Quick Reference
Quick reference • Miller Indices Calculator
Intercepts → reciprocals → clear fractions → (hkl); cubic d = a/√(h²+k²+l²)Valid range: h, k, l are integers and not all zero; a must be positive for a physical d spacing.
Common Values
⚠ Watch Out
- •Do not use (000); it is not a valid crystallographic plane.
- •Do not apply the cubic d-spacing formula to hexagonal, tetragonal, or lower-symmetry cells.
- •Reduce reciprocals to the smallest integer triplet when deriving hkl from intercepts.
- •Use consistent units: a and d will have the same length unit.
Pro Tips
- →Check d/a values first to catch arithmetic mistakes before entering a.
- →Use measured lattice parameters for diffraction comparisons instead of idealized radii.
- →Write barred indices clearly when any intercept reciprocal is negative.
- →Pair d spacing with Bragg's law, nλ = 2d sinθ, to analyze X-ray diffraction peaks.
FAQs
What do Miller indices (hkl) mean?
They identify the orientation of a crystal plane. The integers h, k, and l are proportional to the reciprocals of the plane intercepts with the crystallographic axes.
How do I get (110) from intercepts?
For intercepts (1, 1, ∞), take reciprocals to get (1, 1, 0). No fraction clearing is needed, so the Miller plane is (110).
What does a zero Miller index mean?
A zero index means the plane is parallel to that axis. Mathematically, the axis intercept is at infinity and its reciprocal is zero.
What is the cubic interplanar spacing formula?
For cubic crystals, dₕₖₗ = a / √(h² + k² + l²), where a is the lattice parameter and h, k, l are the Miller indices.
Can I use this for hexagonal crystals?
No. Hexagonal crystals use a different spacing equation and often four-index Miller-Bravais notation. This calculator is limited to cubic crystals.
Why is (000) invalid?
The indices (000) would mean a plane parallel to all axes with infinite intercepts, which does not define a finite crystallographic plane family.