# Quasicrystals and Other Aperiodic Structures in Mineralogy

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Before Crystallography

## 3. The Crystallographic Paradigm and “Dissident” Minerals

_{1−x}Ag

_{x}Te

_{2}) revealed an astonishingly complex crystal morphology defined by more than 90 different crystallographic forms [39,40] (Figure 5). The first attempts to index all calaverite faces required the use of different lattices, which led questioning the universality of the law of rational indexes for the first time. Far from clarifying the “calaverite problem”, X-ray diffraction data showed satellite spots, which indicated that calaverite had a complex superstructure whose explanation was still elusive. Subsequent electron diffraction patterns confirmed the existence of satellite diffraction spots. A detailed analysis of such spots revealed that calaverite has a modulated structure [41]. Such a modulation results from both the displacement of Te and the occupation of Ag atoms (which partially substitute the Au atoms), resulting in a deviation from the average C2/m monoclinic structure of calaverite [42]. Since the period of modulation is not an integral number of lattice translations, the structure of calaverite is defined as incommensurate. Remarkably, the incommensurability of the calaverite structure has a morphological expression consisting in the coexistence of many crystal forms, whose indexation was conducted by the simultaneous use of different lattices. Recently, the problems associated with using this cumbersome and artificial indexation have been overcome by the use of face Miller indexes consisting of four numbers, where the fourth number describes the modulation [43,44,45,46], and references therein].

## 4. Quasicrystals and Minerals

_{6}Hg

_{3}As

_{4}S

_{12}), tantalite ((Fe,Mn)Ta

_{2}O

_{6}), and gratonite (Pb

_{9}As

_{4}S

_{15}). In 2007, Luca Bindi began to 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 Al

_{63}Cu

_{24}Fe

_{13}was named icosahedrite for its icosahedral symmetry (with probable space group Fm$\overline{3}\overline{5}$) 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 Al

_{71}Ni

_{24}Fe

_{5}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 the fivefold and tenfold symmetries of the quasicrystals found in the Khatyrka meteorite. The fact that the only natural quasicrystals found to date have a meteoritic origin, together with recent shock-induced synthesis experiments, suggests that the formation of quasicrystals in nature may be the result of asteroid collisions [16,56].

_{3}) is the endmember of extensive solid solution series in which cobalt can be substituted by nickel and minor amounts of iron (skutterudites with Fe: (Co + Ni) ratios higher than 1 have not been found in nature). Depending on both the extent of the cationic substitution and the arsenic content, different mineral names are used: skutterudite sensu stricto, with the formula (Co,Ni,Fe)As

_{3}(with Fe < 12%), and the arsenic-deficient varieties smaltite and chloanthite with a general formula (Co,Ni)As

_{3−x}and variable Co:Ni ratios [57].

_{2}, 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.

_{12}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 precursor of crystalline skutterudite. Furthermore, Gévay and Szederkény [15] indicated that rapid cooling in magmatic systems and shock hardening due to meteoritic impacts may generate appropriate conditions for quasicrystal formation. In view of the recent discoveries of natural quasicrystals in meteorites [11,12,13,16], it seems clear that the search for quasicrystalline forms of skutterudite in natural environments such as Sudbury’s igneous complex is worthwhile.

_{2}) structure by ideally replacing Fe by Co and S

_{2}by As–S pairs (Figure 9b). But only in the case of a complete As–S disorder would cobaltite be isostructural with pyrite. The ordering of the As–S pairs reduces symmetry, and therefore the orthorhombic space group Pca2

_{1}has been assigned to cobaltite [72,73,74]. Despite this, cobaltite crystals with pyritohedral, elongated pyritohedral, and icosahedral shapes are relatively frequent (Figure 10). In addition, “flame textures” similar to those found in skutterudite are also common in cobaltite samples. Both the singular morphologies and the “flame textures” of cobaltite again suggest the existence of mineral precursors, some of which might correspond to a quasicrystalline state.

_{1}. 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 Pca2

_{1}[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 $\overline{3}$ twin axis parallel to the [1$\overline{1}$1] direction of the orthorhombic Pca2

_{1}cobaltite unit cell (Figure 11).

## 5. Conclusions

## Acknowledgements

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Forms with non-crystallographic pentagonal external symmetries by Romé de L’Isle. (

**a**) Regular dodecahedron; (

**b**) Elongated or pyramidal dodecahedron; (

**c**) Regular triacontahedron; (

**d**) Icosahedron. Illustration adapted from Table II (Le Cube ou L’Hexaèdre et ses Modifications) in Volume IV of Cristallographie [1].

**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.

**Figure 3.**Crystal morphologies reported by Romé de L’Isle [1] and the corresponding stereographic projections of their symmetry elements. (

**a**) Forms belonging to the m$\overline{35}$ icosahedral quasicrystal class (from top to bottom: dodecahedron, icosahedron, and triacontahedron); (

**b**) elongated dodecahedron showing the symmetry of the $\overline{5}$m2 quasicrystal class; and (

**c**) cube belonging to m$\overline{3}$m crystal class. Symbols: ellipses, triangles, squares, and pentagons indicate the orientations of the twofold, threefold, fourfold, and fivefold axes respectively. The full lines in the stereographic projections represent mirror planes.

**Figure 4.**(

**a**) Scanning electron microscopy image of an Al

_{62.2}Cu

_{25.3}Fe

_{12.5}quasicrystal with the shape of an elongated dodecahedron (reproduction from [28]); (

**b**) Scanning electron microscopy image of a limonitised pyrite with the approximate shape of an elongated dodecahedron, similar to the marcassite described by Romé de L’Isle [1].

**Figure 6.**Incommensurate structure of quartz (modified from Putnis [48]). This structure is formed during the transformation from high to low quartz. Both regions of lattice distortion (+ and –) and shear on the Dauphiné twin boundaries (↑↓ and ↓↑) oscillate. Since the oscillations (represented by the waves a and b) are not an integral multiple of the translational periodicity of the quartz lattice, the structure is termed incommensurate and shows a “periodicity” of ≈150 Å.

**Figure 7.**Plot of $\overline{\mathrm{Q}}$ versus |Δ| of X-ray diffraction patterns listed in the ICDD-PDF. The main cluster of grey dots corresponds to crystalline materials and the open circles correspond to the known synthetic quasicrystals with icosahedral symmetry. The black circle represents the new mineral icosahedrite, whose $\overline{\mathrm{Q}}$ and |Δ| values are within the cluster of quasicrystals. Reproduced with permission from Bindi [54].

**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 the cobaltite structure showing the As–S pairs and the cobalt atoms (blue). Cobaltite and pyrite are isostructural only when the As–S pairs are fully disordered.

**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 11.**Model of the cobaltite multiple twin according to [73]. Projection along the [111] direction. In this cell, there are 52 atoms (4 Co, 24 S, and 24 As).

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Pina, C.M.; López-Acevedo, V.
Quasicrystals and Other Aperiodic Structures in Mineralogy. *Crystals* **2016**, *6*, 137.
https://doi.org/10.3390/cryst6110137

**AMA Style**

Pina CM, López-Acevedo V.
Quasicrystals and Other Aperiodic Structures in Mineralogy. *Crystals*. 2016; 6(11):137.
https://doi.org/10.3390/cryst6110137

**Chicago/Turabian Style**

Pina, Carlos M., and Victoria López-Acevedo.
2016. "Quasicrystals and Other Aperiodic Structures in Mineralogy" *Crystals* 6, no. 11: 137.
https://doi.org/10.3390/cryst6110137