Gévay, Gábor

Szederkényi, Tibor

English

Quasicrystals are a recently (1984) discovered family of structures producing X-ray diffraction diagrams with five-fold symmetry. This non-crystallographic symmetry can be explained by supposing that the strict periodic order is replaced here by a weaker order, called quasiperiodicity. Some of their properties as well as the possibility of formation in natural geological processes are discussed briefly in the paper

Gévay Gábor

Szederkényi Tibor

University of Szeged

Acta Mineralogica-Petrographica, Szeged, XXIX, 5—12,1987—1988 Q U A S I C R Y S T A L S A N D T H E I R S P O N T A N E O U S F O R M A T I O N P O S S I B I L I T I E S I N T H E N A T U R E G. GÉVAY, T. SZEDERKÉNYI Department of Mineralogy, Geochemistry and Petrography Attila József University ABSTRACT Quasicrystals are a recently (1984) discovered family of structures producing X-ray diffraction diagrams with five-fold symmetry. This non-crystallographic symmetry can be explained by supposing that the strict periodic order is replaced here by a weaker order, called quasiperiodicity. Some of their properties as well as the possibility of formation in natural geological processes are discussed briefly in the paper. A BRIEF INTRODUCTION TO QUASICRYSTALS Quasicrystals were discovered by SHECHTMAN et ah (1984). A puzzling aspect of this discovery is that the BRAGG diffraction diagram of quasicrystals consists of sharp spots that indicates long-range order, however, it exhibits five-fold rotational symmetry. It is well known that this kind of symmetry is forbidden by the funda-mental laws o f classical crystallography (BARLOW'S theorem, COXETER, 1961). Thus it is supposed that we are facing a new type of order, called quasiperiodicity (ROMERIO, 1971). Indeed, as it turned out, the mathematics of diffraction does not require strict periodicity in order to produce sharp peaks, but quasiperiodicity is sufficient (LEVINE and STEINHARDT, 1984). This latter means, in a descriptive sense, that a quasiperiodic structure is built of two (or more) different unit cells instead of one, such that the ratio of their corresponding geometric parameters (e.g. volume) is irrational number (and so is their relative frequency, i.e. occurrence in at least in principle, an infinite structure). Thus, the translational order is lacking here. On the other hand, the long-range rational order is replaced by a long-range "orientational" order. This means that the characteristic bonding directions fit to some straight lines of different orien-tation. A two-dimensional model of such a quasiperiodic structure is visualized in Fig. 1. It is directly seen that picking out any lattice point and considering the densest packed straight lines passing through it, these lines succeed one after the other by an angle 360/5=72°. "Thus, the symmetry of this "quasilattice" reveals itself in such an indirect way (or, as mentioned above, through the symmetry of its diffraction pattern). In three dimensions, taking another such pentagyre through the same point, further five-fold axes are induced, thus we have altogether six pentagyres. Actually, ten three-fold axes and fifteen two-fold axes will be generated at the same time. This configuration of symmetry elements corresponds just to the point group consist-H—6701 Szeged, P.O.Box: 651, Hungary. 5 Fig. 1. A two-dimensional quasiperiodic structure: the Penrose pattern ing of all rotations, of an icosahedron (Fig. 2) , or of a regular pentagonal dodecahed-ron (Fig. 3). So, these two regular polyhedra (Platonic solids), exorcised from the classical crystallography, appear in the geometry of quasicrystals. The symmetric axes listed before can be more easily perceived on a combination of these two Platonic solids (Fig. 4), which is an intermediate form recalling the cubooctahedron of the classical crystals. The stereographic projection of the point group I—235 considered is shown in Fig. 5 '. This may be regarded as à hemihedral "quasicrystal class" "and, as ï t contains only;rotational axes, an.enäntiomorphic one. Adding inversion centré to it,- fifteen additional mirror planes are induced and the holohedral class Ih -r-' m33. is obtained (Fig. 6), which is the full symmetry group of the icosahedron. The. corre-: sponding general (i.e. having faces of maximal number) "quasicrystal form" consists of 120 faces ("hecatonicosahedron", Fig. 7). This formis