Review Quasicrystals and Other Aperiodic Structures in Mineralogy
Carlos M. Pina 1,2,* and Victoria López-Acevedo 1,3
1 Department of Crystallography and Mineralogy, Faculty of Geological Sciences, Complutense University of Madrid, 28040 Madrid, Spain; [email protected] 2 Geosciences Institute IGEO (Universidad Complutense de Madrid-Consejo Superior de Investigaciones Científicas), 28040 Madrid, Spain 3 Museum of Geology at the Complutense University of Madrid, 28040 Madrid, Spain * Correspondence: [email protected]; Tel.: +34-91-394-4879; Fax: +34-91-394-4872
Academic Editor: Helmut Cölfen Received: 1 October 2016; Accepted: 20 October 2016; Published: 27 October 2016
Abstract: In this article, we first present and discuss eighteenth-century descriptions of minerals that contributed decisively to the development of crystallography. Remarkably, these old crystallographic descriptions included morphologies with symmetries incompatible with an internal periodic order of atoms, which, however, have been recognised to be characteristics of quasicrystals. Moreover, we also review a number of studies of minerals with aperiodic crystal structures, including recently reported natural quasicrystals of extra-terrestrial origin. Finally, we discuss the current investigations addressing the search for new quasicrystalline minerals in nature.
Keywords: quasicrystals; aperiodic mineral structures; crystal and quasicrystal morphologies; quasicrystalline minerals; skutterudite; cobaltine
1. Introduction The development of crystallography has been closely interwoven with mineralogical investigation. Minerals were the first accessible crystalline solids that scientists were able to study and measure systematically in order to derive the first crystallographic laws. In the eighteenth century, mineralogists already realised that the external polyhedral forms of minerals were constant and determined by an internal order of their constituting atoms or molecules [1,2]. Despite this remarkable finding, it took about a century to fully establish the crystallographic principles that govern such an order [3–6]. These principles were derived theoretically on the basis of the concept of periodicity; that is, it was assumed that the order of crystal structures is periodic in all three space dimensions. At the beginning of the twentieth century, internal periodicity of crystals was nicely demonstrated by X-ray diffraction experiments using minerals [7,8]. Notwithstanding this, diffraction studies soon showed that the structures of a number of minerals are occasionally not strictly periodic (e.g., calaverite, quartz, and feldspars). These anomalous structures usually resulted from phase transformations and were interpreted as being more or less complex modulations of average periodic lattices. By doing this, the crystallographic paradigm was preserved for some time. However, the discovery of quasicrystals, whose pentagonal symmetries are noticeably incompatible with a periodic order of atoms [9], definitively challenged the fundamentals of crystallography. The scientific community was then forced to review the traditional concepts of crystal and crystal lattices to include internally ordered solids without translational symmetry. Consequently, in 1991, the International Union of Crystallography stated that “a crystal is any solid having an essentially discrete diffraction diagram”, a definition which intentionally avoids the term “periodic” [10]. In this case, the new developments in crystallography and the subsequent redefinition of crystal resulted from the study not of minerals but of synthetic alloys.
Crystals 2016, 6, 137; doi:10.3390/cryst6110137 www.mdpi.com/journal/crystals Crystals 2016, 6, 137 2 of 16
Nevertheless, in 2009, the first mineral with a quasicrystalline structure, named icosahedrite, was found within a meteorite from Khatyrka (eastern Russia), and in 2015 a new natural quasicrystal with decagonal symmetry was also identified in the same meteorite [11–13]. Although mineral quasicrystals formed on Earth have not been recorded to date, it is quite conceivable that some minerals may exist in a quasicrystalline state on our planet. Synthesis experiments and crystallochemical analyses suggest that a number of minerals could be transformed into quasicrystals under extreme pressure and temperature conditions on Earth [14–16]. Hence, finding such mineral quasicrystals of terrestrial origin is relevant to the investigation of the formation conditions of aperiodic crystals in nature and would constitute a new contribution of mineralogical studies to the advancement of crystallography. The aim of this article is to review some of the main contributions of mineralogy to the study of aperiodic and quasicrystalline structures and to discuss mineral structures that could be related to the formation of quasicrystals on Earth.
2. Before Crystallography The foundation of crystallography as a modern science was the result of several centuries of careful observations and mathematical thinking about the origin of the polyhedral shapes of minerals. Initially, the researchers did not make any distinction between minerals, rocks, sediments, and fossils, and the term fossilis referred to all materials that can be found under the ground [17,18]. It was not until the end of the sixteenth century that the difference between fossils (of organic origin) and minerals with polyhedral shapes (formed by the addition of material) began to be recognised [19]. Once the inorganic origin of minerals became apparent, the interest of scientists focussed on their regular external morphologies. As early as 1669, Nicolaus Steno realised that the angles between analogous faces of quartz crystals were always identical [20]. This important observation was subsequently generalised by Romé de L’Isle [21], who measured a large number of interfacial angles of many different crystals and formulated the first law of crystallography: The Law of the Constancy of Interfacial Angles. But we also owe to him one of the earliest and broadest inventories of crystal forms. In his book Cristallographie [1], he described up to 746 crystals of different minerals and presented ten tables with drawings of observed and idealised crystal morphologies. Some of them were also carefully reproduced in baked clay and complemented his book. Crystal morphologies reported by Romé de L’Isle can be divided into two categories. The first of these categories includes both crystal shapes that he apparently observed in minerals from his own collection (At present, part of the mineral collection of Romé de L’Isle can be found at the Muséum National d’Histoire Naturelle in Paris (France)) and descriptions of minerals reported by other scientists [22,23]. The second category of morphologies described in Cristallographie [1] is constituted by several idealised crystal shapes obtained after virtual cutting of vertices and/or bevelling of edges of forms considered by Romé de L’Isle as “primitives” (i.e., parallelepipeds, prisms, and pyramids). Interestingly, in both categories of crystal shapes, some forms with pentagonal symmetries (i.e., morphologies incompatible with a periodic internal order of crystal structures) can be found (Figure1). Although Romé de L’Isle could only see crystal forms with approximately fivefold symmetries, it is worth considering his descriptions in detail. According to him, all the morphologies with pentagonal faces shown in Figure1 are variants of the cube and they can be considered as characteristic of the minerals pyrite and marcassite (the name commonly given in the seventeenth and eighteenth centuries to any pyrite with a high content of sulphur and variable amounts of zinc, copper, and other metals). In the case of the dodecahedron and the icosahedron, it seems obvious that Romé de L’Isle tried to reconcile his observations to the forms of the Platonic solids, which fascinated scientists and philosophers in ancient times. Similarly, the description of the triacontahedron and the pyramidal dodecahedron were attempts by Romé de L’Isle to approximate observed mineral forms to regular polyhedra. While Romé de L’Isle admitted in his Cristallographie that he never observed an isolated triacontahedron, he claimed to have a marcassite crystal with the shape of a pyramidal dodecahedron in Crystals 2016, 6, 137 3 of 16
Crystals 2016, 6, 137 3 of 16
his personal mineral collection. Marcassite crystals of similar shape to that described by Romé de L’Isle wereFigure also reported 1. Forms bywith Démeste non-crystallographic [22], indicating pentagonal that the external morphologies symmetries with by Romé apparent de L’Isle. pentagonal (a) Regular dodecahedron. (b) Elongated or pyramidal dodecahedron. (c) Regular triacontahedron. (d) symmetriesCrystals 2016, 6, are 137 not infrequent in this mineral (Figure2). 3 of 16 Icosahedron. Illustration adapted from Table II (Le Cube ou L’Hexaèdre et ses Modifications) in Volume IV of Cristallographie [1].
