Miller Indices Calculator: (hkl) from Intercepts, d-Spacing and Bragg Angle

How it works

Miller indices (h k l) = reciprocals of the intercepts (x/a, y/b, z/c), cleared of fractions and reduced to the smallest integers; an intercept of ∞ gives 0 · cubic d-spacing d = a ÷ √(h² + k² + l²) · Bragg’s law λ = 2d sin θ · angle between planes cos φ = (h₁h₂ + k₁k₂ + l₁l₂) ÷ (√(h₁² + k₁² + l₁²) √(h₂² + k₂² + l₂²))

To name a crystal plane, write where it cuts the x, y and z axes as multiples of the cell edges (a plane parallel to an axis cuts it at ∞), take the reciprocals, then multiply by the smallest number that clears every fraction. A plane cutting at ½, 1 and ∞ gives reciprocals 2, 1 and 0, so it is (2 1 0). A negative intercept gives a negative index, written with a bar over the number. If the plane goes through the origin, move the origin to another corner first; the calculator tells you when that is needed. You can also start from (h k l) to see the intercepts. For cubic crystals the spacing between neighboring (hkl) planes is d = a ÷ √(h² + k² + l²). With the X-ray wavelength (Cu Kα by default) the calculator applies Bragg’s law to give the diffraction angle 2θ. It also checks the structure-factor rule for your lattice: in BCC only planes with h + k + l even reflect, in FCC h, k and l must be all odd or all even, and diamond cubic adds h + k + l = 4n for the all-even case. When a plane is absent it shows the first allowed higher order, and a table lists the first reflections you would see in a powder pattern. The family {hkl} and its multiplicity count the symmetry-equivalent planes. Unlike the equation-of-a-plane calculator in geometry, this tool uses crystallographic intercepts and lattice constants. All formulas assume a cubic cell.

Worked example

A plane in copper (FCC, a = 3.615 Å) cuts the axes at ½, 1 and ∞. The reciprocals are 2, 1 and 0, already whole numbers, so the Miller indices are (2 1 0); the family {2 1 0} has 24 equivalent planes. h² + k² + l² = 5, so d = 3.615 ÷ √5 = 1.6167 Å. With Cu Kα X-rays (λ = 1.5406 Å), θ = asin(1.5406 ÷ 3.2334) = 28.455°, so 2θ = 56.91°. But in an FCC lattice 2, 1 and 0 mix odd and even, so this reflection is absent; the first allowed order is (4 2 0) with d = 0.8083 Å at 2θ = 144.71°. The plane is 26.57° from (1 0 0).

Sources: Omni Calculator: Miller indices calculator (a = 2 Å, (2 0 1) example) · King Saud University lecture notes: positions, directions and planes (worked intercept examples) · ConductScience: Miller indices d-spacing calculator

Frequently asked questions