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 (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.

Crystal-lattice translational-invariance is described by a set of unit-cell direct-lattice (contra-variant or polar) basis-vectors a, b, c, or in essence by the magnitudes of these vectors (the lattice parameters a, b and c) and the angles between them (namely 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 or Ångstroms.

These lattice-vectors are indexed by listing one (often integral) multiplier for each basis-triplet component, generally to be placed between either square [] or angle 〈〉 brackets. Thus the direct lattice-vector s_uvw or [u,v,w] is defined as ua+vb+wc. Angle brackets are used when you don't want to refer to a specific lattice vector, but to a symmetrically equivalent class of lattice vectors. In the case of a cubic lattice, for instance, 〈100〉 represents [100], [010] and [001] because each of these vectors is symmetrically equivalent.

The term zone-axis, more specifically, refers to only the direction of a direct-space lattice-vector. For example, since the [120] and [240] lattice-vectors share a common direction, 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 zone-axis is "lattice-plane normal" or "g-vector direction". Reciprocal-lattice (one-form or axial) vectors are Miller-indexed using the reciprocal-lattice basis-triplet (a*, b*, c*) instead, generally between either round () or curly {} brackets. Curly brackets are used when you don't want to refer to a specific reciprocal-lattice vector, but to a symmetrically equivalent class of reciprocal-lattice vectors.

Here as usual a* ≡ b×c/V_c, b* ≡ c×a/V_c, and c* ≡ a×b/V_c, where the unit cell volume is V_c = a•(b×c). Thus reciprocal-lattice vector g_hkl or (h,k,l) = ha*+kb*+lc* has a direction perpendicular to the (hkl) crystallographic plane, and a magnitude g_hkl = 1/d_hkl equal to the reciprocal of the spacing between (hkl) planes, measured in spatial-frequency units e.g. of cycles per Ångstrom.

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Wikipedia