Although Romé de L’Isle could only see crystal forms with approximately fivefold symmetries, it is worth considering his descriptions in detail. According to him, all the morphologies with pentagonal faces shown in Figure 1 are variants of the cube and they can be considered as characteristic of the minerals pyrite and marcassite (the name commonly given in the seventeenth and eighteenth centuries to any pyrite with a high content of sulphur and variable amounts of zinc, copper, and other metals). In the case of the dodecahedron and the icosahedron, it seems obvious that Romé de L’Isle tried to reconcile his observations to the forms of the Platonic solids, which fascinated scientists and philosophers in ancient times. Similarly, the description of the triacontahedron and the pyramidal dodecahedron were attempts by Romé de L’Isle to approximate observed mineral forms to regular polyhedra. While Romé de L’Isle admitted in his Cristallographie that he never observed an isolated triacontahedron, he claimed to have a marcassite crystal with the shapeFigure of a pyramidal 1. FormsForms with withdodecahedron non-crystallographic non-crystallographic in his personal pentagonal pentagonal mineral external external collection. symmetries symmetries Marcassite by byRomé Romé crystals de L’Isle. de L’Isle.of (similara) shape(Regulara )to Regular that dodecahedron. described dodecahedron; by ( bRomé) (Elongatedb) Elongated de L’Isle or pyramidal or were pyramidal also dodecahedron. reported dodecahedron; by (Démestec) Regular (c) Regular [22],triacontahedron. triacontahedron; indicating that(d) the morphologies(Icosahedron.d) Icosahedron. with Illustration apparent Illustration adapted pentagonal adapted from fromTable symmetries TableII (Le Cube II (areLe ou Cubenot L’Hexaèdre infrequent ou L’Hexaèdre et ses in Modifications this et ses mineral Modifications) in (Figure Volume) in 2). VolumeIV of Cristallographie IV of Cristallographie [1]. [1].
Although Romé de L’Isle could only see crystal forms with approximately fivefold symmetries, it is worth considering his descriptions in detail. According to him, all the morphologies with pentagonal faces shown in Figure 1 are variants of the cube and they can be considered as characteristic of the minerals pyrite and marcassite (the name commonly given in the seventeenth and eighteenth centuries to any pyrite with a high content of sulphur and variable amounts of zinc, copper, and other metals). In the case of the dodecahedron and the icosahedron, it seems obvious that Romé de L’Isle tried to reconcile his observations to the forms of the Platonic solids, which fascinated scientists and philosophers in ancient times. Similarly, the description of the triacontahedron and the pyramidal dodecahedron were attempts by Romé de L’Isle to approximate Figure 2. (a) Photograph of a limonitised pyrite from Jarapalos, Málaga (Spain), showing a pyramidal observed mineral forms to regular polyhedra. While Romé de L’Isle admitted in his Cristallographie dodecahedron-like morphology (size of the crystal: ≈2 cm). Collection and picture from J.M. Bruguera. that he never observed an isolated triacontahedron, he claimed to have a marcassite crystal with the This crystal is similar to the marcassite reported by Romé de L’Isle in his book Cristallographie [1]. Figure 2. (a) Photograph of a limonitised pyrite from Jarapalos, Málaga (Spain), showing a shape(b of) Photograph a pyramidal of andodecahedron eighteenth-century in his bakedpersonal clay mineral model of collection. a pyramidal Marcassite dodecahedron crystals from of the similar pyramidal dodecahedron-like morphology (size of the crystal: ≈ 2 cm). Collection and picture from shapecollection to that described of the Geology by Romé Museum de L’Isle at the were Complutense also reported University by Démeste of Madrid [22], (size indicating of the model: that the J.M. Bruguera. This crystal is similar to the marcassite reported by Romé de L´Isle in his book morphologies2.5 cm × 2.3 with cm). apparent This model pentagonal reproduces symmetries a single crystal are of notmarcassite infrequentfrom in Romé this demineral L’Isle’s (Figure personal 2). Cristallographie [1]. (b) Photograph of an eighteenth-century baked clay model of a pyramidal mineral collection. Photograph by Toya Legido. dodecahedron from the collection of the Geology Museum at the Complutense University of Madrid (size of the model: 2.5 cm × 2.3 cm). This model reproduces a single crystal of marcassite from Romé Ade symmetryL’Isle’s personal analysis mineral of collection. the morphologies Photograph by described Toya Legido. by Romé de L’Isle shows that the dodecahedron, the icosahedron, and the triacontahedron have identical symmetry, that is, six fivefold roto-inversion axes, ten threefold roto-inversion axes, fifteen twofold axes, fifteen mirror planes, and a centre of symmetry. Differently, the symmetry content of the pyramidal dodecahedron only contains one fivefold roto-inversion axis, five twofold axes, five mirror planes, and a centre of symmetry. Using an extended Hermann-Mauguin notation, the dodecahedron, icosahedron, and triacontahedron belong to the m35 quasicrystal class, while the pyramidal dodecahedron belongs to the 5m2 quasicrystal class [24–26] (Figure3).
Figure 2. (a) Photograph of a limonitised pyrite from Jarapalos, Málaga (Spain), showing a pyramidal dodecahedron-like morphology (size of the crystal: ≈ 2 cm). Collection and picture from J.M. Bruguera. This crystal is similar to the marcassite reported by Romé de L´Isle in his book Cristallographie [1]. (b) Photograph of an eighteenth-century baked clay model of a pyramidal dodecahedron from the collection of the Geology Museum at the Complutense University of Madrid (size of the model: 2.5 cm × 2.3 cm). This model reproduces a single crystal of marcassite from Romé de L’Isle’s personal mineral collection. Photograph by Toya Legido.
Crystals 2016, 6, 137 4 of 16
A symmetry analysis of the morphologies described by Romé de L´Isle shows that the dodecahedron, the icosahedron, and the triacontahedron have identical symmetry, that is, six fivefold roto-inversion axes, ten threefold roto-inversion axes, fifteen twofold axes, fifteen mirror planes, and a centre of symmetry. Differently, the symmetry content of the pyramidal dodecahedron only contains one fivefold roto-inversion axis, five twofold axes, five mirror planes, and a centre of symmetry. Using an extended Hermann-Mauguin notation, the dodecahedron, icosahedron, and triacontahedron belong to the m35 quasicrystal class, while the pyramidal dodecahedron belongs to theCrystals 5 m22016 quasicrystal, 6, 137 class [24–26] (Figure 3). 4 of 16
Figure 3. Crystal morphologies reported by Romé de L’Isle [1] and the corresponding stereographic Figureprojections 3. Crystal of their morphologies symmetry elements. reported (a by) Forms Romé belonging de L’Isle [1] to theand m the35 icosahedralcorresponding quasicrystal stereographic class projections(from top to of bottom: their symmetry dodecahedron, elements. icosahedron, (a) Forms and belonging triacontahedron); to the m 35 ( b ) elongatedicosahedral dodecahedron quasicrystal classshowing (from the top symmetry to bottom: of the dodecahedron5m2 quasicrystal, icosahedron, class; and ( c)and cube triacontahedron); belonging to m3 m(b) crystal elongated class. dodecahedronSymbols: ellipses, showing triangles, the squares,symmetry and of pentagonsthe 5 m2 quasicrystal indicate the class; orientations and (c) ofcube the belonging twofold, threefold, to m3 m crystalfourfold, class. and Symbols: fivefold axesellipses, respectively. triangles, Thesquares, full linesand pentag in theons stereographic indicate the projections orientations represent of the twofold,mirror planes. threefold, fourfold, and fivefold axes respectively. The full lines in the stereographic projections represent mirror planes. Interestingly, recent investigations have shown that the morphologies with symmetries described aboveInterestingly, are commonly recent exhibited investigations by quasicrystals have shown [27–29] (Figurethat the4a). morphologies Obviously, Romé with desymmetries L’Isle did describednot predict above the existence are commonly of quasicrystals exhibited but by onlyquasicrystals described [27–29] some forms (Figure that 4a he). Obviously, considered plausibleRomé de L’Islefor natural did not crystals predict (Figure the existence4b). In fact, of quasicrystals icosahedral quasicrystalbut only described classes some (e.g., mforms35 and that5 m2)he considered and cubic plausiblecrystal classes for natural (e.g., m crystals3m and (Figure m3) are 4b). symmetrically In fact, icosahedral and topologically quasicrystal not soclasses different (e.g. fromm35 eachand 5 other, m2) andsince cubic they crystal are related classes to the(e.g. most m3 m efficient and m ways3 ) are ofsymmetrically filling the three-dimensional and topologically space not so with different atoms. fromFurthermore, each other, it is nowsinceknown they are that related a cubic to lattice the mo andst anefficient icosahedral ways quasi-latticeof filling the can three-dimensional both be obtained spaceby projecting with atoms. a six-dimensional Furthermore, it hypercube is now known on the that three-dimensional a cubic lattice and space an icosahedral [30]. The fundamental quasi-lattice candifference both be between obtained the by two projecting types of a projectionssix-dimensio isnal that hypercube periodicity on is the preserved three-dimensional in the former space and [30]. lost Crystals 2016, 6, 137 5 of 16 Thein the fundamental latter. difference between the two types of projections is that periodicity is preserved in the former and lost in the latter.
