Zone Axis Calculator
Find the zone axis shared by two crystal planes, check the Weiss zone law, and list all planes belonging to a given zone.
🔷 What is a Zone Axis?
A zone axis is a crystallographic direction, written [u v w] in square brackets, that lies parallel to the line where two or more crystal planes intersect. The whole set of planes sharing that direction is called a zone, and the direction itself is the zone axis. The idea goes back to Christian Samuel Weiss in the early nineteenth century, long before X-ray diffraction existed, because mineralogists noticed that the faces of a natural crystal fall into families whose intersection edges are all parallel. That purely geometric observation turns out to be exactly the relation that governs modern electron diffraction.
Zone axes matter in several everyday laboratory situations. In transmission electron microscopy, tilting a specimen until the electron beam runs down a low-index zone axis produces a symmetric spot pattern, and the pattern is labelled by that zone axis, for example a [001] pattern or a [110] pattern. Every diffraction spot visible in such a pattern comes from a plane that belongs to that zone, so the Weiss zone law hu + kv + lw = 0 becomes a hard consistency test for any proposed indexing. In electron backscatter diffraction, zone axes appear as the bright points where several Kikuchi bands cross. In metallurgy, the intersection line of two slip planes is a zone axis and controls how dislocations cross-slip.
A frequent misconception is that the direction [h k l] is always perpendicular to the plane (h k l). That is true only in cubic crystals, where the direct and reciprocal lattices happen to line up. In tetragonal, hexagonal, orthorhombic, monoclinic, and triclinic crystals, the direction with the same three integers is generally not the plane normal, and treating it as one produces silently wrong answers. Another common slip is forgetting to reduce the cross product, [2 0 −2] and [1 0 −1] describe the same line, and only the reduced form is conventional.
This calculator covers the three questions that come up in practice. It finds the zone axis shared by two planes using the cross product of their Miller indices, it tests whether a given plane belongs to a given zone using the Weiss zone law, and it lists every distinct plane family that belongs to a zone up to a maximum index you choose. Every mode shows the full arithmetic and none of them needs any lattice parameters, because the zone relations are pure integer geometry valid in every crystal system.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Zone Axis of (111) and (100)
Plane 1 = (1 1 1), plane 2 = (1 0 0), a pairing that appears in almost every FCC diffraction pattern
Example 2 — A Cross Product That Needs Reducing
Plane 1 = (2 0 2), plane 2 = (1 1 1)
Example 3 — Weiss Zone Law Test
Does the plane (1 −1 0) belong to the zone [1 1 1]?
Example 4 — Every Plane in the [001] Zone
Zone axis [0 0 1], searching all Miller indices up to 2
❓ Frequently Asked Questions
🔗 Related Calculators
What is a zone axis in crystallography?
A zone axis is a crystallographic direction [u v w] that is parallel to the line of intersection of two or more crystal planes. All planes sharing that direction form a zone. In transmission electron microscopy the zone axis is the beam direction that produces a symmetric diffraction pattern, so it labels the orientation of the specimen.
How do you find the zone axis from two planes?
Take the cross product of the two Miller index triples: u = k1l2 minus l1k2, v = l1h2 minus h1l2, w = h1k2 minus k1h2. Then divide u, v and w by their greatest common divisor. For (111) and (100) the result is [0 1 -1].
What is the Weiss zone law?
The Weiss zone law states that a plane (h k l) belongs to the zone [u v w] if and only if hu + kv + lw = 0. It is a dot product between a reciprocal lattice vector and a direct lattice vector, and it holds in every crystal system regardless of symmetry, including triclinic.
Why must the zone axis indices be reduced?
Direction indices describe an orientation, not a length, so [2 0 -2] and [1 0 -1] denote the same direction. Dividing by the greatest common divisor gives the conventional reduced form, which makes results directly comparable with tables, software output, and published patterns.
What does it mean when the cross product is zero?
It means the two planes are parallel, for example (111) and (222), or (110) and (-1 -1 0). Parallel planes never intersect in a single line, so they do not define a unique zone axis. This calculator reports that case as an error rather than returning [0 0 0], which is not a valid direction.
How many planes belong to one zone?
Infinitely many in principle, since any integer combination of two planes in the zone is also in the zone. In practice only low-index planes matter, so the calculator lists every distinct plane family up to a maximum index you choose. For zone [0 0 1] with indices up to 2 there are 8 distinct families.
Why are planes with a common factor excluded from the list?
Because (2 2 0) has the same orientation as (1 1 0) and would be a duplicate entry in an orientation list. Only primitive index triples, those whose greatest common divisor is 1, are listed, which is the standard convention for naming plane families.
How is a zone axis used to index an electron diffraction pattern?
Every spot in a single-crystal selected-area diffraction pattern comes from a plane in the same zone, and the zone axis is the beam direction. Index two spots, take the cross product to get the zone axis, then verify every remaining spot satisfies hu + kv + lw = 0. Any spot that fails means the assignment is wrong.
What is the difference between [uvw] and (hkl) notation?
Square brackets [u v w] denote a direction, a vector in the direct lattice. Round brackets (h k l) denote a plane, described by Miller indices in the reciprocal lattice. Angle brackets and curly braces denote the symmetry-equivalent families of directions and planes respectively.
In a cubic crystal, is [hkl] perpendicular to (hkl)?
Yes, but only in the cubic system. Cubic symmetry makes the direct and reciprocal lattices parallel, so the direction [1 1 1] is normal to the plane (1 1 1). In every other crystal system the direction with the same indices is generally not perpendicular to the plane, which is a very common source of confusion.
Can zone axis indices be negative?
Yes. A negative index is usually printed with an overbar, but here you simply type a minus sign. The calculator normalises the sign so the first non-zero index is positive, since [1 0 -1] and [-1 0 1] are the same line through the crystal traversed in opposite senses.
Does the Weiss zone law depend on the lattice parameters?
No. The relation hu + kv + lw = 0 involves only the integers, never a, b, c or the interaxial angles. That is why it works identically for cubic, hexagonal, and triclinic crystals, and why this calculator needs no unit cell dimensions in any of its three modes.