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Quasicrystals Volume 106, Number 6, November–December 2001 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 106, 975–982 (2001)] Quasicrystals Volume 106 Number 6 November–December 2001 John W. Cahn The discretely diffracting aperiodic crystals Key words: aperiodic crystals; new termed quasicrystals, discovered at NBS branch of crystallography; quasicrystals. National Institute of Standards and in the early 1980s, have led to much inter- Technology, disciplinary activity involving mainly Gaithersburg, MD 20899-8555 materials science, physics, mathematics, and crystallography. It led to a new un- Accepted: August 22, 2001 derstanding of how atoms can arrange [email protected] themselves, the role of periodicity in na- ture, and has created a new branch of crys- tallography. Available online: http://www.nist.gov/jres 1. Introduction The discovery of quasicrystals at NBS in the early Crystal periodicity has been an enormously important 1980s was a surprise [1]. By rapid solidification we had concept in the development of crystallography. Hau¨y’s made a solid that was discretely diffracting like a peri- hypothesis that crystals were periodic structures led to odic crystal, but with icosahedral symmetry. It had long great advances in mathematical and experimental crys- been known that icosahedral symmetry is not allowed tallography in the 19th century. The foundation of crys- for a periodic object [2]. tallography in the early nineteenth century was based on Periodic solids give discrete diffraction, but we did the restrictions that periodicity imposes. Periodic struc- not know then that certain kinds of aperiodic objects can tures in two or three dimensions can only have 1,2,3,4, also give discrete diffraction; these objects conform to a and 6 fold symmetry axes. With no exceptions, each mathematical concept called almost-1 or quasi-periodic- crystal was found to conform to one of only 32 ways of ity [3]. By definition all quasi-periodic objects diffract combining these symmetry axes, the so-called discretely, even though they are not periodic. Quasiperi- “crystallographic” point group symmetries. External odic objects can have any of the infinite set of point forms of periodic crystals were found to be limited to group symmetries listed as non-crystallographic in the combinations of only 47 forms (32 general and 15 spe- International Tables for Crystallography [4]; because cial) made of symmetrically arranged bounding planes they have a single rotation axis of order 5, or one greater [5]. Cubes, octahedra, and tetrahedra, for instance, are than or equal to 7, or have icosahedral symmetry with its examples of special forms belonging to the cubic point six intersecting 5-fold axes. groups, octahedra to point groups 432, m3, and m3m, tetrahedra to 23 and 43m, and cubes to all five. In the nineteenth century each known crystal could be fit into 1 For reason discussed below we need not be concerned with almost periodicity. one (or more) of these 32 point groups by the examina- tion of its external form. That no additional 975 Volume 106, Number 6, November–December 2001 Journal of Research of the National Institute of Standards and Technology form was found could be taken as proof that all crystals Interim Commission on Aperiodic Crystals wrote “by are periodic. Regular icosahedra and dodecahedra are ‘crystal’ we mean any solid having an essentially dis- special forms of both icosahedral point groups, 235 and crete diffraction pattern, and by ‘aperiodic crystal’ we m35. All icosahedral forms have fifteen intersecting 2- mean any crystal in which three-dimensional lattice pe- and ten intersecting 3-fold axes, as well as six intersect- riodicity can be considered to be absent” [9]. By this ing non-crystallographic 5-fold axes. latest definition, our solid is a crystal, albeit an aperiodic With the assumption of periodicity, the mathematical one. It is a “quasiperiodic crystal” or quasicrystal for aspects of crystallography were set and completely short, a word coined by Levine and Steinhardt [10]. worked out in the 19th century; that aspect became an Our surprising discovery created quite a stir and has almost closed field. In two and three dimensions the influenced research in many fields, not just crystallogra- number of crystal systems, point groups, and plane or phy, but also materials science, physics, mathematics space groups were all enumerated. When the allowed [11,12], biology [13,14], and even art. There have been symmetry axes are combined with translations, it was about 10 000 papers in these fields and many conference shown that there are only 230 space groups in three proceedings [15]. Hundreds of quasicrystals have been dimensions. In two dimensions there are only ten point found since, some with non-crystallographic axial sym- groups and seventeen plane groups. An elementary metries, pentagonal, octagonal [16], decagonal [17], proof why this listing contains every case allowed by