FigureFigure 4.4. (a) Scanning electron microscopy imageimage of an Al62.2CuCu25.325.3FeFe12.512.5 quasicrystalquasicrystal with with the the shape shape of ofan anelongated elongated dodecahedron dodecahedron (reproduction (reproduction from from [28]). [28]); (b ()b Scanning) Scanning electron electron microscopy microscopy image image of of a alimonitised limonitised pyrite pyrite withwiththe the approximate approximate shape shape of an of elongated an elongated dodecahedron, dodecahedron, similar tosimilar the marcassite to the describedmarcassite bydescribed Romé de by L’Isle Romé [1 de]. L’Isle [1].
Romé de L’Isle was not the only scientist interested in the shapes of crystals at that time. His contemporary, René Just Haüy, also devoted much effort to describing and studying the morphology of minerals. Haüy presented most of his observations and ideas about crystals in his Traité de Mineralogie [2], which also contains numerous drawings of natural crystals and geometrical constructions. From many observations and measurements, Haüy deduced that crystals are built up from fundamental units with a parallelepipedic shape, repeated in three dimensions. Such repetitions imply that the internal order of crystals is periodic. Haüy’s measurements of angles between crystal faces are consistent with a periodic stacking of crystal unit cells (or molécules intégrantes, according to Haüy´s definition) and allowed him to formulate the second law of crystallography, the law of rational indexes, which states that the intercepts of the natural faces of a crystal form with the unit-cell axes are inversely proportional to prime integers. As a consequence of this law, external crystal morphologies with real pentagonal symmetries must be considered impossible. Haüy was aware of the incompatibility of pentagonal symmetries with the law of rational indexes. Accordingly, in his review of the forms described by Romé de L’Isle, Haüy explicitly discarded the dodecahedron, the icosahedron, and the triacontahedron as morphologies exhibited by real crystals. After that, we had to wait almost two centuries until scientists reconsidered such forms as the result of a highly ordered (but not periodic) arrangement of atoms within solids. The law of rational indexes paved the way for the development of crystallography in the following two centuries by excluding the possibility of any internal crystal order other than periodic. However, as we will see in the next section, the impressive achievements of the research in crystallography in the nineteenth and twentieth centuries were also accompanied by the discovery of mineralogical cases in which violation of the periodic internal order of crystals was evident. Some of these cases were found to be difficult or impossible to explain within the classical crystallographic paradigm and were often ignored or forgotten.
3. The Crystallographic Paradigm and “Dissident” Minerals The hypothesis that crystals are formed by a periodic arrangement of unit cells was the starting point for the development of modern crystallography. In the decades following the publication of Haüy’s works, scientists reported descriptions and goniometric measurements of a large number of minerals with polyhedral shapes [31–33]. These investigations led to the establishment of the geometrical principles of crystallography. One of the first contributions to so-called geometrical crystallography was a demonstration by Hessel [34,35] that all crystal shapes can be grouped into 32 symmetry classes; that is, there are only 32 possible combinations of symmetry elements consistent with solids with internal periodic order. A few years later, Bravais [3] derived the possible three-dimensional lattices that describe the periodic repetition of unit cells within the crystals, that is, the so-called 14 Bravais
Crystals 2016, 6, 137 5 of 16
Romé de L’Isle was not the only scientist interested in the shapes of crystals at that time. His contemporary, René Just Haüy, also devoted much effort to describing and studying the morphology of minerals. Haüy presented most of his observations and ideas about crystals in his Traité de Mineralogie [2], which also contains numerous drawings of natural crystals and geometrical constructions. From many observations and measurements, Haüy deduced that crystals are built up from fundamental units with a parallelepipedic shape, repeated in three dimensions. Such repetitions imply that the internal order of crystals is periodic. Haüy’s measurements of angles between crystal faces are consistent with a periodic stacking of crystal unit cells (or molécules intégrantes, according to Haüy’s definition) and allowed him to formulate the second law of crystallography, the law of rational indexes, which states that the intercepts of the natural faces of a crystal form with the unit-cell axes are inversely proportional to prime integers. As a consequence of this law, external crystal morphologies with real pentagonal symmetries must be considered impossible. Haüy was aware of the incompatibility of pentagonal symmetries with the law of rational indexes. Accordingly, in his review of the forms described by Romé de L’Isle, Haüy explicitly discarded the dodecahedron, the icosahedron, and the triacontahedron as morphologies exhibited by real crystals. After that, we had to wait almost two centuries until scientists reconsidered such forms as the result of a highly ordered (but not periodic) arrangement of atoms within solids. The law of rational indexes paved the way for the development of crystallography in the following two centuries by excluding the possibility of any internal crystal order other than periodic. However, as we will see in the next section, the impressive achievements of the research in crystallography in the nineteenth and twentieth centuries were also accompanied by the discovery of mineralogical cases in which violation of the periodic internal order of crystals was evident. Some of these cases were found to be difficult or impossible to explain within the classical crystallographic paradigm and were often ignored or forgotten.
3. The Crystallographic Paradigm and “Dissident” Minerals The hypothesis that crystals are formed by a periodic arrangement of unit cells was the starting point for the development of modern crystallography. In the decades following the publication of Haüy’s works, scientists reported descriptions and goniometric measurements of a large number of minerals with polyhedral shapes [31–33]. These investigations led to the establishment of the geometrical principles of crystallography. One of the first contributions to so-called geometrical crystallography was a demonstration by Hessel [34,35] that all crystal shapes can be grouped into 32 symmetry classes; that is, there are only 32 possible combinations of symmetry elements consistent with solids with internal periodic order. A few years later, Bravais [3] derived the possible three-dimensional lattices that describe the periodic repetition of unit cells within the crystals, that is, the so-called 14 Bravais lattices. Finally, Fedorov, Schoenflies, and Barlow [4–6] independently derived the 230 space groups, that is, the combinations of symmetry elements compatible with a periodic order which describe all possible crystal structures. The 32 crystal classes, the 14 Bravais lattices, and the 230 space groups constitute the fundamentals of crystallography that allow us to explain both the external symmetries and the structures of all crystalline materials. The discovery of the diffraction of X-rays by crystals in 1912 and the subsequent use of diffraction techniques provided an experimental support for the abovementioned fundamentals of crystallography [7,8]. Moreover, diffraction data allowed scientists to determine in a few years the structure of a multitude of minerals and synthetic compounds. This led to unprecedented knowledge of the solid matter and the principles that govern its internal order. The structure of the mineral halite (NaCl) was the first crystal structure determined by X-ray diffraction, that is, the positions of the sodium and chlorine atoms within a cubic lattice [36]. After that, the number of structures solved increased rapidly [37,38]. Currently, there are more than 50,000 known crystal structures. Among them are about 4500 minerals. Crystals 2016, 6, 137 6 of 16 lattices. Finally, Fedorov, Schoenflies, and Barlow [4–6] independently derived the 230 space groups, that is, the combinations of symmetry elements compatible with a periodic order which describe all possible crystal structures. The 32 crystal classes, the 14 Bravais lattices, and the 230 space groups constitute the fundamentals of crystallography that allow us to explain both the external symmetries and the structures of all crystalline materials. The discovery of the diffraction of X-rays by crystals in 1912 and the subsequent use of diffraction techniques provided an experimental support for the abovementioned fundamentals of crystallography [7,8]. Moreover, diffraction data allowed scientists to determine in a few years the structure of a multitude of minerals and synthetic compounds. This led to unprecedented knowledge of the solid matter and the principles that govern its internal order. The structure of the mineral halite (NaCl) was the first crystal structure determined by X-ray diffraction, that is, the positions of the sodium and chlorine atoms within a cubic lattice [36]. After that, the number of structuresCrystals 2016 ,solved6, 137 increased rapidly [37,38]. Currently, there are more than 50,000 known crystal6 of 16 structures. Among them are about 4500 minerals. As the knowledge of mineral structures and external morphologies of crystals increased, mineralogistsAs the knowledgeand crystallographers of mineral discovered structures that and some external crystals morphologies showed features of crystals that were increased, clearly inconsistentmineralogists with and the crystallographers idea of a perfect discovered and period thatic internal some crystals atomic showed order. Investigations features that were conducted clearly atinconsistent the beginning with