Interplanar Angle Calculator
Calculate the angle between two crystal planes from their Miller indices in cubic, tetragonal, orthorhombic, and hexagonal crystals.
📐 What is an Interplanar Angle?
The interplanar angle is the angle between two families of crystal planes, each identified by its Miller indices (h k l). By crystallographic convention it is measured between the two plane normals rather than between the plane surfaces themselves, because the normal is the direction that actually appears in diffraction geometry. Every plane's normal is a reciprocal lattice vector, so calculating an interplanar angle is really a dot product between two reciprocal lattice vectors, normalised by their lengths. That single idea produces all four formulas on this page, one for each crystal system, and explains why the expressions get progressively more complicated as symmetry is lost.
Interplanar angles are the working currency of several everyday laboratory tasks. When indexing a selected-area electron diffraction pattern, the measured angles between diffraction spots are compared with the calculated angles for candidate index assignments until one set matches. When interpreting an electron backscatter diffraction map or a Kikuchi pattern, the band intersections encode interplanar angles directly. In physical metallurgy, slip and twinning systems are described by the angles between {111} planes and {110} planes and directions, and Schmid factor calculations use exactly these angles. Mineralogists use the constancy of interfacial angles, first noted by Nicolas Steno in 1669, to identify minerals from crystal morphology alone.
A common misconception is that interplanar angles depend on the size of the unit cell. In the cubic system they do not, the edge length a cancels completely, so silicon, gold, diamond, and rock salt all share identical angles between the same index pairs. Once symmetry is lower, the axial ratios matter enormously, and a tetragonal or hexagonal angle can shift by tens of degrees as c/a changes. A second misconception is that a hexagonal calculation needs the four-index Bravais-Miller form. It does not, because the third index i is fixed by i = −(h + k), so this calculator takes the compact three-index form.
This calculator returns the angle in degrees and radians, the raw cosine, and the supplementary angle, together with a full breakdown of the reciprocal-space arithmetic. It also builds a table of angles from your first plane to twelve common low-index planes, which turns pattern indexing into a lookup rather than a dozen separate calculations, and for the non-cubic systems it charts how the angle responds to the axial ratio c/a.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Cubic (111) Against (100)
Any cubic crystal, plane 1 = (1 1 1), plane 2 = (1 0 0)
Example 2 — Tetragonal Rutile TiO₂, (110) Against (101)
Rutile a = 4.594 Å, c = 2.959 Å, plane 1 = (1 1 0), plane 2 = (1 0 1)
Example 3 — Hexagonal Zinc, Basal Plane Against a Pyramidal Plane
Zinc a = 2.665 Å, c = 4.947 Å, plane 1 = (0 0 1) basal, plane 2 = (1 0 1) pyramidal
Example 4 — Orthorhombic Cell, (110) Against (011)
Orthorhombic a = 5 Å, b = 6 Å, c = 7 Å, plane 1 = (1 1 0), plane 2 = (0 1 1)
❓ Frequently Asked Questions
🔗 Related Calculators
What is the interplanar angle in crystallography?
The interplanar angle is the angle between the normals of two families of crystal planes, identified by their Miller indices (h1 k1 l1) and (h2 k2 l2). It is the angle you measure between diffraction spots in an electron diffraction pattern and between facets on a crystal, so it is the main geometric fingerprint used to index unknown patterns.
How do you calculate the angle between two planes in a cubic crystal?
Use cos(phi) = (h1h2 + k1k2 + l1l2) / sqrt((h1 squared + k1 squared + l1 squared)(h2 squared + k2 squared + l2 squared)). For (111) and (100) this gives cos(phi) = 1/sqrt(3) = 0.5774, so phi = 54.7356 degrees. The lattice parameter cancels out completely.
Why does the cubic formula not need a lattice parameter?
In a cubic crystal all three axes are equal and orthogonal, so the lattice parameter a appears as a common factor in the numerator and in both terms under the square root, and cancels exactly. Every cubic material, from silicon to sodium chloride to gold, therefore has identical interplanar angles for the same pair of indices.
What is the tetragonal interplanar angle formula?
cos(phi) = [(h1h2 + k1k2)/a squared + l1l2/c squared] divided by the square root of [((h1 squared + k1 squared)/a squared + l1 squared/c squared)((h2 squared + k2 squared)/a squared + l2 squared/c squared)]. Setting a equal to c recovers the cubic formula exactly.
What is the hexagonal interplanar angle formula?
cos(phi) = [h1h2 + k1k2 + 0.5(h1k2 + h2k1) + (3a squared/4c squared) l1l2] divided by the square root of the two matching self terms. The half-cross term comes from the 120 degree angle between the a1 and a2 axes, which makes the hexagonal reciprocal metric non-diagonal.
Do I enter three or four indices for a hexagonal crystal?
Enter three: h, k, and l. The four-index Bravais-Miller notation (h k i l) used for hexagonal crystals carries a redundant third index defined by i = minus (h + k), so it adds no information. A plane written (1 0 minus-1 1) is entered here as h = 1, k = 0, l = 1.
What is the angle between (111) and (110) in a cubic crystal?
35.2644 degrees. cos(phi) = (1 + 1 + 0) / sqrt(3 x 2) = 2/sqrt(6) = 0.8165, so phi = 35.2644 degrees. This pair appears constantly in FCC slip-system geometry, where {111} slip planes meet {110} directions and planes.
Is the interplanar angle the angle between planes or between normals?
Between the plane normals, which is the standard crystallographic convention and the quantity that matches diffraction geometry. Two planes physically meet at either that angle or its supplement (180 degrees minus it), so this calculator reports both values side by side.
Can Miller indices be negative in this calculator?
Yes. Negative indices are conventionally printed with an overbar, but here you simply type a minus sign, for example h = 1, k = 1, l = -1 for the plane usually written (1 1 bar-1). Sign matters for interplanar angles, unlike for d-spacing, because the cross terms in the numerator change sign.
What does the common-plane angle table show?
It lists the angle between your first plane and twelve widely used low-index planes, from (100) through (311), computed with the same crystal system and lattice parameters you entered. It is a fast lookup when you are indexing a diffraction pattern and need to test several candidate planes at once.
Why does the angle change with the c/a ratio?
In any non-cubic system the reciprocal lattice is stretched along one or more axes, so the direction of a plane normal depends on the axial ratio. The chart plots the interplanar angle for your chosen pair of planes as c/a varies, which shows how sensitive an indexing assignment is to the lattice parameters you assumed.
Can two different planes have an interplanar angle of zero?
Yes, when the two index triples are integer multiples of each other, such as (111) and (222), because they describe parallel families of planes with different spacings. The calculator returns 0 degrees in that case, which is correct, the normals point the same way even though the d-spacings differ.