### Elementary crystallography; an introduction to the fundamental geometrical features of crystals

If D is infinite, the function is called almost periodic. We have so far not been concerned with almost periodicity, since in any experiment D is less than or equal to the number of observed reflections, and thus is finite. The Fourier transform consists of two delta functions. If b is rational, f is periodic, the two delta functions can be indexed with a single reciprocal lattice vector.

Note that there would be no diffuse scattering from a quasiperiodic object with f as its density function. Figure 1 shows the first diffraction pattern taken from a quasicrystal oriented along the 5. Note first the discrete diffraction and the apparent fold symmetry.

Note that there are no systematic rows; spots twice or three times as far as a bright spot are much weaker if seen at all. Lastly note that it is impossible to index this pattern with just three reciprocal lattice vectors. The first electron diffraction pattern from a quasicrystal [ 1 ]. Note the forbidden, fold axis, the absence of systematic rows, and the need for more than three vectors to index all the spots. Our brains often take us to higher dimensions to simplify what is seen. Every triplet of rhombs meeting at a triple corner in the Penrose tiling in Fig. In five dimensions this tiling finally becomes simple and unambiguous with each edge along a specific one of five orthogonal axes and each rhomb becomes a square with a unique orientation.

A zigzag path along the lines of the tiling becomes a Cartesian path in five dimensions, and a five-index coordinate system specifies each corner. In five dimensions the Penrose tiling is confined to the set of all the lattice points within a slice bounded by two parallel plane hypersurfaces with irrational orientations. Penrose tilings are quasiperiodic. In five dimensions this ambiguity is removed, all line segments can be orthogonal, an then this entire pattern will fit between two parallel hyperplanes.

Since the plane of the primitive cubic lattice is the two-dimensional hexagonal lattice, the three-dimensional hexagonal lattice can be considered as the plane of a four-dimensional cubic lattice [ 19 ].

This rational cut can simplify the understanding of indexing hexagonal structures. Distances between two such points can be computed more easily in the 4-index notation. A physical example of a two-dimensional quasiperiodic object is the surface obtained by cutting a three-dimensional crystal by an irrational plane.

## Elementary Crystallography: An Introduction to the Fundamental Geometric Features of Crystals

In this example the three basis vectors of the periodic three-dimensional crystal are needed to describe this two-dimensional quasiperiodic surface. Because the cutting plane is irrational the surface cannot be periodic; it will never go through exactly the same point in two different unit cells. Yet when the plane comes close to the same point in some distant unit cell, another plane through that point will be very close all the way out to infinity. The aperiodic structures these planes represent will superimpose with little error all the way to infinity. That distance between the points is an approximate translation vector, whose existence depends on the specification of how small a superposition error we require.

For a periodic function the superposition would be exact; the translation can be repeated indefinitely, and thus form a lattice. For a quasiperiodic function, repetition of any translation increases the mismatch, and eventually the error becomes too large; thus the translations in a quasiperiodic structure do not form a lattice, but what is called a quasilattice. But the existence of these translations is an important property of quasiperiodic functions and of quasicrystals. There are many kinds of defects in periodic structures that have their analogs in quasiperiodic structures.

Let us begin by examining how defects in a three-dimensional periodic crystal would appear on a two-dimensional aperiodic surface. Such metallic crystals commonly have internal boundaries, called antiphase boundaries, separating domains in which the site occupations are reversed.

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Such boundaries break the translational symmetry in an otherwise periodic crystal. Now consider a cut of such a crystal on an irrational plane. Although this cut surface is aperiodic, the domain boundary, a translation defect, would clearly be visible in the quasiperiodic surface.

Dislocation lines in three dimensions, intersecting the surface, would show up in the surface as points with associated Burgers vectors. Since, apart from some small strains, the three-dimensional structure is perfect away from the dislocation, so is the quasiperiodic surface. Thus we can detect translational faults and imperfections in quasiperiodic objects. Defects in quasicrystals can be understood as defects in a higher-dimensional periodic crystal. Ordering can occur in icosahedral quasicrystals, giving rise to antiphase boundaries that are five-dimensional hypersurfaces in the six-dimensional crystal and seen as surfaces in the three-dimensional quasicrystal [ 20 ].