of the the ideatwentieth of a perfect century and already periodic dem internalonstrated atomic that order.crystals Investigations usually exhibit conducted more or lessat the severe beginning alterations of the of twentieth their internal century periodicity. already demonstratedThese could affect that single crystals atomic usually positions, exhibit rows more of or atoms,less severe or even alterations reticular of planes. their internal While some periodicity. of these These violations could affectof the singlecrystallographic atomic positions, order were rows interpretedof atoms, or as even simple reticular “crystal planes. defects,” While others some posed of these really violations serious ofproblems the crystallographic for the crystallographic order were paradigm,interpreted thereby as simple warranting “crystal defects”,further analysis. others posed really serious problems for the crystallographic paradigm,In the therebyearly twentieth warranting century, further morphologi analysis. cal studies of the mineral calaverite (Au1-xAgxTe2) revealedIn the an early astonishingly twentieth century,complex morphological crystal morphology studies ofdefined the mineral by more calaverite than (Au 901 −differentxAgxTe2 ) crystallographicrevealed an astonishingly forms [39,40] complex(Figure 5). crystal The first morphology attempts to definedindex all bycalaverite more faces than required 90 different the usecrystallographic of different lattices, forms [ 39which,40] (Figureled questioning5). The first the attempts universality to index of the all calaveritelaw of rational faces requiredindexes thefor usethe firstof different time. Far lattices, from clarifying which led the questioning “calaverite the problem,” universality X-ray of diffraction the law of data rational showed indexes satellite for the spots, first whichtime. Farindicated from clarifying that calaverite the “calaverite had a complex problem”, supe X-rayrstructure diffraction whose data explanation showed satellite was still spots, elusive. which Subsequentindicated that electron calaverite diffraction had a complex patterns superstructure confirmed the whose existence explanation of satellite was still diffraction elusive. Subsequent spots. A detailedelectron diffractionanalysis of patternssuch spots confirmed revealed the that existence calaverite of satellite has a diffraction modulated spots. structure A detailed [41]. analysisSuch a modulationof such spots results revealed from that both calaverite the displacement has a modulated of Te structure and the [41 occupation]. Such a modulation of Ag atomsresults (which from partiallyboth the displacementsubstitute the of Au Te andatoms), the occupationresulting in of Aga deviation atoms (which from partially the average substitute C2/m the monoclinic Au atoms), structureresulting inof acalaverite deviation from[42]. theSince average the period C2/m monoclinicof modulation structure is not of an calaverite integral [42 number]. Since theof lattice period translations,of modulation the is notstructure an integral of numbercalaverite of lattice is defined translations, as theincommensurate. structure of calaverite Remarkably, is defined the as incommensurabilityincommensurate. Remarkably, of the calaverite the incommensurability structure has a ofmorphological the calaverite expression structure has consisting a morphological in the coexistenceexpression consistingof many crystal in the coexistence forms, whose of many indexa crystaltion forms,was conducted whose indexation by the simultaneous was conducted use by theof differentsimultaneous lattices. use ofRecently, different the lattices. problems Recently, associated the problems with using associated this withcumbersome using this and cumbersome artificial indexationand artificial have indexation been overcome havebeen by the overcome use of face by Miller the use indexes of face consisting Miller indexes of four consisting numbers, ofwhere four thenumbers, fourth wherenumber the describes fourth number the modula describestion [43–46], the modulation and references [43–46 therein].], and references therein].
Figure 5. Morphology of calaverite. (a) Calaverite twinned crystals; (b) Indexed morphology of Figurecalaverite 5. Morphology according to Jannerof calaverite. and Dam (a) [ 45Calaverite]. Modified twinned and redrawn crystals. from (b) Chapuis Indexed [46 morphology]. of calaverite according to Janner and Dam [45]. Modified and redrawn from Chapuis [46].
Far from being exceptional, modulated structures are relatively common in minerals. In all cases, transmission electron microscopy (TEM) observations have shown that structural modulations result in the presence of satellite reflections and irregular arrangements of Bragg maxima in the diffraction patterns. As for the mineral calaverite, modulated structures are commonly described by a basic (average) periodic structure to which periodic lattice distortions are introduced by means of one or more modulation vectors. When modulation vectors can be expressed by a linear combination of the lattice vectors of the basic structure, superstructures are formed and an ordinary space group can be assigned to them. Otherwise, modulations are incommensurate and crystal structures are aperiodic along one or more crystallographic directions. The origin of periodic lattice distortions in crystals is diverse, and modulated structures can arise from cationic ordering, exsolution phenomena involving two or more chemical components, transformation twinning, formation of stacking faults, development of antiphase boundary domains, and so on [47,48]. These modulated structures usually appear during transitions from high-temperature Crystals 2016, 6, 137 7 of 16
Far from being exceptional, modulated structures are relatively common in minerals. In all cases, transmission electron microscopy (TEM) observations have shown that structural modulations result in the presence of satellite reflections and irregular arrangements of Bragg maxima in the diffraction patterns. As for the mineral calaverite, modulated structures are commonly described by a basic (average) periodic structure to which periodic lattice distortions are introduced by means of one or more modulation vectors. When modulation vectors can be expressed by a linear combination of the lattice vectors of the basic structure, superstructures are formed and an ordinary space group can be assigned to them. Otherwise, modulations are incommensurate and crystal structures are aperiodic along one or more crystallographic directions. The origin of periodic lattice distortions in crystals is diverse, and modulated structures can arise from cationic ordering, exsolution phenomena involving two or more chemical components, Crystalstransformation2016, 6, 137 twinning, formation of stacking faults, development of antiphase boundary domains,7 of 16 and so on [47,48]. These modulated structures usually appear during transitions from high-temperature to low-temperature phases. Within the mineral world, the complex modulated structures of to low-temperature phases. Within the mineral world, the complex modulated structures of feldspars feldspars due to Al/Si and Ca/Na ordering and the incommensurable structure formed during the due to Al/Si and Ca/Na ordering and the incommensurable structure formed during the polymorphic polymorphic transition of quartz can be considered some of the most remarkable mineralogical cases transition of quartz can be considered some of the most remarkable mineralogical cases of aperiodicity of aperiodicity in crystal structures (Figure 6). in crystal structures (Figure6).
Figure 6. Incommensurate structure of quartz (modified from Putnis [48]). This structure is formed duringFigure the6. Incommensurate transformation fromstructure high of to quartz low quartz. (modified Both from regions Putnis of lattice [48]). distortionThis structure (+ and is formed –) and shearduring on the the transformation Dauphiné twin from boundaries high to (low↑↓ and quartz.↓↑) oscillate.Both regions Since of the lattice oscillations distortion (represented (+ and –) and by ↓↑ theshear waves on the a and Dauphiné b) are not twin an boundaries integral multiple (↑↓ and of the translational) oscillate. Since periodicity the oscillations of the quartz (represented lattice, theby structurethe waves is termeda and b) incommensurate are not an integral and showsmultiple a “periodicity” of the translational of ≈150 Å.periodicity of the quartz lattice, the structure is termed incommensurate and shows a “periodicity” of ≈ 150 Å. The introduction of the concepts of modulation and incommensurability to describe those crystal The introduction of the concepts of modulation and incommensurability to describe those structures that partially depart from periodic order allowed crystallographers to avoid a profound crystal structures that partially depart from periodic order allowed crystallographers to avoid a revision of the crystallographic paradigm for some time. By assuming aperiodicity in crystal structures profound revision of the crystallographic paradigm for some time. By assuming aperiodicity in is somewhat “pathological” and, as a result of more or less extensive deviations from perfect and ideal crystal structures is somewhat “pathological” and, as a result of more or less extensive deviations periodic arrangements of atoms, the main axioms of crystallography could be preserved. However, from perfect and ideal periodic arrangements of atoms, the main axioms of crystallography could be this conservative approach could no longer be maintained when quasicrystals were discovered and preserved. However, this conservative approach could no longer be maintained when quasicrystals scientists realised that their structures could not be described at all in terms of the classical concepts of were discovered and scientists realised that their structures could not be described at all in terms of the unit cell and periodicity. the classical concepts of the unit cell and periodicity. 4. Quasicrystals and Minerals The discovery of quasicrystals by Shechtman in 1984 had a great impact on crystallography [9,49,50].
