JOHN W.CAHN and DENIS GRATIAS

Science evolves with the search for laws of nature, their testing, and ultimately their confident application. Occasionally an established law is contradicted by experiment and the period of ensuing ferment is one of the most rewarding times to be a scientist. This chapter describes such a time, specifically, the period of research that has resulted from the discovery by Daniel S.Shechtman in 1983 of a metallic solid, the discrete electron diffraction pattern (Figure 1) of which exhibited icosahedral point symmetry.^{1}^{,}^{2} This is a symmetry that could not occur in crystals according to the laws of crystallography, yet in many ways this solid behaves like a crystal.

The laws of nature are both a boon and a bane to scientists, who are taught to work within them because they cannot be changed. All of the marvelously rich phenomena of the physical world conform to these laws, and knowledge of the laws is an important guide to invention, by allowing scientists to know which things are possible and which are impossible. Sometimes scientists do not heed the laws and expend great effort in an obviously useless search to do something that would violate, for example, a law of thermodynamics. Nevertheless, some versions of these laws are imprecise, and the exceptions point to their limitations and the need for reformulation. The law that Shechtman’s discovery violated was based not on a fundamental, immutable law of nature but on an axiom of crystallography, namely, that all crystals are periodic.

Materials science is a composite science, containing elements of many other disciplines—including mathematics, physics, chemistry, electronics, mechanics, and crystallography. The continuing search for new materials with different properties makes use of many laws from these disciplines. The laws are often applied in complex and unusual combinations requiring

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Advancing Materials Research
Quasi-Periodic Crystals: A Revolution in Crystallography
JOHN W.CAHN and DENIS GRATIAS
Science evolves with the search for laws of nature, their testing, and ultimately their confident application. Occasionally an established law is contradicted by experiment and the period of ensuing ferment is one of the most rewarding times to be a scientist. This chapter describes such a time, specifically, the period of research that has resulted from the discovery by Daniel S.Shechtman in 1983 of a metallic solid, the discrete electron diffraction pattern (Figure 1) of which exhibited icosahedral point symmetry.1,2 This is a symmetry that could not occur in crystals according to the laws of crystallography, yet in many ways this solid behaves like a crystal.
The laws of nature are both a boon and a bane to scientists, who are taught to work within them because they cannot be changed. All of the marvelously rich phenomena of the physical world conform to these laws, and knowledge of the laws is an important guide to invention, by allowing scientists to know which things are possible and which are impossible. Sometimes scientists do not heed the laws and expend great effort in an obviously useless search to do something that would violate, for example, a law of thermodynamics. Nevertheless, some versions of these laws are imprecise, and the exceptions point to their limitations and the need for reformulation. The law that Shechtman’s discovery violated was based not on a fundamental, immutable law of nature but on an axiom of crystallography, namely, that all crystals are periodic.
Materials science is a composite science, containing elements of many other disciplines—including mathematics, physics, chemistry, electronics, mechanics, and crystallography. The continuing search for new materials with different properties makes use of many laws from these disciplines. The laws are often applied in complex and unusual combinations requiring

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FIGURE 1 Diffraction patterns of the icosahedral Al-Mn crystals aligned along various axes.

