How do I find Miller indices from intercepts?

Write the intercepts as fractions of the unit cell, take reciprocals (∞ gives 0), then multiply by the smallest number that clears the fractions and reduce by any common factor. Intercepts (1, 2, ∞) give reciprocals (1, 1/2, 0), which scale to (210). Intercepts (1/2, −1, 1/3) give (2 1̄ 3).

What are Miller indices?

Miller indices (hkl) are three integers that describe the orientation of a crystal plane relative to the unit cell axes. A bar over an index means a negative value. (hkl) is one plane, {hkl} the family of equivalent planes, [uvw] a direction and ⟨uvw⟩ a family of directions.

How do I convert hexagonal (hkl) to Miller–Bravais (hkil)?

Keep h, k and l and add the third basal index i = −(h + k). So (101) becomes (1 0 1̄ 1), (100) becomes (1 0 1̄ 0) and (001) becomes (0001). The four-index form makes equivalent planes in a hexagonal crystal look alike, for example (1 0 1̄ 0) and (0 1 1̄ 0).

What is the d-spacing formula for cubic and hexagonal crystals?

Cubic: d = a / √(h² + k² + l²). Tetragonal: 1/d² = (h² + k²)/a² + l²/c². Orthorhombic: 1/d² = h²/a² + k²/b² + l²/c². Hexagonal: 1/d² = 4(h² + hk + k²)/(3a²) + l²/c². For magnesium (a = 3.209 Å, c = 5.211 Å) the (1 0 1̄ 1) spacing is 2.452 Å.

Which reflections are allowed in BCC and FCC crystals?

Simple cubic: all (hkl). BCC: only h + k + l even, so (110), (200), (211) appear and (100), (111) do not. FCC: h, k, l all odd or all even, so (111), (200), (220), (311). Diamond cubic (Si, Ge): FCC peaks where all are odd or h + k + l is a multiple of 4, so (200) and (222) are missing.

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

Bragg's law, 2d sin θ = nλ, gives the angle at which planes with spacing d diffract X-rays of wavelength λ. For silicon (111), d = 3.1356 Å and Cu Kα1 (1.5406 Å) gives θ = 14.22°, 2θ = 28.44°. A peak exists only when λ ≤ 2d.

What is a plane family {hkl} and its multiplicity?

A family {hkl} contains all planes made equivalent by the crystal's symmetry; they share one d-spacing. In a cubic crystal {100} has 6 members, {110} 12, {111} 8, {210} 24 and {321} 48. Multiplicity scales the intensity of a powder diffraction peak.

What lattice parameter units should I use?

Use angstroms (Å); d-spacing is returned in the same unit. 1 nm = 10 Å. Examples: silicon a = 5.431 Å, α-iron a = 2.866 Å, aluminium a = 4.046 Å, magnesium a = 3.209 Å and c = 5.211 Å.

Miller Indices Calculator

Enter plane intercepts (decimals, fractions or ∞) or h, k, l to get (hkl), hexagonal (hkil), the {hkl} family and multiplicity, d-spacing for cubic, tetragonal, orthorhombic or hexagonal cells, and the Bragg 2θ for Cu, Mo, Co, Fe, Cr or Ag Kα.

Crystal Plane Inputs

Crystal System

Input Mode

Plane Intercepts (unit-cell fractions)

x (a)

y (b)

z (c)

Accepts decimals, fractions (1/2, -2/3) and inf or ∞ for a plane parallel to that axis (0 is also read as parallel). For hexagonal cells enter intercepts on a₁, a₂ and c.

Lattice Parameters (Å)

Cubic Lattice (reflection rules)

X-ray Wavelength λ (Å)

Common Crystals

Notation & d-Spacing Formulas

(hkl) / {hkl}Plane / family of planes

[uvw] / ⟨uvw⟩Direction / family

(hkil), i = −(h + k)Hexagonal Miller–Bravais

1/d² = (h²+k²+l²)/a²Cubic

1/d² = (h²+k²)/a² + l²/c²Tetragonal

1/d² = h²/a² + k²/b² + l²/c²Orthorhombic

1/d² = 4(h²+hk+k²)/3a² + l²/c²Hexagonal

Enter crystal plane intercepts or indices and click Calculate.

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Allowed Cubic Reflections: SC, BCC, FCC and Diamond

Structure-factor rules decide which (hkl) peaks appear in a powder XRD pattern. N = h² + k² + l², d = a / √N, and multiplicity is the number of equivalent planes in the {hkl} family (cubic m3̄m symmetry).

