Miller Indices Calculator - Crystal Plane (hkl) Notation

What are Miller indices?

Miller indices are a set of three integers, written (h,k,l), that identify a plane in a crystal lattice. A crystal lattice is the repeating, orderly arrangement of atoms found in a solid material. Miller indices tell scientists exactly which slice through that lattice a plane represents, without needing to describe its geometry in words. They were introduced by the British mineralogist William Hallowes Miller in 1839 and remain the standard notation in crystallography today.

This calculator converts the intercepts of a plane on the x, y, and z crystallographic axes into its Miller indices (h,k,l).

Miller indices formula

Miller indices come from the reciprocals of a plane's axis intercepts. If a plane crosses the x, y, and z axes at distances a, b, and c from the origin, the indices satisfy:

$h : k : l = \frac{1}{a} : \frac{1}{b} : \frac{1}{c}$

How to calculate Miller indices from intercepts

1. Find where the plane crosses each axis, measured in units of the unit cell edge. Call these intercepts a, b, and c.
2. Take the reciprocal of each intercept: 1/a, 1/b, 1/c.
3. Multiply all three reciprocals by the same number so each becomes a whole number.
4. Divide the three whole numbers by their greatest common divisor, so they cannot be reduced any further.

The three resulting integers are h, k, and l. A plane that never crosses an axis is treated as intercepting it at infinity, and 1 divided by infinity is 0, so that axis gets a Miller index of 0.

Worked example

A plane intercepts the x-axis at 2, the y-axis at 3, and the z-axis at 6.

1. Intercepts: (2, 3, 6)
2. Reciprocals: (1/2, 1/3, 1/6)
3. Multiply by the lowest common multiple of the denominators, 6: (3, 2, 1)
4. The greatest common divisor of 3, 2, and 1 is 1, so the numbers cannot be reduced further.

The Miller indices are (321).

Other examples

• A plane through (1, ∞, 2) — parallel to the y-axis — has reciprocals (1, 0, 1/2). Multiplying by 2 gives (2, 0, 1). The 0 shows the plane never crosses the y-axis.
• A plane through (-1, 2, 3) has reciprocals (-1, 1/2, 1/3). Multiplying by 6 gives (-6, 3, 2), usually written (6̄32) with a bar over the negative index.
• A plane through (1, ∞, ∞) has reciprocals (1, 0, 0), giving (100). This is one of the six faces of a cube.

Special cases

A plane parallel to an axis never crosses it, so that intercept is treated as infinite and its Miller index is 0. A plane that intercepts an axis on the negative side of the origin gets a negative index, written with a bar over the number in formal notation, or with a minus sign in plain text. A plane passing through the origin itself has no defined Miller indices, because every set of parallel planes in a lattice must be offset from the origin to be indexed; the origin has to be shifted before such a plane can be described. Fractional intercepts are handled the same way as whole-number ones: take the reciprocal of each, then clear the fractions.

Indices are always reduced to the smallest possible integers. (622) and (311) describe the same family of planes, but crystallographers always write (311).

How to use this calculator

Enter the intercepts of the plane on the x, y, and z axes.

• Use a positive number for an intercept on the positive side of the origin, and a negative number for the negative side.
• Type "infinity", "inf", or "∞" for a plane that runs parallel to an axis.
• A zero intercept is not accepted, because a plane passing through the origin cannot be assigned Miller indices with this method.

The calculator computes the indices instantly and shows a diagram of the plane inside the unit cell.

Applications of Miller indices

X-ray diffraction. Each spot or peak in an X-ray diffraction pattern corresponds to a set of lattice planes with a specific (hkl). Bragg's law connects the spacing of those planes to the diffraction angle:

$n\lambda = 2d_{hkl}\sin\theta$

Here d_hkl is the spacing between planes (h,k,l), λ is the X-ray wavelength, and θ is the diffraction angle. Matching observed peaks to (hkl) values, a process called indexing, is a basic step in identifying an unknown crystal structure.

Materials science. Different crystal planes expose different arrangements of atoms, so they have different surface energies, different rates of chemical reaction, and different roles in how a metal deforms. In face-centered cubic metals such as aluminum, atomic planes slip past each other most easily along {111} planes. Silicon wafers for computer chips are usually cut along the (100) or (111) plane, because the choice affects how later manufacturing steps behave.

Mineralogy. The flat faces seen on a natural crystal, and the directions along which a mineral splits cleanly (its cleavage), both correspond to specific low-index planes. Halite (table salt) cleaves along {100} planes, and fluorite cleaves along {111} planes.

Miller-Bravais indices for hexagonal crystals

Hexagonal crystals, such as magnesium, zinc, and titanium, are usually described with a four-index system (hkil) instead of three. The extra index i is not independent; it is always i = -(h+k). This form makes it easier to see that two planes belong to the same symmetry-equivalent family, which is not always obvious from the three-index Miller notation alone in a hexagonal lattice.

History

Miller indices were introduced by William Hallowes Miller in his 1839 book A Treatise on Crystallography. Earlier systems for labeling crystal faces already existed, but Miller's method of using reciprocals of the intercepts made planes parallel to an axis easy to handle, since their intercept is infinite and its reciprocal is simply 0. After Max von Laue discovered X-ray diffraction by crystals in 1912, William Henry Bragg and William Lawrence Bragg showed that Miller indices describe exactly the planes responsible for diffraction peaks, work for which they shared the 1915 Nobel Prize in Physics.

Frequently asked questions

What are Miller indices used for? They identify specific planes in a crystal lattice. They are used to index X-ray and electron diffraction peaks, to describe slip planes in metals, to specify the orientation of semiconductor wafers, and to label the faces and cleavage of minerals.

What does a zero in a Miller index mean? It means the plane is parallel to that axis and never crosses it. For example, (201) is parallel to the y-axis.

What do negative Miller indices mean? They mean the plane crosses that axis on the negative side of the origin. A negative index is written with a bar over the number, such as (2̄11), or with a minus sign, such as (-2,1,1).

Why can't a plane through the origin have Miller indices? Miller indices come from an intercept's reciprocal. A plane through the origin intercepts every axis at 0, and 1/0 is undefined, so the method cannot assign indices unless the plane is shifted.

What is the difference between Miller and Miller-Bravais indices? Miller indices (hkl) use three numbers and work for any crystal system. Miller-Bravais indices (hkil) add a fourth, dependent index and are used mainly for hexagonal crystals, where they make symmetry-equivalent planes easier to recognize.

How does plane spacing relate to Miller indices? In a cubic crystal with edge length a, the spacing between adjacent (hkl) planes is d = a / √(h² + k² + l²). Planes with higher indices are more closely spaced. Other crystal systems use a similar but more complex formula that accounts for unequal axis lengths and non-90° angles.

References

1. Miller, W. H. (1839). A Treatise on Crystallography. Cambridge: J. & J. J. Deighton.
2. Ashcroft, N. W., & Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston.
3. Hammond, C. (2015). The Basics of Crystallography and Diffraction (4th ed.). Oxford University Press.
4. Cullity, B. D., & Stock, S. R. (2014). Elements of X-ray Diffraction (3rd ed.). Pearson Education.
5. International Union of Crystallography. (2016). International Tables for Crystallography, Volume A: Space-group symmetry. Wiley.