School of Physics & Astronomy

Tel Aviv University

Quasicrystals - Introduction

by Ron Lifshitz

What are quasicrystals? Why was their discovery by Dan Shechtman in April of 1982 met by so much disbelief that it took it more than two and a half years to find its way into the scientific literature? Why was there so much fuss about having 5-fold symmetry in the first place? What do we actually mean when we say that a crystal has such a symmetry? 

  Before we answer these questions let us first review some of the background. For centuries crystals were merely thought of as solids which have flat surfaces (facets) that intersect at characteristic angles. This is what we often see at mineral exhibitions in museums of natural history. It is also what we see when we look at certain quasicrystals like the ones shown in the figure below.

Single grains of quasicrystalsFigure 1. Scanning electron micrographs of single grains of quasicrystals: (a) an Aluminum - Copper - Iron alloy which crystallizes in the shape of a dodecahedron. This is one of the five Platonic solids, containing 12 faces of regular pentagons. Its symmetry is the same as that of the icosahedron, one of the other platonic solid with 20 faces of equilateral triangles. (b) An Aluminum - Nickel - Cobalt alloy which crystallizes in the shape of  a decagonal (10-sided) prism. Photographs courtesy of An Pang Tsai, NRIM, Tsukuba, Japan.

  During the 17th century, initial ideas regarding the microscopic structure of crystals began to emerge in the works of scientists such as Johannes Kepler and Robert Hooke (for a brief historical sketch see the book by Marjorie Senechal [4]). These ideas were formalized into a theory of crystallography by René-Just Haüy in the early 19th century. The basic notion of this theory is that crystals are solids which are ordered at a microscopic level. It was assumed that the only way to achieve order is by having periodicity, that is, some basic structural unit which repeats itself infinitely in all directions, filling up all of space. This is very much like the way in which identical square tiles can be used to tile a bathroom floor; like the way in which bees arrange their honeycombs in periodic hexagonal arrays; and like lizards, fish, angels, and devils fill the plane in the popular periodic drawings of the artist M. C. Escher. 
Escher's angels and devils
Figure 2. One of the many periodic drawings of the Dutch artist M. C. Escher. © Copyright Cordon Art B.V.

  The idea that crystals are periodically ordered was amazingly successful. Crystallographers were able to predict all the characteristic angles that could appear between the facets of crystals of any given type. With the discovery of x-ray diffraction in crystals by Max von Laue in 1912 and the subsequent development of x-ray crystallography by William H. and William L. Bragg the theory of crystallography received an unequivocal stamp of approval. During the seventy (!!) years that followed, all observed diffraction diagrams were in complete agreement with the predictions of this theory and with the notion that all crystals achieve their order through periodicity. It is no surprise then that periodicity, though never proven to be a requirement for order, was incorporated into the definition of crystal. Thus, on the eve of the discovery of quasicrystals, everybody "knew" that crystals were solids composed of a periodic arrangement of identical unit cells.
Diffraction diagram   Among the the most well known consequences of periodicity is the fact that the only rotational symmetries that are possible are 2-, 3-, 4-, and 6-fold rotations. Five-fold rotations (and any n-fold rotation for n>6) are incompatible with periodicity. We shall say more about what we mean by symmetry in the next section, but for the meantime let us think of the set of rotations that leave the directions of the facets (Figure 1) unchanged, or the set of rotations that leave the diffraction diagram (Figure 3) unchanged. Thus, on the eve of the discovery of quasicrystals, everybody "knew" that crystals and their diffraction diagrams cannot have 5-fold symmetry. One can only imagine what Shechtman felt when on April 8, 1982, while performing an electron diffraction experiment on an alloy of Aluminum and Manganese he observed a diffraction diagram similar to the one in Figure 3.  By orienting the alloy in different directions he found that it had the symmetry of an icosahedron, containing six axes of 5-fold symmetry, along with ten axes of 3-fold symmetry and 15 axes of 2-fold symmetry.
Figure 3. Typical diffraction diagram of a quasicrystal, exhibiting 5-fold or 10-fold rotational symmetry. Source unknown.