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new formulations or theorems that are more directly applicable. The search for new materials also provides opportunities for formulating and testing new laws, which then move into the other disciplines. Thus, Shechtman’s discovery in materials science is having repercussions in mathematical and chemical crystallography and in solid-state physics. It challenged a law that was mature, well tested, and confining in a way that could have been foreseen but was not.
CLASSICAL CRYSTALLOGRAPHY
As is so often the case in science, crystallography began by focusing on what was then observable—the external form of crystals. These forms obeyed certain rules, and the apparently limitless shapes were in fact found to be combinations of only a few forms: the form of any given crystal was a combination of forms conforming to 1 of 32 point groups. Only 51 different forms were found, and all were shown to be predictable from an axiom that the internal structure of crystals was composed of repeating units arranged with three-dimensional translational periodicity. Translational periodicity permits only 1-, 2-, 3-, 4-, and 6-fold rotational axes, and 32 crystallographic point groups are the only combinations allowed.
Throughout the nineteenth century, before x-ray diffraction in 1912 confirmed3 the periodicity, this axiom allowed rapid development of crystallography. Many aspects of crystal physics were established. For instance, J.W.Gibbs and G.Wulff calculated the equilibrium shape of a crystal in terms of orientation-dependent surface free energies. Gibbs also formulated thermodynamic barriers to crystal growth resulting from the difficulty of adding a new layer to an existing crystal. The axiom of periodicity worked with few exceptions, and these exceptions could be fitted into the laws by adding such concepts as composite crystals—for example, two or more crystals grown together (called twins or multiple twins) to give rise to objects with additional symmetries. The translational periodicity ceases where the boundary between crystals is crossed.
X-ray diffraction not only confirmed the internal periodicity but also produced methods for identifying the arrangement of atoms within the unit cells. External form lost its preeminence in crystallography, and today crystals are identified almost exclusively by their diffraction patterns, not by their form. The axiom of periodicity not only became a definition of what is a crystal, but it also took on the force of a law of nature.
Soon after x-ray diffraction came into use, deviations from periodicity were discovered.4 Among the first of these were modulated structures in which modulations of distortion, composition, order, and magnetic or electric polarization were superimposed on an otherwise periodic structure. The wavelength of these modulations was soon suspected to be incommensurate with

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the lattice periodicity, opening up the possibility that these structures were not periodic. Because the modulations were small distortions and the underlying lattice otherwise conformed to the laws of crystallography, the lack of periodicity could be ignored.
Shechtman’s icosahedral solid exhibited a forbidden rotational symmetry. It also exhibited discrete diffraction, indicating some kind of translational regularity.
FERMENT IN RELATED FIELDS
By the time of Shechtman’s research, several important developments had occurred in related fields that had immediate bearing on his discovery.
Mathematicians realized early that aperiodic functions could have discrete Fourier spectra. Fourier analysis of periodic functions led to the exploration of a special kind of aperiodic function called almost-periodic functions and quasi-periodic functions.5,6 Both of these functions have Fourier spectra with individual discrete peaks. In one variable,
If the function is periodic, the kn are all multiples of a single wave number related to the wavelength λ:
kn=2πn/λ.
If the kn are not of this form, then the function is quasi-periodic if there is a finite set of lengths λ1, λ2,…, λn such that each kn can be expressed as a sum of integer multiples of 2π/λi:
If N is infinite, the function is almost periodic.
The quasi-periodic functions are most interesting. In particular they can be considered a cut of a higher-dimensional periodic function. For instance the quasi-periodic function
is a cut of the two-dimensional periodic function
f(x,y)=sin x+sin y
along the line
This property of quasi-periodic functions was exploited by crystallographers to analyze modulated structures.7–9 The suggestion that a singly mod-

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FIGURE 2 A quasi-periodic two-dimensional tiling generated by sectioning a three-dimensional cubic lattice along an irrational plane.
ulated three-dimensional crystal could be described as a three-dimensional cut of a four-dimensional crystal was actively pursued.
Since we live in a three-dimensional world, most people find this hard to visualize. On the other hand, two-dimensional quasi-periodic structures are readily exhibited by an irrational cut of a three-dimensional periodic crystal. Figure 2 is a drawing of an array of cubes intercepted by an irrational plane. The resultant structure is quasi-periodic and has a sharp diffraction pattern. Two-dimensional modulated structures are frequently seen in surface physics. But higher-dimensional periodic functions could have rotational symmetries not possible in three-dimensional periodic functions. In particular, 5-, 8-, 10-, and 12-fold axes can occur in four dimensions, and the combination of six 5-fold, ten 3-fold, and fifteen 2-fold axes that defines icosahedral symmetry can first occur with six-dimensional periodicity. Six dimensions also permit 7-, 9-, 14-, and 18-fold axes. Since no new rotational symmetries occur in going from two to three dimensions, the only analogue in our three-dimensional world is that of going from one dimension (2-fold possible) to two dimensions (3-, 4-, and 6-fold added). Figure 3 shows a set of two-dimensional tilings produced by sectioning higher-dimensional cubic lattices.
Icosahedral molecules and icosahedral packing units were observed in materials 30 years ago.10 F.C.Frank pointed out that often the lowest energy arrangement of 13 atoms consisted of 12 atoms arranged on the corners of an icosahedron, surrounding a central atom. Of course, such arrangements could not be packed periodically without losing the 5-fold axes. Yet, this