Once the existence of solids whose diffraction patterns are inconsistent with a periodic internal order of their atoms had been confirmed, crystallographers were forced to admit that ordered matter can exhibit rotational symmetry that violates the crystallographic restriction theorem. This important theorem states the symmetry axis of crystals must fulfil the condition 2cosθ = Z (where Z is an integer and θ is the rotation angle corresponding to the symmetry axis); that is, symmetry axes must be compatible with periodicity. To date, most reported quasicrystals are synthetic alloys that show morphologies and electron diffraction patterns with fivefold and tenfold symmetries, which are incompatible with the crystallographic restriction theorem. Although quasicrystals can be easily recognised by their electron diffraction patterns, currently the number of these patterns is limited. In contrast, there is a published collection of over 80,000 powder diffraction patterns, called “ICDD PDF” [51], which includes synthetic Crystals 2016, 6, 137 8 of 16
4. Quasicrystals and Minerals The discovery of quasicrystals by Shechtman in 1984 had a great impact on crystallography [9,49,50]. Once the existence of solids whose diffraction patterns are inconsistent with a periodic internal order of their atoms had been confirmed, crystallographers were forced to admit that ordered matter can exhibit rotational symmetry that violates the crystallographic restriction theorem. This important theorem states the symmetry axis of crystals must fulfil the condition 2cosθ = Z (where Z is an integer and θ is the rotation angle corresponding to the symmetry axis); that is, symmetry axes must be compatible with periodicity. To date, most reported quasicrystals are synthetic alloys that show morphologies and electron diffraction patterns with fivefold and tenfold symmetries, which are incompatible with the Crystalscrystallographic2016, 6, 137 restriction theorem. Although quasicrystals can be easily recognised by their electron8 of 16 diffraction patterns, currently the number of these patterns is limited. In contrast, there is a published collection of over 80,000 powder diffraction patterns, called "ICDD PDF" [51], which inorganic and organic compounds as well as about 9000 files of mineral phases. Unfortunately, powder includes synthetic inorganic and organic compounds as well as about 9000 files of mineral phases. diffraction patterns do not allow us to directly detect quasicrystal symmetry. Therefore, some scientists Unfortunately, powder diffraction patterns do not allow us to directly detect quasicrystal symmetry. considered the possibility that some of the listed powder diffraction patterns in the ICDD-PDF could Therefore, some scientists considered the possibility that some of the listed powder diffraction correspond in reality to quasicrystals [52]. patterns in the ICDD-PDF could correspond in reality to quasicrystals [52]. In order to identify possible “hidden” icosahedral quasicrystals within the ICDD-PDF, Lu et al. [52] In order to identify possible “hidden” icosahedral quasicrystals within the ICDD-PDF, Lu et al. [52] proposed a searching method based on two figures of merit: Q and |∆|. While Q quantifies the match proposed a searching method based on two figures of merit: Q and ||. While Q quantifies the between the wave vectors of a given diffractogram and the wave vectors of an ideal icosahedral match between the wave vectors of a given diffractogram and the wave vectors of an ideal diffraction pattern, |∆| measures the match between the corresponding relative intensities [52–54]. icosahedral diffraction pattern, || measures the match between the corresponding relative Figure7 shows a plot of the distribution of the quantities Q and |∆| for about 60,000 diffraction intensities [52–54]. Figure 7 shows a plot of the distribution of the quantities Q and || for about patterns in the ICDD-PDF (grey dots), including the currently known icosahedral quasicrystals (circles). 60,000 diffraction patterns in the ICDD-PDF (grey dots), including the currently known icosahedral As can be seen in this figure, all quasicrystals are plotted far away from the dense cluster of ordinary quasicrystals (circles). As can be seen in this figure, all quasicrystals are plotted far away from the crystalline compounds. This allows one to identify quasicrystals from their powder diffraction dense cluster of ordinary crystalline compounds. This allows one to identify quasicrystals from their patterns alone. powder diffraction patterns alone.
FigureFigure 7.7. Plot of QQ versusversus| |∆|| of X-ray diffraction patterns listed in the ICDD-PDF. The main cluster ofof greygrey dotsdots correspondscorresponds toto crystallinecrystalline materialsmaterials and thethe openopen circlescircles correspondcorrespond toto thethe knownknown syntheticsynthetic quasicrystalsquasicrystals withwith icosahedralicosahedral symmetry.symmetry. The The black circle represents the newnew mineralmineral icosahedrite,icosahedrite, whosewhose QQ andand ||∆| values values are are within the clustercluster ofof quasicrystals.quasicrystals. Reproduced withwith permissionpermission fromfrom BindiBindi [[54].54].
FromFrom inspectioninspection of of the the figures figures of merit, of merit,Q and Q | ∆and|, Lu | et|, al.Lu [52 et] proposedal. [52] proposed a list of 19 a quasicrystal list of 19 candidatesquasicrystal within candidates the ICDD-PDF, within the that ICDD-PDF, is, materials that whose is, materialsQ and | ∆whose| values Q and plot closer|| values to the plot cluster closer of quasicrystalsto the cluster shownof quasicrystals in Figure 7shown. This listin Figure included 7. This the following list included minerals: the following aktashite minerals: (Cu 6Hg aktashite3As4S12), 6 3 4 12 2 6 9 4 15 tantalite(Cu Hg As ((Fe,Mn)TaS ), tantalite2O6), and((Fe,Mn)Ta gratoniteO (Pb), and9As 4gratoniteS15). In 2007, (Pb As LucaS Bindi). In 2007, began Luca to study Bindi samples began ofto these minerals from the Mineralogical Collection of the Museo di Storia Naturale, Università di Firenze. Unfortunately, he concluded one year later that none of these minerals are quasicrystals. Then, he and his collaborators focussed their search for natural quasicrystals on materials of extra-terrestrial origin. After a few years of investigations, they discovered the first natural quasicrystal within a meteorite found in 1979 in the Khatyrka region of the Koryak Mountains in the Kamchatka Peninsula (Russia) and which has been stored in the Florence Museum since 1990 [11,55]. This quasicrystalline mineral of ideal composition Al63Cu24Fe13 was named icosahedrite for its icosahedral symmetry (with probable space group Fm35) and its name was approved by the Commission on New Minerals, Nomenclature and Classification, International Mineralogical Association (2010-042). More recently, a second natural quasicrystal with the composition Al71Ni24Fe5 and decagonal symmetry has been found in the same meteorite from the Koryak Mountains [12,13]. Figure8 shows two electron diffraction patterns, Crystals 2016, 6, 137 9 of 16
study samples of these minerals from the Mineralogical Collection of the Museo di Storia Naturale, Università di Firenze. Unfortunately, he concluded one year later that none of these minerals are quasicrystals. Then, he and his collaborators focussed their search for natural quasicrystals on materials of extra-terrestrial origin. After a few years of investigations, they discovered the first natural quasicrystal within a meteorite found in 1979 in the Khatyrka region of the Koryak Mountains in the Kamchatka Peninsula (Russia) and which has been stored in the Florence Museum since 1990 [11,55]. This quasicrystalline mineral of ideal composition Al63Cu24Fe13 was named icosahedrite for its icosahedral symmetry (with probable space group Fm3 5 ) and its name was approved by the Commission on New Minerals, Nomenclature and Classification, International
CrystalsMineralogical2016, 6, 137 Association (2010-042). More recently, a second natural quasicrystal with9 ofthe 16 composition Al71Ni24Fe5 and decagonal symmetry has been found in the same meteorite from the Koryak Mountains [12,13]. Figure 8 shows two electron diffraction patterns, which nicely revealed whichthe fivefold nicely and revealed tenfold the symmetries fivefold and of tenfold the quasicryst symmetriesals found of the quasicrystalsin the Khatyrka found meteorite. in the Khatyrka The fact meteorite.that the only The natural fact that quasicrystals the only natural found quasicrystals to date have found a tometeoritic date have origin, a meteoritic together origin, with together recent withshock-induced recent shock-induced synthesis experiments, synthesis experiments, suggests that suggests the formation that the formation of quasicrystals of quasicrystals in nature in may nature be maythe result be the of result asteroid of asteroid collisions collisions [16,56]. [16,56].