  The crystal that Shechtman discovered, as well as scores of other crystals that have been discovered since 1982, have been named "quasicrystals," which is short for "quasiperiodic crystals," by Levine and Steinhardt in 1984, in a first of a series of papers from the University of Pennsylvania that set up much of the initial theoretical foundations of the field . Quasicrystals share many of the characteristics of their periodic siblings: they can exhibit facets (Figure 1); they produce diffraction diagrams with sharp peaks (Figure 3, see more detail below); and those which are thermodynamically stable, like the AlPdMn quasicrystal in Figure 4, can be grown to very large dimensions and have a degree of microscopic order which surpasses even that of the most perfect periodic crystals. Quasicrystals are also very different from periodic crystals. They may possess rotational symmetry which is incompatible with periodicity. To this date, quasicrystals have been found that have the symmetry of a tetrahedron, a cube, an icosahedron [Figure 1(a)], and that of 5-sided, 8-sided, 10-sided [Figure 1(b)], and 12-sided prisms. They also possess unique physical properties that are currently under vigorous study, most notably the fact that even though they are all alloys of two or three metals they are very poor conductors of electricity and of heat.
Figure 4. A single Aluminum - Palladium - Manganese quasicrystal with icosahedral symmetry. This single crystal, grown at the IFF Forschungszentrum Jülich, is more than 6 cm long. Photograph courtesy of M. Feuerbacher, M. Beyss, and B. Grushko, Jülich, Germany. Al-Pd-Mn single crystal

On diffraction and the new definition of "crystal"

Before we end this introduction we should say a few words about diffraction experiments and the current definition of "crystal." A diffraction experiment directly probes the degree of order in a solid by measuring density correlations, namely, what are the chances of finding an atom at a certain location if we know that there is another atom at some other given location. [In more technical terms: The diffraction experiment displays the Fourier transform of the two-point density correlation function of the solid, also known as the "Patterson function."] If there is order the diffraction diagram shows a set of spots, called "Bragg peaks," in an otherwise essentially dark background. The longer the range of the correlations in the solid the sharper these peaks are. 

  In 1991, the International Union of Crystallography decided to redefine the term "crystal" to mean any solid having an essentially discrete diffraction diagram. Within the family of crystals we distinguishes between "periodic crystals," which are periodic on the atomic scale, and "aperiodic crystals" which are not. This broader definition reflects our current understanding that microscopic periodicity is not necessary for achieving order, yet it is sufficiently vague to reflect our uncertainty as to what are the necessary requirements for order. The new definition is based on the outcome of an experiment rather than a set of microscopic rules. 

  One can also use the diffraction diagram to distinguish between periodic crystals and quasicrystals as follows: Each Bragg peak in the discrete diffraction diagram defines a vector which points from the center of the diagram to that peak. In a diffraction diagram of a periodic crystal (in three dimensions) one can always find three peaks, corresponding to three vectors b1, b2, and b3, which can be used to index all the other peaks. This means that any other peak can be generated as a vector sum with three integer coefficients (h,k,l) as  hb1+ kb2+ lb3. In the case of a quasicrystal one needs more than three wave vectors to generate all the peaks and therefore more than three integers to index each peak.

  This is it for the introduction. We still need to explain the notion of symmetry for quasiperiodic crystals. In order to do that we shall take a careful look at the effect of rotations on quasiperiodic tilings, like the famous Penrose tilings, which are used as models of quasicrystals. 

Upcoming Conference in Israel

October 14-19, 2007, Tel Aviv, Israel.

Suggested Further Reading

  1. Somewhat technical but this is the paper that started it all:
  2. D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn, "Metalic phase with with long-range orientational order and no translational symmetry," Phys. Rev. Lett. 53 (1984) 1951-1953.

  3. The paper where the term "quasicrystal" was first defined:
  4. D. Levine and P.J. Steinhardt, "Quasicrystals: A New Class of Ordered Structures," Phys. Rev. Lett. 53 (1984) 2477 - 2480.

  5. Personal recollections on the discovery of quasicrystals:
  6. D. Shechtman and C. Lang, "Quasiperiodic materials: Discovery and recent developments," MRS Bulletin Vol. 22 No. 11 (1997) 40-42.

  7. On the discovery of quasicrystals as a Kuhnian Scientific Revolution:
  8. J. W. Cahn, "Epilogue," Proceedings of the 5th International Conference on Quasicrystals, Ed. C. Janot and R. Mosseri (World Scientific, Singapore 1995) 807-810.

  9. A good informal introduction to crystallography:
  10. M. Senechal, "Crystalline symmetries: An informal mathematical introduction" (Adam Hilger, Bristol, 1990).

  11. One of the first general textbooks on quasicrystals:
  12. C. Janot, "Quasicrystals: A primer," 2nd edition (Clarendon Press, Oxford, 1994).
    [Click here for a book review by Walter Steurer.]

  13. A textbook on the mathematics of quasicrystals:
  14. M. Senechal, "Quasicrystals and geometry" (Cambridge University Press, Cambridge, 1995).

Some of My Related Publications

Other Quasicrystal Sites

Slightly modified: January 2, 2007 by Ron Lifshitz.
This page has been visited  times since February 20, 1999.