Zone axis

The translational invariance of a crystal lattice is described by a set of unit cell, direct lattice basis vectors (contravariant[1] 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).[2][3] 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 ${\displaystyle \{{\bf {a}},{\bf {b}},{\bf {c}}\}}$ and is generally placed between square brackets []. Thus a direct lattice vector ${\displaystyle {\bf {s}}_{uvw}}$, or ${\displaystyle [uvw]}$, is defined as ${\displaystyle u{\bf {a}}+v{\bf {b}}+w{\bf {c}}}$. 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], [100], [010] and [001] 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., ${\displaystyle [{\overline {1}}00]=-{\bf {a}}}$).

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. 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[4][5] 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[6] or axial) are Miller-indexed using coordinates in the reciprocal lattice basis ${\displaystyle \{{\bf {a}}^{*},{\bf {b}}^{*},{\bf {c}}^{*}\}}$ 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 ⟨⟩ for classes of direct lattice vectors.

Here, ${\displaystyle {\bf {a}}^{*}\equiv (b\times c)/V_{c}}$, ${\displaystyle {\bf {b}}^{*}\equiv (c\times a)/V_{c}}$, and ${\displaystyle {\bf {c}}^{*}\equiv (a\times b)/V_{c}}$, where the unit cell volume is ${\displaystyle V_{c}={\bf {a}}\cdot ({\bf {b}}\times {\bf {c}})}$ (${\displaystyle \cdot }$ denotes a dot product and ${\displaystyle \times }$ a cross product). Thus a reciprocal lattice vector ${\displaystyle {\bf {g}}_{hkl}}$ or ${\displaystyle (hkl)=h{\bf {a}}^{*}+k{\bf {b}}^{*}+l{\bf {c}}^{*}}$ has a direction perpendicular to a crystallographic plane and a magnitude ${\displaystyle g_{hkl}=1/d_{hkl}}$ equal to the reciprocal of the spacing ${\displaystyle d_{hkl}}$ between those ${\displaystyle (hkl)}$ planes, measured in spatial frequency units, e.g. of cycles per angstrom (cycles/Å).

Bracket conventions for direct lattice and reciprocal lattice Miller indices in crystallography

Object	Equivalence Class	Specific Vector	Units	Transformation
zone or lattice-vector suvw	${\displaystyle \langle {uvw\rangle }\,}$	${\displaystyle [uvw]\,}$	direct space, e.g. [meters]	contravariant or polar
plane or g-vector ghkl	${\displaystyle \{hkl\}\,}$	${\displaystyle (hkl)\,}$	reciprocal space, e.g. [cycles/m]	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 [uvw] (or zone axis) to be perpendicular to a reciprocal lattice vector (hkl) can be written with a dot product as ${\displaystyle [uvw]\cdot (hkl)=uh+vk+wl=0}$. This is true even if, as is often the case, the basis vector set used to describe the lattice is not Cartesian.

By extension, a [uvw] zone-axis pattern (ZAP) 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 [uvw]. Because of their small wavelength λ, high energy electrons used in electron microscopes have a very large Ewald sphere radius (1/λ), so that electron diffraction generally "lights up" diffraction spots with g-vectors (hkl) that are perpendicular to [uvw].[7]

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 (g-values) and hence large lattice periodicities (d-spacings). A possible intuition behind this is that in electron microscopy, for electron beams to be directed down wide (i.e. easily visible) tunnels between columns of atoms in a crystal, directing the beam down a low-index (and by association high-symmetry) zone axis may help.[8][9][10][11]

• Crystallography
• Dual basis
• Reciprocal lattice
• Miller index
• Diffraction
• Electron diffraction
• Transmission electron microscopy

• International Tables for Crystallography