Figure 8. Electron diffraction patterns of quasicrystals found in the Khatyrka meteorite. Figure 8. Electron diffraction patterns of quasicrystals found in the Khatyrka meteorite. (a) (a) icosahedrite [11] and (b) quasicrystal with decagonal symmetry [12]. icosahedrite [11] and (b) quasicrystal with decagonal symmetry [12].
EvenEven thoughthough both both the the findings findings of meteoriticof meteoritic quasicrystals quasicrystals and theand conclusions the conclusions drawn drawn from recent from shockrecent experimentsshock experiments may shed may light shed on light the origin on the of origin natural of quasicrystals, natural quasicrystals, a fundamental a fundamental question remains:question Isremains: there any Is placethere onany Earth place with on suitableEarth with conditions suitable for conditions the formation for the of quasicrystals?formation of Consideringquasicrystals? the Considering diversity of thethe chemicaldiversity compositions of the chemical of minerals compositions and the vastof minerals temperature–pressure and the vast rangestemperature–pressure in the geological ranges systems, in the it seemsgeological reasonable systems, to thinkit seems that thereasonable formation to ofthink quasicrystal that the mineralsformation could of quasicrystal also occur in minerals our planet. could Obviously, also occur occurrences in our of planet. quasicrystals Obviously, must beoccurrences rare on Earth of andquasicrystals should be must related be to rare very specificon Earth geochemical and should and be formation related environments.to very specific Therefore, geochemical the search and forformation them cannot environments. be random Therefore, but must focusthe search on those for knownthem cannot minerals be withrandom compositional but must focus and structural on those characteristicsknown minerals that couldwith developcompositional quasicrystalline and structural order under characteristics certain conditions. that could In this develop regard, mineralsquasicrystalline containing order transition under certain metals conditions. in their formulas In this andregard, showing minerals external containing and internal transition (structural metals and/orin their textural) formulas features and showing with apparent external dodecahedral and intern oral icosahedral(structural symmetriesand/or textural) can be features considered with as goodapparent candidates dodecahedral for continuing or icosahedral the search symmetrie for naturals quasicrystals. can be considered Among theseas good mineral candidates candidates, for somecontinuing sulphides the search and arsenides for natural of cobalt, quasicrystals. iron, and Among nickel, these and theirmineral solid candidates, solutions, showsome asulphides number ofand morphological, arsenides of cobalt, textural, iron, and and diffraction nickel, and peculiarities their solid thatsolutions, deserve show special a number attention. of morphological, In this article, wetextural, will focus and ourdiffraction analysis peculiarities on two minerals: that deserve skutterudite special and attention. cobaltine. In this article, we will focus our analysisSkutterudite on two minerals: (CoAs3) skutterudite is the endmember and cobaltine. of extensive solid solution series in which cobalt can be substitutedSkutterudite by (CoAs nickel3 and) is the minor endmember amounts of of extensive iron (skutterudites solid solution with series Fe: (Co in which + Ni) ratioscobalt highercan be thansubstituted 1 have notby nickel been foundand minor in nature). amounts Depending of iron (skutterudites on both the extent with ofFe: the (Co cationic + Ni) ratios substitution higher than and the1 have arsenic not been content, found different in nature). mineral Depending names areon both used: the skutterudite extent of thesensu cationic stricto substitution, with the formula and the (Co,Ni,Fe)As 3 (with Fe < 12%), and the arsenic-deficient varieties smaltite and chloanthite with a general formula (Co,Ni)As3−x and variable Co:Ni ratios [57]. Regardless of compositional variations, skutterudite is considered cubic and the space group Im3 has been assigned to its structure [58–61]. However, skutterudites frequently exhibit “flame textures” and an anomalous optical anisotropy inconsistent with a cubic symmetry, which can be attributed to exsolution and phase transition phenomena. Alternatively, such conspicuous textures and optical anisotropy could be the result of partial substitution of skutterudites by the related minerals safflorite and clinosafflorite, (Co,Ni,Fe)As2, which crystallise in the orthorhombic and monoclinic systems respectively. In any case, it is clear that skutterudites experience major structural and/or compositional rearrangements when P-T conditions change. Crystals 2016, 6, 137 10 of 16
arsenic content, different mineral names are used: skutterudite sensu stricto, with the formula (Co,Ni,Fe)As3 (with Fe < 12%), and the arsenic-deficient varieties smaltite and chloanthite with a general formula (Co,Ni)As3-x and variable Co:Ni ratios [57]. Regardless of compositional variations, skutterudite is considered cubic and the space group Im3 has been assigned to its structure [58–61]. However, skutterudites frequently exhibit “flame textures” and an anomalous optical anisotropy inconsistent with a cubic symmetry, which can be attributed to exsolution and phase transition phenomena. Alternatively, such conspicuous textures and optical anisotropy could be the result of partial substitution of skutterudites by the related minerals safflorite and clinosafflorite, (Co,Ni,Fe)As2, which crystallise in the orthorhombic and Crystalsmonoclinic2016, 6, 137systems respectively. In any case, it is clear that skutterudites experience major structural10 of 16 and/or compositional rearrangements when P-T conditions change. Undoubtedly,Undoubtedly, the the most most striking striking feature feature of of the the structure structure of of skutterudite skutterudite is is the the existence existence of of two two icosahedralicosahedral empty empty cavities cavities per per unit unit cell, cell, which which are are defined defined by by the the positions positions of of arsenic arsenic atoms atoms (Figure (Figure9a). 9a). Recently,Recently, these these icosahedral icosahedral voids voids have have attracted attracted much much attention attention in in the the scientific scientific community community due due to to the the possibilitypossibility of of filling filling them them with with a largea large variety variety of of atoms atom (e.g.,s (e.g. rare rare earths, earths, alkaline alkaline earths), earths), which which provide provide interestinginteresting electronic, electronic, optic, optic, andand thermoelectricthermoelectric proper propertiesties to to the the so-called so-called filled filled skutterudites skutterudites [62,63] [62,63 and] andreferences references therein. therein.