obtained from a single face 6. Fig. 2. Icosahedron, representingquasicrystalline symmetry situated so that it is. orthogonal, to none of the symmetry elements of Ih: in this case each element of the point group will multiply it.). For completeness, we note that the 3-dimensional quasilattice of icosahedral symmetry is built also of two unit cells,' acute and obtuse rhombohedra with edges of equal length (just as the two-dimensional pentagonal quasilattice is built of acute and obtuse rhombuses, cf. Fig. 1}. Moreover, any section through a lattice point and orthogonal to a five-fold axis provides just the PENROSE pattern (for further details of this quasilattice see e.g. GINGL, 1988). As a matter; of fact, the "shechtmanite" AJ0 86Mn0.14 described by SHKCHTMAN etal. (1984) exhibits this icosahedral symmetry, and phases like that have been called 7, Fig 4. A combination of icosahedron (unshaded faces) and regular pentagonal dodecahedron (shaded faces). The central points of shaded faces, unshaded faces and of edges between two unshaded faces are the emergence points of five-fold, three-fold and two-fold axes, respectively Fig. 5. Stereographic projection of the point group 1—235. since then icosahedral phase (IP). One should note, however, that this is not a unique type of quasicrystals : there are other phases which are characterized by ten-fold (BENDERSKY, 1985), moreover, by twelwefold (ISHIMASHA et al. 1985) rotational axis. CONSIDERATIONS ON THE FORMATION OF QUASICRYSTALS The first publications on quasicrystals described preparations produced in laboratory conditions. Naturally arises the question, in what circumstances can quasicrystals be expected to form spontaneously in Nature. Such a possibility cannot 8 Fig. 6. Stereographic projection of the point group Ib—m33. Fig. 7. Hecatonicosahedron, the form of maximal face number corresponding to the point group Ib—m35. It has 120 faces, 180 edges and 62 vertices. It is an icosahedral anal ogueof the hexakis octahedron of cubic symmetry be excluded ab ovo, as it is indicated by some analogies in connection with other "exotic" structures, as follows. In a sense, silicon atoms in octahedral position can be considered strange enough. However, stishovite, a high pressure modification of Si02 known from the Arizona meteor crater (STRUNZ, 1966) contains [Si06] octahedra (type rutil). In extreme con-ditions, silicon atoms are forced into such a coordination (formation at 1200— 9 1400 °C, >160,000 atm), though, here Js-rio-indication of violating the strict peri-odicity. A real deviation from strict periodicity can be observed in a continuously broadening group of structures which are called modulated structures. Modulation means that a periodic perturbation occurs in the lattice (GUGGENHEIM and EGGLE-TON, 1987). These phases were discovered in the sixties on the basis of the observation that in their diffraction diagram extra (satellite) reflections could' be found. The satelli-te reflections cannot be indexed in the usual way, actually, they often have irrational indices, violating the classical law of rationality (DE WOLFF, 1974). The irrationality is considered as a consequence of incommensurability between the translational and modulation period (i.e. their ratio cannot be represented as a ratio of two integer numbers). Since the discovery, a number of modulated crystals have been found and described (see e.g. the review by GUGGENHEIM and EGGLETON. 1987), indicating that the world of minerals is more abundant than one might imagine merely on the basis of classical crystallography. The relation of modulated (incommensurate) crystals and quasicrystals is still a debated problem. However, it is worth pondering on the possibility of proper quasicrystals among minerals. For example, BOISEN JR. and GIBBS (1985) speculate on a possible quasicrystalline form of skutterudite. In the skutterudite (Co4As12), there are icosahedral units As12 which, combining with interstitial Co atoms, form a lattice of cubic (m3) symmetry. The authors cited refer to the fact that the domi-nant crystal form of skutterudite, the pyritohedron, closely resembles the regular pentagonal dodecahedron. Their supposition that this mineral may have passed through a quasicrystalline state prior to its final crystallization, is in accordance with OSTWALD'S rule. By this rule, a stable modification often does not form directly, but the action progresses