text: 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 &alpha; (between b and c), &beta; (between c and a), and &gamma; (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 &lang;&rang; 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, &lang;100&rang; 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 &lang;120&rang; 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 (one-form or axial) are Miller-indexed using coordinates in the reciprocal lattice basis  instead, generally between round brackets () (similar to square brackets [] for direct lattice vectors). Curly brackets {} (not to be confused with a mathematical set) are used to refer to a symmetrically equivalent class of reciprocal lattice vectors, similar to angle brackets &lang;&rang; for classes of direct lattice vectors.
  Here, , , and , where the unit cell volume is  ( denotes a dot product and  a cross product). 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 (cycles/Å).
  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 [uvw] (or zone axis) to be perpendicular to a reciprocal lattice vector (hkl) 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.