Crystallographers name planes of atoms with three whole numbers, the Miller indices (hkl). To find them, note where the plane cuts the x, y and z axes in units of the lattice constant, take the reciprocal of each intercept, and multiply through by the smallest number that turns them all into integers. A plane parallel to an axis meets it at infinity, so its index is 0; a plane cutting an axis on the negative side gets a bar over that index, written here as 1̄. The braces {hkl} denote the whole family of equivalent planes, for example the six faces of a cube are {100}.
For cubic crystals the spacing between neighbouring parallel planes is simply d = a/√(h² + k² + l²), so higher indices mean more closely spaced, less densely populated planes. That spacing is what X-ray diffraction measures: Bragg’s law λ = 2d·sin θ gives the angle at which each set of planes reflects. Not every set of planes produces a peak, though – in body-centred and face-centred lattices the extra atoms cancel some reflections (systematic absences), which is how a diffraction pattern reveals the lattice type.
The material menu fills in room-temperature lattice constants for common cubic metals and semiconductors, or you can enter your own.
How to use the Miller Indices Calculator
- Start from
- Intercepts on x, y, z (in units of a) — Separate with commas; use fractions like 1/2, negatives like -1, and ∞ or inf for a plane parallel to an axis.
- h
- k
- l
- Material (lattice constant)
- Lattice constant (a)
- Lattice type (for allowed reflections)
- X-ray wavelength λ (for the Bragg angle) — Cu Kα₁ = 1.5406 Å, Mo Kα₁ = 0.7093 Å.
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How it works
Enter the three intercepts (numbers, fractions such as 1/2, or ∞) or the indices themselves. The calculator takes reciprocals, clears fractions with the least common multiple, divides by any common factor, then uses the lattice constant to compute d, the first-order Bragg angle and the extinction rule for the chosen lattice.
Worked example
A plane cuts the axes at x = 1, y = 1 and z = 1/3 (in units of a). The reciprocals are 1, 1 and 3, already whole numbers, so the plane is (113). In copper, a = 3.615 Å and h² + k² + l² = 11, so d = 3.615 ÷ √11 = 1.0900 Å. With Cu Kα₁ X-rays (λ = 1.5406 Å), sin θ = 1.5406 ÷ (2 × 1.0900) = 0.7067, θ = 44.97° and 2θ = 89.94°. Because 1, 1 and 3 are all odd, the reflection is allowed for FCC copper. Intercepts 2, 3, ∞ would instead give (320): mixed odd and even indices, so that reflection is absent in copper.
How to interpret the result
Low indices such as (100), (110) and (111) are the most densely packed planes and usually form the natural faces and cleavage planes of crystals. In FCC metals the (111) planes are closest packed, which is why slip happens on them; in BCC metals the (110) planes are densest. The table of first allowed reflections lists the order in which peaks appear in a powder X-ray pattern, from the largest d (smallest angle) upwards.
Formula
(hkl) = reciprocals of the intercepts (in units of a), cleared of fractions and reduced · d = a ÷ √(h² + k² + l²) · Bragg: λ = 2d·sin θ
Worked example
Inputs — Start from: Axis intercepts of the plane; Intercepts on x, y, z (in units of a): 1, 1, 1/3; Material (lattice constant): Copper – FCC, 3.615 Å; X-ray wavelength λ (for the Bragg angle): 1.5406 Å.
Results — Miller indices (hkl) = (113); Plane = (113); Interplanar spacing d = 1.0900e-10; Interplanar spacing d = 1.0900e-10; Indices as numbers (h, k, l) = (1, 1, 3); Family of planes = {113} (sign and order ignored).
Key terms
- Miller indices
- Three integers (hkl) that name a lattice plane by the reciprocals of its intercepts.
- Interplanar spacing (d)
- Perpendicular distance between neighbouring parallel planes of the same family.
- Lattice constant (a)
- Edge length of the cubic unit cell.
- Bragg's law
- nλ = 2d·sin θ, the condition for constructive interference of X-rays reflected by lattice planes.
- Systematic absence
- A reflection with zero intensity because of the centring of the lattice.
Frequently asked questions
How do you find Miller indices from intercepts?
Take the reciprocals of the intercepts and clear the fractions. Intercepts 2, 3, ∞ give 1/2, 1/3, 0; multiplying by 6 gives (320).
What does a zero Miller index mean?
The plane is parallel to that axis – it meets it only at infinity, and 1/∞ = 0. (100) is parallel to the y and z axes.
How are negative indices written?
With a bar over the number, e.g. (1̄10). The plane cuts that axis on the negative side.
What is the difference between (hkl) and {hkl}?
Round brackets name one specific plane; braces name all symmetry-equivalent planes. In a cubic crystal {100} includes (100), (010), (001) and their negatives.
How do I calculate interplanar spacing?
For cubic lattices d = a/√(h² + k² + l²). Copper (a = 3.615 Å) has d₁₁₁ = 3.615/√3 = 2.087 Å.
Why are some reflections missing in FCC and BCC?
The extra atoms scatter out of phase for certain planes. BCC shows only peaks with h + k + l even; FCC only with h, k, l all odd or all even; diamond additionally needs h + k + l divisible by 4 when all are even.
What if the plane passes through the origin?
Its intercepts are zero, which has no reciprocal. Shift the origin to another lattice point (for example to the next cell along one axis) and read the intercepts again.
Is (222) the same plane as (111)?
They are parallel, but (222) describes planes half as far apart – d₂₂₂ = d₁₁₁ ÷ 2. Diffraction uses such indices for higher-order reflections, so the direct-entry mode does not reduce them.