through successive steps in which intermediate phases appear. Its application here is justified by the observation that formation of quasi-crystals is a non-equilibrium process and they are generally in metastable state. It is also worth mentioning that skutterudite has a metalloid structure and the quasi-crystalline substances (according to our present knowledge) are generally alloys. Thus the crystal chemical resemblance also supports the idea due to the above authors. Let us see now the laboratory formation circumstances of quasicrystals. By now a series of methods is known, see RONCHETTI (1987). From our point of view, the most important methods are those in which rapid melt cooling (quenching) is applied (with often a simultaneous spinning of melt). A common and important feature of these and other methods is the high nucleation rate in comparison with growth rate. This seems to be a rather characteristic condition for the formation of quasicrystals. In what follows we try to give some geological examples where physico-chemical conditions mentioned just above might be expected to favour formation of natural quasicrystals. A. Rapid cooling resulting in highly supercooled melt in which crystallization is characterized by high nucleation rate is a typical feature of magma intrusion. An additional parameter intensifying nucleation is the strong convective motion of the melt: the intruding magma is forced in the magma chamber into an intense motion controlled by several interconnected thermodynamical parameters. Actually, at least three types of convection are distinguished (MORSE, 1988) thermal convection (due to thermal gradient, especially that between floor and roof of the chamber), two-phase convection (involving crystal+liquid suspensions that can move much faster than small individual crystals can settle), and compositional convection (due 10 to rejection of solute by growing cumulates). Study of layered intrusions show that competition between nucleation, growth, and cooling rates (THY et al. 1988) can often .lead to an oscillatory behaviour of the multisaturated liquid, which is reflected in the rhyhtmic layering. (MAAL0E, 1978) of the product phases. Though nucleation theories aiming at explaining rhythmic layering are, at least partly, in controversy with each other, they admit that abrupt changes in parameters controlling nucleation can occur, giving rise to burst of nucleation. This is the very point which is important to our discussion: the magmatic dynamic patterns may create appropriate physico-chemicar conditions for the formation of quasicrystals. B. Shock-hardened meteorites: in these substances a number of specific trans-formations can be traced. Hardening, multiple twinning and polymorphic transition are typical alterations induced by terrestrial or preterrestrial shock. From the point of view of solid state physics, hardening is due to blocking the motion of dislocations (KITTEL, 1978). This can bé achieved, for example, by homo-génizing the dislocation density, a transformation which results'e.g. from explosive deformation. A related process that also blocks the motion of dislocations is the for-mation-of short-range order. As quasip.eriodicity is an intermediate case between long-range (periodic) order and total disorder, a factor of hardening may be such partial disordering, i.e. formation of quasicrystalline domains. In this respect, a phenomenon called "preferred disorder" respectively "preferred orientation" (JAIN and LIPSCHUTZ, 197Ó) may be worthy of attention. This roughly means that disorder-ing is confined to certain directions and it is . revealed by radial streaks in X-ray diffraction photographs, which is called asterism. In a more recent work by HORZ et al. (1986), Debye—Scherrer patterns of experimentally shocked augite and nat-urally shocked pyroxenes from the Shergotty meteorite are shown. As it is noted there, substantial disturbance of long-range order and preferred orientation are indi-cated, the latter by discontinuous and streaky reflections. Moreöver, what is mainly essential from our point of view, the asterism exhibits nearly 14-fold symmetry (Fig. 1 in the paper referred to; the relevant photographs cannot be adequately repro-duced here because of their small dimension).. These figures originate from discrete grains of 220—250 |im diameter and are said to be hybrids between single crystals and fine-grained