Physics: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.

Zone-axis indexing

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 angstroms.

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 suvw 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 ,  and  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  and  lattice vectors share a common direction, their orientations both correspond the  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/Vc, b* ≡ c×a/Vc, and c* ≡ a×b/Vc, where the unit cell volume is Vc = a•(b×c). Thus reciprocal-lattice vector ghkl or (h,k,l) = ha*+kb*+lc* has a direction perpendicular to the (hkl) crystallographic plane, and a magnitude ghkl = 1/dhkl equal to the reciprocal of the spacing between (hkl) planes, measured in spatial-frequency units e.g. of cycles per angstrom.

Bracket conventions for direct-lattice and reciprocal-lattice (Miller) indices in crystallography
Object Generic Type Specific Instance Units Transformation
zone or lattice-vector suvw $\displaystyle{ \langle{uvw\rangle}\, }$ $\displaystyle{ [uvw]\, }$ direct space e.g. [meters] contra-variant or polar
plane or g-vector ghkl $\displaystyle{ \{hkl\}\, }$ $\displaystyle{ (hkl)\, }$ reciprocal space e.g. [cycles/m] co-variant or axial

A useful and quite general rule of crystallographic "dual vector spaces in 3D" is that the condition for a direct lattice-vector [uvw] to have a direction (or zone-axis) perpendicular to the reciprocal lattice-vector [hkl] is simply hu+kv+lw = 0. 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, then, 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, the 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], i.e. for which hu+kv+lw=0. Diamond FCC silicon atom-cluster (left) viewed down the  zone, with the corresponding zone-axis pattern (right).

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). Put another way, in electron microscopy if you want your electron beam to be directed down wide (hence 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 to this end.

Footnotes

1. J. M. Ziman (1972 2nd ed) Principles of the theory of solids (Cambridge U. Press, Cambridge UK).
2. Zbigniew Dauter and Mariusz Jaskolski (2010) "How to read (and understand) Volume A of International Tables for Crystallography: an introduction for nonspecialists", J. Appl. Cryst. 43, 1150-1171 pdf
3. George Arfken (1970) Mathematical methods for physicists (Academic Press, New York).
4. E. W. Nuffield (1966) X-ray diffraction methods (John Wiley, NY).
5. B. E. Warren (1969) X-ray diffraction (Addison-Wesley, paperback edition by Dover Books 1990) ISBN:0-486-66317-5.
6. cf. Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (1973) Gravitation (W. H. Freeman, San Francisco CA).
7. John M. Cowley (1975) Diffraction Physics (North-Holland, Amsterdam).
8. P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977) Electron microscopy of thin crystals (Butterworths/Krieger, London/Malabar FL) ISBN:0-88275-376-2
9. J. W. Edington (1976) Practical electron microscopy in materials science (N. V. Philips' Gloeilampenfabrieken, Eindhoven) ISBN:1-878907-35-2
10. Ludwig Reimer (1997 4th ed) Transmission electron microscopy: Physics of image formation and microanalysis (Springer, Berlin) preview.
11. David B. Williams and C. Barry Carter (1996) Transmission electron microscopy: A textbook for materials science (Plenum Press, NY) ISBN:0-306-45324-X