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FIGURE 3 Quasi-periodic tilings with 8-, 10-,…, 22-fold diffraction rotation axes produced by sectioning, respectively, 4-, 5-,…, 11-dimensional cubic lattices.
arrangement began to become an important feature in descriptions of the structures of liquids and glasses. Distorted icosahedra were also frequently seen in crystals with large unit cells. The idea that liquids and glasses might be understood as simpler structures in a higher-dimensional curved space has been widely explored.11 In a computer simulation of freezing of a liquid, a tendency for icosahedral bond orientational order was noted, but in such small model systems periodic boundary conditions were imposed, making strict icosahedral symmetry impossible.12
Along with crystallographers, mathematicians studying infinite tilings had made the assumption of periodicity. Aperiodic tilings suddenly appeared, the most famous of which were discovered by Richard A.F.Penrose and had 10-fold symmetry.13 It was not immediately appreciated that these were quasi-periodic and could be generated by cuts of five-dimensional cubic structures.
More than five years before, an architect14 found that geodesic domes having exact icosahedral symmetry could be infinitely extended. He noted that this phenomenon was an example of three-dimensional aperiodic tiling and attributed it to earlier work of J.Kepler and E.S.Federov. More than a decade later, Mackay15 rediscovered this structure and recognized its implications for an aperiodic tiling of three-dimensional space. He coined the word quasilattice and demonstrated empirically that it might have a discrete diffraction pattern.
It has been possible to prove by theoretical counterexamples that the lowest energy packing of atoms and molecules, the ground state, is not always

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periodic.16 For certain theoretical interaction models, the ground state has been shown to be quasi-periodic. In some of these models there are no equilibrium periodic states at any temperature. Many two-dimensional surface phase diagrams show widespread existence of quasi-periodic phases extending to low temperatures. There is no thermodynamic basis for requiring periodicity.
In 1977 an attempt to use Landau theory to predict crystallization of a liquid yielded the rather surprising conclusion that an icosahedral arrangement of density waves was a likely possibility, but since this was obviously inconsistent with translational periodicity, the result was an enigma.17
RAPID SOLIDIFICATION AND METASTABLE PHASES
Metastable phases abound in the natural world. Natural diamonds form as equilibrium crystals at high pressure, and they survive metastably at ambient pressure because the rate of conversion to the equilibrium state, graphite, is unmeasurably slow. More generally, metastable phases form and survive without ever being stable because the stabler phases form slowly. Materials scientists create nonequilibrium materials through processing cycles designed to capitalize on such kinetic differences, thereby vastly increasing the range of available materials. The heat-treatment cycles of steel discovered 3,000 years ago create carbide particles instead of the equilibrium graphite and thereby give steel its unusual properties. One of the more widely used processing techniques that create metastable materials is rapid solidification, developed by Pol Duwez in the late 1950s.18 Crystals vary by many orders of magnitude in their crystallization rates. Such diverse elements as nickel and phosphorus can crystallize from their melts at rates of 10 m/s. Quartz crystallization from its melt can be 10 orders of magnitude less. A rate-limiting factor in the crystallization of liquid mixtures is diffusion. If no crystal can form with the same composition as the melt, crystal growth rate is limited to about 10 cm/s. Rapid solidification thus favors phases with wide solid solution ranges over intermetallic compounds with narrow composition ranges. When during rapid solidification a liquid is cooled below its melting point, it becomes metastable and can crystallize to metastable solids that are more stable than the liquid. The degree of metastability is small, approximately 0.01 eV/atom, but there are an amazingly large number of different metallic phases with almost the same energy that become accessible. The simplest are the solid, supersaturated solutions, which play an important role in twentieth-century metallurgy.
Solid solubilities change with temperature, and the excess solutes can be made to precipitate from the solid state to give desirable precipitation-hardened alloys. Precipitation hardening, discovered this century,19,20 has been