This boundary can also be seen in an imperfect ordering of a Penrose tiling in which adjacent corners alternate black and white. Dislocation lines in icosahedral quasicrystals arise from a four-dimensional defect surface in six dimensions. Mechanical deformation of quasicrystals is a most interesting subject. Away from the dislocation line, the quasicrystal is perfect, as it would be with dislocations in periodic crystals.

Although no new symmetry axes appear in going from two to three dimensions, higher dimensions allow new symmetries to be consistent with periodicity. For the axial groups a n -fold symmetry axis first becomes possible with translational symmetry if the dimensionality equals the totient of n , which is the number of positive integers less than or equal to n which are relatively prime no common factors to n [ 21 ]. Two has a totient of 1; three, four, and six have totients of 2; none have 3; five, eight, ten, and twelve have 4, etc.

Thus five, eight, ten, and twelve-fold rotations first appear in four-dimensional periodicity. Icosahedral symmetry with its intersecting five-fold axes requires six dimensions. Each of the six axes in an isometric six-dimensional lattice meets the five others at right angles, giving rise to six fold axes. Because there is no point group with more than one fold axes in three dimensions, the cuts by irrational planes can only preserve the six 5-fold or the six 5-fold inversion axes of the icosahedral symmetry.

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The study of quasicrystals benefited greatly from prior research in the mathematical subjects of quasiperiodic functions, aperiodic tilings, and hyperspace crystallography. The latter had already been applied in the study of modulated crystals [ 22 ]. Modulated structures had been found long before the discovery of quasicrystals and had provided some well-documented and understood exceptions to periodic crystals. Because they could be considered as small incommensurate distortions of periodic structures with a crystallographic point group, they could be fit into the schemes of crystallography.

But the incorporation of the modulation wavelength as an additional length provided an impetus for the development of hyperspace crystallography which allowed a periodic indexing in the higher dimension. Modulated structures could then be treated as cuts on irrational planes, and sometimes as projections of a slice, of a four or higher dimensional periodic structure. In an ideal modulated structure, each spot, including the satellites, is a Bragg peak, indexed with more than three, usually four, numbers.

Consider an icosahedral structure to be an irrational cut of a six-dimensional cubic structure with a single lattice parameter. Indexing requires six numbers, which is obvious in six dimensions, but is also true for three. Indexing of icosahedral powder patterns is also straightforward; the ambiguities resulting from superpositions such as and in bcc powder patterns are infrequent.

After a lattice parameter has been selected, indexing for all six numbers for single crystals is unambiguous in either three or six dimensions. Using synchrotron radiation from a single AlCuFe quasicrystal, Moss and coworkers have measured intensities of about crystallographically distinct peaks, every peak found using a single icosahedral quasi lattice parameter and a six-parameter icosahedral indexing [ 24 ].

Structure determinations would seem like a hopeless task. Has one to describe the structure of an aperiodic solid out to infinity?

Because there is periodicity in the higher dimensions, one needs only to describe the content of one unit cell hypercell in the higher dimensional space. Structure determination in six dimensions is not very different from what it is in three. Once the diffraction peaks from single crystals or lines from powders have been indexed in six dimensions, or in three with six basis vectors, and their corrected intensities measured, the diffraction pattern can be considered either on a three-dimensional reciprocal quasilattice or on a six-dimensional periodic lattice.

They are completely equivalent to another, but standard methods of crystallographic structure determination for periodic structures are applicable with little modification to the six-dimensional data. Indexing allows Patterson functions to be directly obtainable in three or six dimensions from powder data. They have the directional information lost in a radial distribution function. Although the three-dimensional Patterson functions are aperiodic and complicated with many peaks, near the origin they bear striking similarities to Patterson functions of related periodic approximants, large cell periodic crystals with compositions slightly different from quasicrystals [ 25 ].

Thus the local atom packing of quasicrystals are found to be very similar to that of corresponding periodic phases. Patterson functions in six dimensions are usually found to be much simpler, with only a few peaks in each unit cell.

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Actual structure determinations have now been carried out for several quasicrystals with very good residuals [ 26 , 27 ]. Atom positions are described in the six-dimensional unit cell by three-dimensional surfaces; the intersections of these surfaces periodically repeated in six dimensions by the irrational three-dimensional plane are the points that describe the atom positions in the three-dimensions of quasiperiodic structures. In analogy with the finding of three dimensional periodic structures by fitting balls with atomic radii together, plausible quasiperiodic structures have been constructed by fitting atomic surfaces into six dimensional unit cells [ 28 ].