The Miller indices calculator above does the two things that come up constantly in crystallography teaching and practice. It converts a set of axial intercepts into the reduced integer triplet that names a lattice plane, and it turns that triplet plus the lattice parameters into the perpendicular spacing between successive planes. It then applies Bragg's law to give the angle at which that family of planes will diffract.
Arb Digital publishes free calculators that show each step rather than just the answer. Miller indexing is a short procedure with three places to slip: taking reciprocals in the wrong order, mishandling a plane that never meets an axis, and forgetting to clear the fractions to the smallest integers. The output panel here reports the intermediate reciprocals so you can see where a mismatch came from.
What This Miller Indices Calculator Does
A Miller index triplet names a whole family of parallel, equally spaced lattice planes. Everything about how a crystal diffracts, cleaves, grows and slips is described in that language, so being able to move between a geometric description and the index notation is basic equipment.
In intercept mode the calculator takes the three axial intercepts, inverts them, scales to integers and divides by the greatest common divisor. In index mode you type the triplet directly. Either way it then computes the interplanar spacing from the crystal system you select and the lattice parameters you supply, using the standard closed-form expressions for cubic, tetragonal, orthorhombic and hexagonal cells. The grid completes the picture with the Bragg angle, the detector angle and the sum of squares that indexes a cubic powder pattern.
Two boundaries are worth stating clearly. The live Bragg's law calculator solves nλ = 2d sinθ and takes the d-spacing as an input value; this page is what produces that d-spacing from indices and lattice parameters, so the two are designed to be used in sequence. Separately, the unit cell calculator describes what a repeating cell contains — its volume, its atom count, its packing efficiency and density. This page describes the orientation and spacing of planes cut through it. Contents versus orientation is the whole distinction.
How to Use It
- Choose your input. Intercepts if you are working from a drawing, indices if you already have the triplet.
- Enter intercepts in cell-edge units. A plane crossing a at one cell edge and b at two has intercepts of 1 and 2, whatever those edges measure in ångströms.
- Use inf for a parallel axis. A plane that never meets the c axis has an intercept of infinity, which inverts to an index of zero.
- Pick the crystal system and fill in only the lattice parameters it needs. The unused boxes disappear.
- Read the detector angle if you are comparing against a diffraction pattern. Powder patterns are almost always plotted against 2θ, not θ.
The Method and How It Is Calculated
Indexing runs in four steps. Find where the plane crosses each crystallographic axis, in units of the cell edge. Take the reciprocal of each intercept, so a plane parallel to an axis gives zero. Multiply the three reciprocals by whatever common factor makes them all integers. Divide by their greatest common divisor to reduce to the smallest set. The result is written in round brackets without commas, and a negative index is written with a bar over it, which is conventionally typed as a leading minus sign.
A worked example: intercepts of 1, 2 and 3 give reciprocals of 1, 1/2 and 1/3. Multiplying by six gives 6, 3 and 2, whose greatest common divisor is one, so the plane is (632). A plane crossing a at 1 and running parallel to both b and c gives reciprocals of 1, 0 and 0, which is the (100) plane, the cube face.
Spacing follows from the crystal system. For cubic cells 1/d² = (h² + k² + l²)/a²; for tetragonal, (h² + k²)/a² + l²/c²; for orthorhombic, h²/a² + k²/b² + l²/c²; and for hexagonal, (4/3)(h² + hk + k²)/a² + l²/c². These are the standard forms tabulated in the LibreTexts section on lattice plane orientation and Miller indices.
Checking the default: aluminium is cubic with a = 4.0495 Å, so the (111) spacing is 4.0495/√3 = 2.3380 Å. With copper K-alpha at 1.5406 Å, Bragg's law gives sinθ = 1.5406/(2 × 2.3380) = 0.3295, so θ = 19.24 degrees and the detector angle is 38.47 degrees. That is the position of the aluminium (111) reflection in a real laboratory pattern. Characteristic X-ray wavelengths can be looked up in the NIST X-ray transition energies search form.
Planes, Directions and Brackets
Crystallography uses four bracket types and they are not decorative. Round brackets (hkl) name a specific plane or family of parallel planes. Curly brackets {hkl} name the whole set of planes that symmetry makes equivalent: in a cubic crystal {100} covers the six cube faces. Square brackets [uvw] name a direction, not a plane, and the numbers mean something entirely different — they are vector components, not reciprocals. Angle brackets <uvw> name a family of symmetry-equivalent directions.
Writing (111) when you meant [111] is the single most common notational error in the subject. In a cubic system the two happen to coincide geometrically, because the [111] direction is perpendicular to the (111) plane, and that coincidence teaches a habit that fails in every other crystal system. In a tetragonal or hexagonal cell the [110] direction is not normal to the (110) plane, and assuming it is will produce a wrong answer with no warning.
The bar notation matters too. A plane written with a bar over the k index cuts the b axis on the negative side. Because a plane family extends infinitely in both directions, (111) and the all-negative version describe the same set of planes, but the sign is not free to change on individual indices.
Hexagonal Cells and the Four-Index System
Hexagonal crystals are conventionally described with four indices rather than three, written (hkil), where the third index is redundant and fixed by i = −(h + k). The reason is that the three-index system hides the symmetry: the three equivalent prismatic faces of a hexagonal prism come out as (100), (010) and the mixed-sign form, which do not look related. In four-index notation they become permutations of the same digits, which makes the symmetry obvious at a glance.