{hkl}	N	Multiplicity	d / a	SC	BCC	FCC	Diamond
{100}	1	6	1.0000	Yes	—	—	—
{110}	2	12	0.7071	Yes	Yes	—	—
{111}	3	8	0.5774	Yes	—	Yes	Yes
{200}	4	6	0.5000	Yes	Yes	Yes	—
{210}	5	24	0.4472	Yes	—	—	—
{211}	6	24	0.4082	Yes	Yes	—	—
{220}	8	12	0.3536	Yes	Yes	Yes	Yes
{221}	9	24	0.3333	Yes	—	—	—
{310}	10	24	0.3162	Yes	Yes	—	—
{311}	11	24	0.3015	Yes	—	Yes	Yes
{222}	12	8	0.2887	Yes	Yes	Yes	—
{320}	13	24	0.2774	Yes	—	—	—
{321}	14	48	0.2673	Yes	Yes	—	—
{400}	16	6	0.2500	Yes	Yes	Yes	Yes
{410}	17	24	0.2425	Yes	—	—	—
{330}	18	12	0.2357	Yes	Yes	—	—
{331}	19	24	0.2294	Yes	—	Yes	Yes
{420}	20	24	0.2236	Yes	Yes	Yes	—
{422}	24	24	0.2041	Yes	Yes	Yes	Yes
{333}	27	8	0.1925	Yes	—	Yes	Yes
{511}	27	24	0.1925	Yes	—	Yes	Yes
{440}	32	12	0.1768	Yes	Yes	Yes	Yes
{531}	35	48	0.1690	Yes	—	Yes	Yes

Rock salt (NaCl) follows the FCC rule; its all-odd peaks are weak because Na⁺ and Cl⁻ scatter out of phase. Example: Si (diamond, a = 5.431 Å) has its first peak at (111), d = 3.1356 Å, 2θ = 28.44° with Cu Kα1; (200) is absent.

Worked Examples

Intercepts 1, 2, ∞: reciprocals 1, 1/2, 0 → multiply by 2 → (210). In aluminium (a = 4.046 Å) d = 4.046 / √5 = 1.8094 Å.
Intercepts 1/2, −1, 1/3: reciprocals 2, −1, 3 → (2 1̄ 3), family {321}, 48 equivalent planes.
Hexagonal intercepts a₁ = 1, a₂ = ∞, c = 1: (hkl) = (101), i = −(1 + 0) = −1 → (1 0 1̄ 1). For magnesium (a = 3.209, c = 5.211 Å) 1/d² = 4/3 × 1/3.209² + 1/5.211² → d = 2.4521 Å, 2θ = 36.62° with Cu Kα1.

Summary

The Miller Indices Calculator converts plane intercepts (decimals, fractions such as 1/2, or ∞ for a parallel axis) into reduced (hkl) indices with bar notation for negatives, or into four-index Miller–Bravais (hkil) notation for hexagonal crystals with i = −(h + k). It computes the interplanar spacing d for cubic, tetragonal, orthorhombic and hexagonal cells, the {hkl} family and cubic multiplicity, whether the reflection is allowed in simple cubic, BCC, FCC or diamond lattices, and the Bragg angle θ and 2θ for a chosen X-ray wavelength. A server-rendered table lists the first cubic reflections with N = h² + k² + l², multiplicity and extinction rules.

How it works

Choose the crystal system: cubic, tetragonal, orthorhombic or hexagonal.
Enter intercepts as decimals, fractions (1/2) or inf / ∞ for a parallel axis, or switch to direct mode and type h, k, l.
The tool takes reciprocals, clears fractions and divides by the greatest common divisor to get the smallest integers (hkl). For hexagonal cells it adds i = −(h + k).
Enter the lattice parameters (a, and b or c when the system needs them) or pick a preset such as Al, Si, α-Fe, Mg or rutile.
Pick the X-ray tube (Cu Kα1 1.5406 Å by default) to get θ and 2θ from Bragg's law, 2d sin θ = nλ.
For cubic crystals, select SC, BCC, FCC or diamond to see whether the (hkl) peak is allowed or systematically absent.

Use cases

Determine Miller indices of crystal planes from intercepts in materials science coursework.
Convert hexagonal (hkl) to four-index Miller–Bravais (hkil) for HCP metals such as Mg, Zn and Ti.
Calculate d-spacing and 2θ to index peaks in a powder XRD pattern.
Check which reflections appear in BCC, FCC and diamond-cubic diffraction patterns.
Find the multiplicity of a cubic {hkl} family for intensity calculations.
Compute interplanar spacing for tetragonal or orthorhombic phases such as rutile TiO₂.

Frequently Asked Questions

Last updated: 2026-10-09 · Reviewed by Nham Vu