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responsible for most modern structural alloys, and rapid solidification promised to increase the solute content available for precipitation.
This was indeed the purpose of the study that Shechtman joined at the National Bureau of Standards. He was spending his sabbatical there on a cooperative program with the Johns Hopkins University, funded by the Air Force. He began looking at rapidly solidified aluminum alloys with transition elements. His tool was the transmission electron microscope. He encountered the expected increase in the solubility of manganese in face-centered aluminum and was pushing to higher manganese content when he encountered a new phase unlike any that had ever been seen. He called it the Z-phase.
CRYSTALLOGRAPHY ON THE ELECTRON MICROSCOPE
The modern electron microscope is an extraordinarily versatile tool. It not only gives a magnified image with a 2-angstrom resolution but also permits a host of physical measurements to be made with monoenergetic focused and collimated electrons. Chemical analysis can be made by x-ray emission spectroscopy from volumes as small as 10–20 m3. A wide variety of microelectron diffraction techniques can be made on columnar volumes as small as a few nanometers in diameter and as long as the specimen is thick—10 to 100 nm.
The sample Shechtman examined contained crystals that were microns in diameter. A flip of a switch revealed that the diffraction pattern of these crystals contained an apparent 10-fold symmetry. Rotating the specimen revealed quickly instead that the symmetry was one of six 5-fold inversions axes (Figure 1). The overall symmetry was icosahedral, forbidden for translational periodicity. The diffraction patterns revealed sharp, discrete spots, which we now know result from quasiperiodicity.
The initial reaction to this finding was that it must be multiple twinning of ordinary periodic crystals, but several transmission electron microscopy techniques rule this out unambiguously. Among them is dark-field microscopy, in which an image is formed from the electrons that have been diffracted into a single spot. Only that portion of the specimen that contributes to the diffraction lights up. The individual crystals of any multiple twin are easily revealed by this technique. Microtwinning is ruled out by convergent-beam micro-microdiffraction, which shows that all spots come from the smallest regions in the crystal.1 The convergent-beam technique not only has confirmed the 5-fold symmetry but also has been used to determine which of the two point groups and 235 belonging to the Laue groups is the correct one for these alloys.21 High-resolution transmission electron microscopy22–27 and field-ion microscopy28 further confirm the symmetry and that these are not periodic crystals, multiply twinned.
Without these tools, these crystals would not have been discovered until