powders, and the lattice is considered as mechanically disaggregated by shock. It is stressed by the authors that "crystallographic control becomes imprac-tical for such highly fragmented samples". However, despite of the difficulties, such and similar phenomena perhaps would be worthy to be subjected to further investiga-tion, in our opinion. . As for the other, two phenomena, viz; multiple twinning and polymorphism in meteorites, here we only refer to the fact that there exists an alternative notion of quasicrystals emphasized by LINUS PAULING (1987), according to which the properties of quasicrystals can be satisfactorily explained by supposing certain multiple twinning structures. The debate has not been concluded yet (WOLNY et al., 1988). CONCLUDING REMARKS Of course, the train of thought in A and B is a tentative one and even does not pretend to be anything more. This is because the knowledge on quasicrystals is in an accumulation period and theoretical (e.g. geometric) description of their structure is not completed yet, in contrast to classical crystals for which at least the geometric point lattice theory essentially was completed at the beginning of the century. On the other hand, actually in any geological sample having more or less 1.1 peculiarity in its formation circumstances one may find (at least small domains of) quasicrystals. Perhaps a number of researchers have already met without recognizing-them: surely all of us are more or less confined to scientific "dogmas", like for example the strict crystallographic prohibition of five-fold symmetry. A C K N O W L E D G E M E N T One of the authors (G. G. ) is grateful to Prof. J. MEZOSI for inspiring discussions. REFERENCES BENDERSKY, L. (1985): Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis. Phys. Rev. Lett. 55, 1461—1463. BOISEN Jr., M. B. and GIBBS, G. V. (1985): Mathematical Crystallography. In: Reviews in Mineralogy (Ed. P. H. Ribbe). Vol. 15. COXETER, H. S. M. (1961): Introduction to Geometry. Wiley, New York. GINGL, Z. (1988): Local structure of quasicrystals. Degree Work. J ATE Institute of Mineralogy, Geochemistry and Petrography, Szeged (in Hungarian). GUGGENHEIM, S. and EGGLETON, R. A. (1987): Moduled 2:1 layer silicates: Review, systematics and predictions, Amer. Mineral., 72, 724—738. HORZ, F., HANSS, R. and SERNA, C. (1986): X-ray investigations related to the shock history of the Shergotty achondrite. Geochim. Cosmochim. Acta 50, 905—908. ISHIMASHA, T., NISSEN, H . U . and FUKANO, Y . (1985): N e w ordered state between crystalline and amorphous in Ni-Cr particles. Phys. Rev. Lett. 55, 511—513. JAIN, A. V. and LIFSCHUTZ, M. E. (1970): On preferred disorder and the shock history of chemical group IVA meteorites. Geochim. Cosmochim. Acta 34, 883—892. KITTEL, C. (1978): Introduction to Solid State Physics, Wiley, New York. LEVINE, D. and STEINHARDT, P. J. (1984): Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477—2480. MAALOE, S. (1978): The origin of rhythmic layering. Mineral. Mag. 42, 337—345. MORSE, S. A. (1988): Motion of crystals, solute and heat in layered intrusions. Can. Mineral. 26, 209—224. PAULING, L. (1987): So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal. Phys. Rev. Lett. 58, 365. ROMERIO, M. W. (1971): Almost periodic functions and the theory of disordered systems. J. Math. Phys. 12 (3),'552—562. RONCHEITI, M. (1987): Quasicrystals. An introductory overview. Phil. Mag. B. 56 (2), 237—249. SHECHTMAN, D . , BLECH, I., GRATIAS, D . and CAHN, J. W. (1984): Metall ic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951—1953. STRUNZ, H. (1966): Mineralogische Tabellen. Akademische Verlagsgesellschaft Geest und Portig Ki-G. Leipzig. THY, P., JAKOBSEN, N. N. and WILSON, J. R. (1988): Fine-scale graded layers in the Fongen—Hyllin-gen gabbroic complex. Norway. Can. Mineral. 26, 235—243. DE WOLFF, P. M. (1974): The pseudo-symmetry of modulated crystal structures. Acta Cryst. A30 777—785. ' WOLNY, J., PYTLIK, L. and LEBECH, B. (1988): Quasi-crystals-random structures on twins? J. Phys. C.: Solid State Phys. 21, 2267—2277. Manuscript received, 5 September, 1988 12 

Quasicrystals and their spontaneous formation possibilities in the nature

http://acta.bibl.u-szeged.hu/24753/1/mineralogica_029_005-012.pdf

Quasiciystals and their spontaneous formation possibilities in the nature

Abstract

Similar works

Full text

Available Versions

University of Szeged