Figure 9. (a) Projection of the skutterudite structure showing icosahedral voids (green) defined by
the positions of the arsenic atoms (not represented) and the cobalt atoms (blue); (b) Projection of theFigure cobaltite 9. (a structure) Projection showing of the theskutterudite As–S pairs structure and the cobaltshowing atoms icosahedral (blue). Cobaltitevoids (green) and pyritedefined are by isostructuralthe positions only of the when arsenic the As–S atoms pairs (not are re fullypresented) disordered. and the cobalt atoms (blue). (b) Projection of the cobaltite structure showing the As–S pairs and the cobalt atoms (blue). Cobaltite and pyrite are Despiteisostructural the possibility only when that the As skutterudite–S pairs are icosahedralfully disordered. voids allow the synthesis of materials with new properties, such voids can also be relevant for the research on the formation of quasicrystals and otherDespite non-periodic the possibility structures. that skutterudite Skutterudite icosahedr icosahedralal voids cages allow formedthe synthesis by 12 of arsenic materials atoms with arenew presumably properties, quite such stable voids structuralcan also be entities relevant that for seem the toresearch be related on the to theformation skutterudite of quasicrystals 503 and 530and diffraction other non-periodic peaks, that structures. is, corresponding Skutterudite to close-packed icosahedral planes cages of formed icosahedral by 12 cages. arsenic atoms are presumablyIcosahedral quite structural stable structural entities such entities as those that found seem into skutteruditebe related to are the characteristic skutterudite of503 the and local 530 orderdiffraction in amorphous peaks, that materials is, corresponding [64,65]. In the to liquid close-packed state, icosahedral planes of geometry icosahedral ensures cages. a dense packing of atoms,Icosahedral and icosahedral structural clusters entities have such even as been those detected found in metallicskutterudite glasses are formed characteristic by extremely of the fastlocal coolingorder [in66 ].amorphous Although thesematerials clusters [64,65]. are energeticallyIn the liquid favourable state, icosahedral in liquids, ge theirometry fivefold ensures symmetry a dense preventspacking theof formationatoms, and of icosahedral large structures clusters with have periodic even been long-range detected order in metallic (i.e., crystals) glasses whenformed the by liquid–solidextremely fast transitions cooling [66]. occur. Although This isthese referred clusters to ar ase theenergetically “geometrical favourable frustration” in liquids, of icosahedral their fivefold ordering,symmetry which prevents seems the to formation be related of to large vitrification structures processes with periodic [67–69 long-range]. Apparently, order such (i.e., a crystals) geometrical when frustrationthe liquid–solid can be overcometransitions and occur. structures This is withreferred icosahedral to as the order“geometrical (i.e., quasicrystals) frustration” can of formicosahedral only underordering, very which specific seems solidification to be related conditions. to vitrification processes [67–69]. Apparently, such a geometrical frustrationSkutterudite can be and overcome its solid solutionsand structures can be with typically icosahedral found inorder hydrothermal (i.e., quasicrystals) veins associated can form with only magmaticunder very systems, specific including solidification the exceptional conditions. igneous complex of Sudbury (Canada), where magma was produced by the impact of a huge meteorite 1800 million years ago [70,71]. Considering the chemical variability of skutterudite, the singularity of its structure, and the variable pressures and cooling rates expected in the magmatic and hydrothermal systems where skutterudite appears, the formation of a natural quasicrystalline form of this mineral is plausible. Remarkably, as early as 1985 the possible existence of quasicrystalline polymorphs of skutterudite was already the object of speculation. Boisen and Gibbs [14] noticed the icosahedral As12 units in the skutterudite structure and suggested that the relatively frequent pyritohedral morphology of skutterudite crystals could reflect a previous quasicrystalline state. A few years later, Gévay and Szederkény [15] pointed out that the hypothesis proposed by Boisen and Gibbs [14] is in accordance with Ostwald’s rule and, therefore, a possible quasicrystalline form of skutterudite could be considered as a metastable Crystals 2016, 6, 137 11 of 16
Skutterudite and its solid solutions can be typically found in hydrothermal veins associated with magmatic systems, including the exceptional igneous complex of Sudbury (Canada), where magma was produced by the impact of a huge meteorite 1800 million years ago [70,71]. Considering the chemical variability of skutterudite, the singularity of its structure, and the variable pressures and cooling rates expected in the magmatic and hydrothermal systems where skutterudite appears, the formation of a natural quasicrystalline form of this mineral is plausible. Remarkably, as early as 1985 the possible existence of quasicrystalline polymorphs of skutterudite was already the object of speculation. Boisen and Gibbs [14] noticed the icosahedral As12 units in the skutterudite structure and suggested that the relatively frequent pyritohedral morphology of skutterudite crystals could reflect a previous quasicrystalline state. A few years later, Gévay and Szederkény [15] pointed out that the hypothesis proposed by Boisen and Gibbs [14] is in accordance with Ostwald´s rule and, Crystals 2016, 6, 137 11 of 16 therefore, a possible quasicrystalline form of skutterudite could be considered as a metastable precursor of crystalline skutterudite. Furthermore, Gévay and Szederkény [15] indicated that rapid coolingprecursor in ofmagmatic crystalline systems skutterudite. and shock Furthermore, hardening Gévay due and to Szederkénymeteoritic [impacts15] indicated may thatgenerate rapid appropriatecooling in magmatic conditions systems for quasicrystal and shock hardening formation. due In to view meteoritic of the impacts recent may discoveries generate appropriateof natural quasicrystalsconditions for in quasicrystalmeteorites [11–13,16], formation. it In seems view ofclear the that recent the discoveries search for ofquasicrystalline natural quasicrystals forms of in skutteruditemeteorites [ 11in– natural13,16], itenviro seemsnments clear that such the as search Sudbury’s for quasicrystalline igneous complex forms is worthwhile. of skutterudite in natural environmentsCobaltite such(CoAsS) as Sudbury’s is another igneous cobalt complex mineral is worthwhile. that frequently appears associated with skutteruditeCobaltite in (CoAsS)high-temperature is another hydrothermal cobalt mineral deposi that frequentlyts. As in the appears case of associated skutterudite, with cobalt skutterudite atoms canin high-temperature be partially substituted hydrothermal by iron deposits. and nickel As in in the casecobaltite of skutterudite, structure, resulting cobalt atoms in extensive can be partially solid solutionsubstituted series by (e.g. iron cobaltite–gersdorffite). and nickel in the cobaltite Nonetheless, structure, the resulting structure in of extensive cobaltite soliddiffers solution from that series of skutterudite(e.g., cobaltite–gersdorffite). and it can be derived Nonetheless, from the thepyrite structure (FeS2) structure of cobaltite by differsideally from replacing that of Fe skutterudite by Co and Sand2 by itAs–S can bepairs derived (Figure from 9b). the But pyrite only(FeS in the2)structure case of aby complete ideally replacingAs–S disorder Fe by would Co and cobaltite S2 by As–S be isostructuralpairs (Figure with9b). Butpyrite. only The in the ordering case of aof complete the As–S As–S pairs disorder reduces would symmetry, cobaltite and be therefore isostructural the orthorhombicwith pyrite. The space ordering group ofPca2 the1 has As–S been pairs assigned reduces to symmetry, cobaltite [72–74]. and therefore Despite the this, orthorhombic cobaltite crystals space withgroup pyritohedral,Pca21 has beenelongated assigned pyritohedral, to cobaltite and [72 icosahedral–74]. Despite shapes this, are cobaltite relatively crystals frequent with (Figure pyritohedral, 10). In addition,elongated “flame pyritohedral, textures” and similar icosahedral to those shapes foun ared relativelyin skutterudite frequent are (Figure also 10common). In addition, in cobaltite “flame samples.textures” Both similar the to singular those found morphologies in skutterudite and arethe also “flame common textures” in cobaltite of cobaltite samples. again Both suggest the singular the existencemorphologies of mineral and the precursors, “flame textures” some of ofwhich cobaltite might again correspond suggest to the a existencequasicrystalline of mineral state. precursors, some of which might correspond to a quasicrystalline state.