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large, single crystals were made. Powder patterns do not reveal the rotational symmetry and can always be fitted to a model if a large enough unit cell is assumed.
Simultaneous with the paper2 announcing this finding there appeared a theoretical paper showing that icosahedral quasicrystals could be understood in terms of cuts of 12-dimensional crystals.29 Because of our residual doubts, the announcement paper received wide circulation before it was submitted for publication and this stimulated an immediate paper by Levine and Steinhardt,30 followed shortly by three other papers31–33 that all showed icosahedral symmetry to be consistent with quasiperiodicity and gave diffraction intensities from a quasilattice that were qualitatively in good agreement with the electron diffraction patterns. In the year since publication we have received more than 100 preprints on many aspects of these new materials. Two other point groups, 10/m (decagonal)34 and 12 (dodecagonal)35 have been reported. Icosahedral diffraction patterns have been reported from many different alloys, and the possibility that a stable icosahedral phase (Al5Li3Cu) exists and that large, single crystals can then be grown seems to have been confirmed.36,37 Our research about the structure and properties of these crystals was hampered by the imperfections of rapidly grown crystals, but it can now proceed.
NOTES
1.
D.Shechtman and I.Blech, Metall. Trans. 16A, 1005 (1985).
2.
D.Shechtman, I.Blech, D.Gratias, and J.W.Cahn, Phys. Rev. Lett. 53, 1951 (1984).
3.
W.Friedrich, P.Knipping, and M.Laue, Ann. Phys. 41, 971 (1913).
4.
C.H.Johansson and J.O.Linde, Ann. Phys. 25, 1 (1936).
5.
H.A.Bohr, Almost Periodic Functions (Chelsea, New York, 1947).
6.
A.S.Besicovitch, Almost Periodic Functions (Cambridge University Press, New York, N.Y., 1932).
7.
P.M.deWolf, in Modulated Structures, NATO ASI Series E 83, edited by Tsakalakos (Martinus Nijhoff, The Hague, 1984).
8.
A.Janner and T.Janssen, Acta Crystallogr. Sect. A 36, 399 (1980).
9.
N.G.de Bruijn, Ned. Akad. Wetensch. A 84, 39 (1981).
10.
F.C.Frank, Proc. R. Soc. London 215, 43 (1952).
11.
D.Nelson and B.I.Halperin, Science 229, 233 (1985).
12.
P.J.Steinhardt, D.R.Nelson, and M.Ronchetti, Phys. Rev. B 28, 784 (1983).
13.
M.Gardner, Sci. Am. 236, 110 (1977).
14.
S.Baer, Zome Primer (Zomework Corp., Albuquerque, 1970).
15.
A.L.Mackay, Physica A 114, 609 (1982).
16.
C.Radin, University of Texas (preprint).
17.
S.Alexander and J.McTague, Phys. Rev. Lett. 41, 702 (1978).
18.
P.Duwez, R.H.Willens, and W.Klement, J. Appl. Phys. 31, 1126 (1960).
19.
A.Wilm, Metallurgie 8, 225 (1911).
20.
P.D.Merica, R.G.Waltenberg, and H.Scott, Scientific papers of the U.S. Bureau of Standards, No. 347 (Washington, D.C., 1919), Vol. 15, p. 271.
21.
L.Bendersky and M.J.Kaufman, Philos. Mag. B 53 (3), L75 (1986).

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22.
D.Shechtman, D.Gratias, and J.W.Cahn, C.R.Acad. Sci. II 300(18), 909 (1985).
23.
K.Hiraga, M.Watanabe, A.Inoue, and T.Masumoto, Sci. R. Ton., A-32, 309 (1985).
24.
L.Bursill and J. Lin, Nature 316, 50 (1985).
25.
R.Portier, D.Shechtman, D.Gratias, and J.W.Cahn, J. Micros. Spectros. Electron. 10, 107 (1985).
26.
K.M.Knowles, A.L.Greer, W.O.Saxton, and W.M.Stobbs, Philos. Mag. B 52(1), L31 (1985).
27.
R.Gronsky, K.M.Krishnan, and L.E.Tanner, in Proceedings of the Electron Microscopy Society of America Annual Meeting (Electron Microscopy Society of America, McLean, Va., 1985).
28.
A.J.Melmed and R.Klein, Phys. Rev. Lett. 56(14), 1478–1481 (1986).
29.
P.Kramer and R.Neri, Acta Crystallogr. Sect. A 40, 580 (1984).
30.
D.Levine and P.J.Steinhardt, Phys. Rev. Lett. 53, 2477 (1984).
31.
M.Duneau and A.Katz, Phys. Rev. Lett. 54, 2688 (1985); “Quasiperiodic patterns and icosahedral symmetry,” Ecole Polytechnique, France (preprint).
32.
P.A.Kalugin, A.Kitaev, and L.Levitov, JETP Lett. 41, 119 (1985); J. Phys. Lett. 46, L601 (1985).
33.
C.L.Henley, J. Non-Cryst. Solids 75, 91–96 (1985).
34.
L.Bendersky, Phys. Rev. Lett. 55, 1461 (1985).
35.
H.U.Nissen, T.Ishimasa, and R.Schlogle, Helv. Phys. Acta 58, 819 (1985).
36.
M.S.Ball and D.J.Lloyd, Scripta Metall. 19, 1065 (1985).
37.
E.Ryba and C.Bartge, Pennsylvania State University (private communication).