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Concept
Zone axis
Summary
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 (as distinct from the reciprocal 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.
Miller index
The translational invariance of a crystal lattice is described by a set of unit cell, direct lattice basis vectors (contravariant or polar) 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 (between b and c), β (between c and a), and γ (between a and b). Direct lattice vectors have components measured in distance units, like meters (m) or angstroms (Å).
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 (i.e. the set of vectors generated by an action of the lattice's symmetry group). In the case of a cubic lattice, for instance, 〈100〉 represents [100], [010], [001], [00], [00] and [00] 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 (e.g., ).
The term "zone axis" more specifically refers to the direction of a direct-space lattice vector. For example, since the [120] and [240] lattice vectors are parallel, their orientations both correspond the 〈120〉 zone of the crystal.
Official source
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