and dodecagonal [18]. The crystals with axial sym- periodicity and why no others are allowed has long been metries are usually periodic along the symmetry axis, available in popular mathematics books [6]. Such com- and quasiperiodic in the basal plane. plete listing are called catalogs. Each one of the seven- Quasiperiodicity is a form of aperiodicity that has teen are beautifully illustrated in etchings by M. Escher many of the attributes of periodicity. As one of their [7], as well as Moorish tilings and Turkish carpets. defining properties, Fourier transforms of quasiperiodic Extensions were developed for color groups and for functions are discrete sets of delta-functions; they can crystallography in higher dimensions. Magnetic struc- always be expressed as a series of sine and cosine terms, tures and their 1609 Shubnikov space groups are an but with incommensurate lengths, or a number of arith- example of such an extension in which spins, up or metically independent basis vectors that exceeds the down (or two colors), are treated as if in a fourth dimen- number of independent variables. Physically, a sion [8]. quasiperiodic object diffracts to give a pattern with With the advent of x-ray diffraction in 1912, external sharp Bragg spots. But whereas diffraction from a peri- form became less important. Crystals became defined as odic object forms a reciprocal lattice that can be indexed periodic arrangements of identical unit cells. The domi- with a set of d reciprocal basis vectors, where d is the nant work of crystallographers became structure deter- dimension, the diffraction pattern from a quasiperiodic minations by diffraction to find the atom content of one object requires a finite number, D > d, independent ba- unit cell. The method depends on an assumed periodic- sis vectors. An important consequence of this is that any ity, and the results usually confirmed it. quasiperiodic function can always be represented as a periodic function in D dimensions. The aperiodic func- tion then is a d-dimensional cut of this periodic function. 2. Discussion If D is infinite, the function is called almost periodic. We have so far not been concerned with almost periodicity, Had we found a crystal? Many definitions of crystals since in any experiment D is less than or equal to the are in use, some have changed over the centuries. Our number of observed reflections, and thus is finite. solid was metallic and thus not a “clear transparent As a simple example consider the one-dimensional mineral.” It can be grown to form “convex solids en- function f(x) = cos x + cos bx. The Fourier transform closed by symmetrically arranged plane surfaces, inter- consists of two delta functions. If b is rational, f is secting at definite and characteristic angles.” According periodic, the two delta functions can be indexed with a to the latter of these older definitions, quasicrystals are single reciprocal lattice vector. If b is irrational, f is crystals. The discovery in 1912 that crystals could dif- quasiperiodic; there are two incommensurate lengths in fract x-rays discretely implied either their periodicity or the Fourier transform; D = 2. The function f(x,y) = cos quasiperiodicity. But as noted above, the subsequent x + cos y is periodic in two dimensions; the quasiperi- structure determinations, seem to have led to the accep- odic one-dimensional f(x) is recovered by setting tance of a definition of crystals based on the periodicity y = bx. Note that there would be no diffuse scattering of their internal structure, and one which unnecessarily from a quasiperiodic object with f as its density func- ruled out quasiperiodicity. But by 1992 the IUCr Ad tion. 976 Volume 106, Number 6, November–December 2001 Journal of Research of the National Institute of Standards and Technology Figure 1 shows the first diffraction pattern taken from and each rhomb becomes a square with a unique orien- a quasicrystal oriented along the 5. Note first the discrete tation. A zigzag path along the lines of the tiling be- diffraction and the apparent 10-fold symmetry. Note that comes a Cartesian path in five dimensions, and a five- there are no systematic rows; spots twice or three times index coordinate system specifies each corner. In five as far as a bright spot are much weaker if seen at all. dimensions the Penrose tiling is confined to the set of all Note that the ratio of distances in any row is the “golden the lattice points within a slice bounded by two parallel mean” ␶,(␶ = 2 cos 360 =(1+͙5)/2 = 1.618034...), plane hypersurfaces with irrational orientations. and that ␶ occurs naturally in the ratios of the magni- Since the (111) plane of the primitive cubic lattice is tudes of vector sums of spots at 360 from one another. the two-dimensional hexagonal lattice, the three-dimen- Lastly note that it is impossible to index this pattern with sional hexagonal lattice can be considered as the (1110) just three reciprocal lattice vectors.
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