Zone axis

Zone axis, a term sometimes used to refer to "high-symmetry" orientations in a crystal, most generally refers to any direction referenced to the direct lattice of a crystal in three dimensions. It is therefore indexed with direct lattice indices, instead of with Miller indices.
High-symmetry zone axes through a crystal lattice, in particular, often lie in the direction of tunnels through the crystal between planes of atoms. This is because, as we see below, such zone axis directions generally lie within more than one plane of atoms in the crystal.

Zone-axis indexing

The translational invariance of a crystal lattice is described by a set of unit cell, direct lattice basis vectors called a, b, and c, or equivalently by the lattice parameters, i.e. the magnitudes of the vectors, called a, b and c, and the angles between them, called α, β, and γ. Direct lattice vectors have components measured in distance units, like meters or angstroms.
A lattice vector is indexed by its coordinates in the direct lattice basis system and is generally placed between square brackets . Thus a direct lattice vector, or, is defined as. Angle brackets ⟨⟩ are used to refer to a symmetrically equivalent class of lattice vectors. In the case of a cubic lattice, for instance, ⟨100⟩ represents,,,, and because each of these vectors is symmetrically equivalent under a 90 degree rotation along an axis. A bar over a coordinate is equivalent to a negative sign.
The term "zone axis" more specifically refers to the direction of a direct-space lattice vector. For example, since the and lattice vectors are parallel, their orientations both correspond the ⟨120⟩ zone of the crystal. Just as a set of lattice planes in direct space corresponds to a reciprocal lattice vector in the complementary space of spatial frequencies and momenta, a "zone" is defined as a set of reciprocal lattice planes in frequency space that corresponds to a lattice vector in direct space.
The reciprocal space analog to a zone axis is a "lattice plane normal" or "g-vector direction". Reciprocal lattice vectors are Miller-indexed using coordinates in the reciprocal lattice basis instead, generally between round brackets . Curly brackets are used to refer to a symmetrically equivalent class of reciprocal lattice vectors, similar to angle brackets ⟨⟩ for classes of direct lattice vectors.
Here,,, and, where the unit cell volume is . Thus a reciprocal lattice vector or has a direction perpendicular to a crystallographic plane and a magnitude equal to the reciprocal of the spacing between those planes, measured in spatial frequency units, e.g. of cycles per angstrom.

Object	Equivalence Class	Specific Vector	Units	Transformation
zone or lattice-vector s_uvw			direct space, e.g.	contravariant or polar
plane or g-vector g_hkl			reciprocal space, e.g.	covariant or axial

A useful and quite general rule of crystallographic "dual vector spaces in 3D", e.g. reciprocal lattices, is that the condition for a direct lattice vector to be perpendicular to a reciprocal lattice vector can be written with a dot product as. This is true even if, as is often the case, the basis vector set used to describe the lattice is not Cartesian.

Zone-axis patterns

By extension, a zone-axis pattern is a diffraction pattern taken with an incident beam, e.g. of electrons, X-rays or neutrons traveling along a lattice direction specified by the zone-axis indices . Because of their small wavelength λ, high energy electrons used in electron microscopes have a very large Ewald sphere radius, so that electron diffraction generally "lights up" diffraction spots with g-vectors that are perpendicular to .
One result of this, as illustrated in the figure above, is that "low-index" zones are generally perpendicular to "low-Miller index" lattice planes, which in turn have small spatial frequencies and hence large lattice periodicities. A possible intuition behind this is that in electron microscopy, for electron beams to be directed down wide tunnels between columns of atoms in a crystal, directing the beam down a low-index zone axis may help.