Figure 10. Cobaltite crystals with (a) elongated pyritohedral shape. Size: 2.5 × 2.3 × 2.2 cm (picture from Rob Lavinsky) and (b) icosahedral shape. Size ≈ 2.5 mm (crystal and picture courtesy of the Museo Geominero (IGME), Madrid, Spain). Figure 10. Cobaltite crystals with (a) elongated pyritohedral shape. Size: 2.5 × 2.3 × 2.2 cm (picture from Rob Lavinsky) and (b) icosahedral shape. Size ≈ 2.5 mm (crystal and picture courtesy of the MuseoAt present, Geominero the idea (IGME), that someMadrid, cobaltites Spain). could have been formed from quasicrystalline precursors is highly speculative. However, in this regard, diffractograms of cobaltite have revealed anomalous featuresAt present, that can bethe relevant idea that for the some search cobaltites for aperiodic could atomic have order been in formed the cobaltite from structure. quasicrystalline The most precursorsstriking of is these highly features speculative. is that the However, most intense in this diffraction regard, diffractograms peaks often correspond of cobaltite to have the planes revealed that anomalousdefine the {120}features cobaltite that pyritohedron.can be relevant Giese for andthe se Kerrarch [72 for] proposed aperiodic that atomic at high order temperatures in the cobaltite (above structure.850 ◦C), AsThe and most S are striking completely of these disordered features in theis that cobaltite. the most As a result,intense cobaltite diffraction crystallises peaks often in the cubic space group Pa3; i.e., it is isomorphic with pyrite. At temperatures lower than 850 ◦C, As and S tend to become ordered and a complete As–S ordering results in the orthorhombic (pseudo-isometric) space group Pca21. This would partially explain the importance of the 120 reflections in the cobaltite diffractograms. Nevertheless, cubic cobaltite has not yet been found in nature, and X-ray diffraction patterns of natural cobaltites also show a number of reflections forbidden by the space group Pca21 [73]. In order to explain the presence of such forbidden reflections in the diffractograms of cobaltite, Bayliss [73] proposed a complex twinning model consisting of six interpenetrated domains related by a 3 twin axis parallel to the [111] direction of the orthorhombic Pca21 cobaltite unit cell (Figure 11). Crystals 2016, 6, 137 12 of 16
correspond to the planes that define the {120} cobaltite pyritohedron. Giese and Kerr [72] proposed that at high temperatures (above 850 °C), As and S are completely disordered in the cobaltite. As a result, cobaltite crystallises in the cubic space group Pa3; i.e., it is isomorphic with pyrite. At temperatures lower than 850 °C, As and S tend to become ordered and a complete As–S ordering results in the orthorhombic (pseudo-isometric) space group Pca21. This would partially explain the importance of the 120 reflections in the cobaltite diffractograms. Nevertheless, cubic cobaltite has not yet been found in nature, and X-ray diffraction patterns of natural cobaltites also show a number of reflections forbidden by the space group Pca21 [73]. In order to explain the presence of such forbidden reflections in the diffractograms of cobaltite, Bayliss [73] proposed a complex twinning model consisting of six interpenetrated domains related by a 3 twin axis parallel to the [11 1] Crystalsdirection2016, 6, 137 of the orthorhombic Pca21 cobaltite unit cell (Figure 11). 12 of 16
Figure 11. Model of the cobaltite multiple twin according to [73]. Projection along the [111] direction. In thisFigure cell, there11. Model are 52 of atoms the cobaltite (4 Co, 24 multiple S, and 24 twin As). according to [73]. Projection along the [111] direction. In this cell, there are 52 atoms (4 Co, 24 S, and 24 As). The cobaltite twin model by Bayliss [73] has been, however, questioned by Fleets and Burns [74]. On the basisThe ofcobaltite new X-ray twin diffraction model by dataBayliss and [73] optical has been observations, however, of questioned “flame textures” by Fleets using and reflected Burns [74]. light,On Fleets the andbasis Burns of new [74] proposedX-ray diffraction a simpler data twin modeland optical in which observations two twin domains of “flame with textures” incoherent using boundariesreflected are light, related Fleets by aand single Burns threefold [74] proposed axis along a simpler the [111] twin direction. model Evenin which though two this twin second domains modelwith seems incoherent to describe boundaries diffraction are data related better by (i.e., a single forbidden threefold reflections), axis along the origin the [111] and exactdirection. nature Even of cobaltitethough complexthis second twinning model have seems not to yet describe been completely diffraction elucidated. data better (i.e., forbidden reflections), the originThe proposed and exact cobaltitenature of twincobaltite models complex resemble twinning the have attempts not yet by been Linus completely Pauling elucidated. to provide an explanationThe proposed for the firstcobaltite recorded twin diffractionmodels resemble patterns the of attempts quasicrystals by Linus within Pauling the framework to provide an of classicalexplanation crystallography. for the first Paulingrecorded claimed diffraction that patte diffractionrns of patternsquasicrystals of materials within the with framework external of icosahedralclassical symmetry crystallography. could be explainedPauling claimed by a model that of multiplediffraction twinning patterns of cubicof materials domains withwith hugeexternal uniticosahedral cell dimensions symmetry and sharing could a threefoldbe explained axis andby a three model mirror of multiple planes [75 twinning–79]. Further of cubic investigations domains with showedhuge that unit Pauling’s cell dimensions twinning and model sharing was nota threef the simplestold axis wayand ofthree explaining mirror diffractionplanes [75–79]. patterns Further withinvestigations fivefold symmetries showed and that the Pauling’s quasicrystal twinning model model imposed was [80 not,81]. the The simplest fact that way X-ray of powder explaining diffractiondiffraction patterns patterns of non-twinned with fivefold cobaltite symmetries crystals and have the quasicrystal not been recorded model toimposed date indicates [80,81]. thatThe fact the proposedthat X-ray cobaltite powder twinning diffraction must patte occurrns atof thenon-twinned nanometre cobaltite or subnanometre crystals have scales, not similarly been recorded to the to complexdate twinningindicates modelthat the described proposed by cobaltite Pauling. twinning Considering must this, occur it isat plausible the nanometre that the or anomalous subnanometre diffractionscales, patternssimilarly of cobaltiteto the complex could be moretwinning easily model explained described through by an Pauling. alternative Considering description ofthis, the it is cobaltiteplausible structure that and,the anomalous particularly, diffraction of the arrangement patterns of andcobaltite ordering could of be S–As more within easily it. explained In this regard, through bothan a detailedalternative TEM description study of cobaltite of the andcobaltite a structural structure analysis and, of particularly, its structure basedof the on arrangement the concepts and of Zintl-Klemm,ordering of S–As Pearson’s within generalised it. In this regard, octet rule, both and a detailed cation substructures TEM study of [82 cobaltite–84], and and references a structural therein]analysis may of be its revealing. structure based on the concepts of Zintl-Klemm, Pearson's generalised octet rule, and cationSkutterudite, substructures cobaltite, [82–84], and related and refe mineralsrences therein] are certainly may verybe revealing. interesting cases of minerals with puzzling structures that might be related to the formation of quasicrystalline atomic ordering under certain conditions. Nevertheless, taking into account the enormous variety of mineral structures and compositions, these minerals are surely not the only candidates to be precursors or approximates of quasicrystals. The search for quasicrystal minerals has just started with the pioneering works by Bindi and collaborators on the Khatyrka meteorite. But there are still numerous minerals and geological environments to be investigated, including those of other planets and planetary bodies that are becoming accessible. However, new insights into the formation of quasicrystals in nature will be only gained by combining mineralogical exploration, detailed structural analysis, and experiments conducted using selected minerals (precursors) as starting materials. Crystals 2016, 6, 137 13 of 16
5. Conclusions The crystallographic paradigm based on a strict periodic atomic order was definitively established thanks to X-ray diffraction by crystals. However, investigations conducted in the twentieth century also showed an increasing number of mineral structures with more or less severe deviations from periodicity. These anomalous structures were found to be difficult to explain within the framework of classical crystallography. Finally, the discovery of quasicrystals led to a profound revision of some axioms and concepts of crystallography to include a new type of ordered matter that it was not possible to describe at all by periodic lattices. Essentially, quasicrystals are solids with highly ordered structures that are rigorously aperiodic along one or more crystallographic directions and often show fivefold and tenfold symmetries. Notably, typical morphologies of reported quasicrystals are identical to those described by some early crystallographers and considered as impossible mineral forms for more than two centuries. After the first quasicrystals were synthesised in the laboratory, some mineralogists asked themselves whether minerals with quasicrystalline structures could be found in nature. Although only two natural quasicrystalline minerals of extra-terrestrial origin have been reported to date, the possibility of discovery of quasicrystals on Earth cannot be discounted. Taking into account some analogies with quasicrystal structures currently reported, some natural alloys, sulphides, and sulpharsenides (e.g., skutterudite, cobaltite) can be considered as suitable candidates for transformation into the quasicrystalline state under conditions still to be determined. Furthermore, some of these minerals display both structural features and external morphologies with striking icosahedral and dodecahedral symmetries. Although coordination polyhedra and crystal morphologies with fivefold symmetries do not provide direct evidence of internal quasiperiodic order, their recognition and analysis may be helpful in the future search for quasicrystals within the mineral world.
Acknowledgments: The authors gratefully acknowledge Xan Guerra at the Museo de Historia Natural (Universidade de Santiago de Compostela, Spain) for giving access to a collection of photographs of crystallographic solids by Haüy. Information about the mineral collection of Romé de l’Isle was kindly provided by the Muséum National d’Histoire Naturelle (MNHN) in Paris (France). The authors would also like to thank Johan Kjellman at the Museum of Evolution, Uppsala University (Sweden) and the staff of the Museo Geominero (IGME) in Madrid (Spain) for providing samples and photographs of the mineral cobaltite. Author Contributions: Carlos M. Pina and Victoria López-Acevedo contributed equally to this work. Conflicts of Interest: The authors declare no conflict of interest.
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