The spacing formula uses only h, k and l, so the redundant index changes nothing in the arithmetic. This calculator reports it alongside the triplet when you select the hexagonal system, so you can move between the two conventions without error. Directions in hexagonal systems have their own four-index convention with a different conversion rule, which is a separate source of confusion and not covered here.
Why Some Planes Never Appear in a Pattern
The most valuable thing to understand about this calculation is that it is necessary but not sufficient. Every (hkl) triplet has a geometric d-spacing and therefore a nominal Bragg angle, but a great many of those reflections have zero intensity and simply do not appear on the detector. The geometry says where a peak could be; the arrangement of atoms inside the cell says whether it is there.
The mechanism is destructive interference between atoms at different positions in the unit cell, formalised as the structure factor. The consequences are systematic. A body-centred cubic lattice extinguishes every reflection where h + k + l is odd, so the first reflections are (110), (200) and (211). A face-centred cubic lattice extinguishes every reflection where h, k and l are not all odd or all even, which is why aluminium's pattern starts with (111) and (200) rather than (100). Screw axes and glide planes produce their own systematic absences, and identifying the space group from that pattern of absences is a standard step in structure determination.
This page computes geometry only. If you index a pattern by matching computed spacings to observed peaks and find perfect matches for reflections that are absent from your data, you have probably not made an error — you have found the extinction rule and learned something about the lattice type. The diffraction grating calculator covers the analogous optical case, where the same interference mathematics applies to a one-dimensional periodic structure.
Where Miller Indices Are Used Outside Diffraction
Cleavage is the most visible application. A crystal splits along the planes with the weakest bonding across them, which is why mica peels into sheets and why diamond is cut on specific faces. Those cleavage planes are identified by their indices, and in mineralogy the index of the dominant cleavage is a diagnostic property.
Plastic deformation in metals runs on slip systems, each one a combination of a plane and a direction within it, and the available systems are what make face-centred cubic metals such as aluminium and copper ductile while hexagonal metals such as magnesium are not. Semiconductor wafers are specified by surface orientation, with silicon (100) and (111) wafers behaving differently in etching and oxidation because the surface atom density differs. Catalysis is orientation-sensitive for the same reason: the number and geometry of exposed atoms per unit area changes from face to face, and the Langmuir isotherm calculator models the adsorption that happens on those exposed sites.
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Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Forgetting to take reciprocals — indices are the inverted intercepts, so a plane cutting a at 2 contributes an index of 1 after clearing fractions, not 2.
- Entering intercepts in ångströms — they are in units of the cell edges, so a plane cutting a halfway along has an intercept of 0.5 regardless of the cell size.
- Confusing (hkl) with [uvw] — the first names a plane through reciprocals, the second names a direction through vector components, and they coincide only in cubic systems.
- Using the wrong spacing formula for the system — applying the cubic form to a tetragonal cell silently gives a wrong d-spacing and therefore a wrong angle.
- Expecting every computed reflection to appear — systematic absences from lattice centring and symmetry elements remove many of them entirely.
Related Free Tools From Arb Digital
Take the spacing from this page into the Bragg's law calculator to solve for wavelength or order, and use the unit cell calculator for the contents, volume and density of the same cell. The lattice energy calculator covers the energetics that hold the crystal together, the diffraction grating calculator handles the optical analogue, and the density calculator checks a computed cell density against a measured one. The full free online tools hub lists everything else.
Frequently Asked Questions
They are three integers that name a family of parallel lattice planes in a crystal. They are obtained by taking the reciprocals of the plane's intercepts with the crystallographic axes, clearing fractions, and reducing to the smallest whole numbers.
Express the intercepts in units of the cell edges, take the reciprocal of each, multiply all three by whatever factor makes them integers, then divide by their greatest common divisor. Intercepts of 1, 2 and 3 become reciprocals 1, one half and one third, which reduce to 632.
It means the plane is parallel to that axis and never intersects it, so the intercept is infinite and its reciprocal is zero. The 100 plane is parallel to both b and c, which makes it a cube face in a cubic crystal.
Round brackets name a plane and are built from reciprocal intercepts. Square brackets name a direction and are vector components. They coincide geometrically in cubic systems only, which is why the two are so often confused.
Use the formula for your crystal system. Cubic uses a divided by the square root of h squared plus k squared plus l squared. Tetragonal, orthorhombic and hexagonal each add separate terms for their distinct axis lengths.
Because the three-index form hides the threefold symmetry of the equivalent prism faces. The four-index form adds a redundant third index equal to minus the sum of the first two, which makes symmetry-related planes appear as permutations of the same digits.
Because of systematic absences. Destructive interference between atoms inside the unit cell removes whole classes of reflections: body-centred lattices lose every one where h plus k plus l is odd, and face-centred lattices lose all mixed odd and even triplets.
Powder diffraction patterns are plotted against the detector angle two theta, which is twice the Bragg angle. Reading a peak position as theta when the axis is two theta halves the angle and gives a d-spacing that is roughly double the true value.
This calculator is provided for education and general reference. It describes how Miller indices and interplanar spacings are computed and is not laboratory or instrument guidance; follow the methods issued by your own institution.
