Insight into the structure of decagonite - the extraterrestrial decagonal quasicrystal

Abstract

A set of X-ray data collected on a fragment of decagonite, Al₇₁Ni₂₄Fe₅, the only known natural decagonal quasicrystal found in a meteorite formed at the beginning of the Solar System, allowed us to determine the first structural model for a natural quasicrystal. It is a two-layer structure with decagonal columnar clusters arranged according to the pentagonal Penrose tiling. The structural model showed peculiarities and slight differences with respect to those obtained for other synthetic decagonal quasicrystals. Interestingly, decagonite is found to exhibit low linear phason strain and a high degree of perfection despite the fact it was formed under conditions very far from those used in the laboratory.

Keywords: X-ray diffraction; atomic structure refinement; decagonal quasicrystals.

Figure 1

Analysis of the diffraction data. (Top) The planar sections through the 3D diffraction pattern and its representation in the domain are presented. The superstructure peak is circled with a yellow-dotted line. The profile of the peak 10111 [indexed in setting (1)] indicates it is in fact assembled out of three peaks that are not separated due to the high width of the diffraction peaks. (Mid and bottom) FWHM and the peak shift with respect to . Large experimental uncertainty of the measured peak characteristics makes quantitative analysis of the phason and phonon strain impossible.

Figure 2

2D section through the 4D electron density map calculated on the basis of the 737 phased diffraction peaks. Two atomic layers of the physical space are superposed here. Four atomic surfaces centered along the [1111] direction are identified. They divide the long-body diagonal of the 4D unit cell into five equal sections. In the magnified picture of the section through one 4D unit cell, the distribution of the electron density within four atomic surfaces is emphasized. The smaller picture with four pentagons of the RPT is given as a guide. The maximal electron density is located in the first and fourth pentagons suggesting the distribution of TM atoms. The parameter a _4D is the edge-length of the 4D unit cell, which in the case of decagonite is equal to 5.478 Å.

Figure 3

(Top) 2D section through the physical space electron density map perpendicular to the tenfold direction. The RPT (gray) and PPT (orange) are plotted over the contour plot. Hiraga clusters (black) are centered at the vertices of the PPT. (Bottom) HRTEM image of the decagonite structure with both RPT and PPT plotted over the image for guidance. Plotted tiling are τ times deflated in comparison with tilings in top contour plots. The isosurfaces of the electron density are compared with the magnified patch of the HRTEM image further supporting the agreement between the obtained ab initio structure solution and the real-space structure. The positions of clusters that are not centered in the PPT are located in the positions of the rearranged tiling: these are phason-flip-equivalent positions.

Figure 4

600 × 600 px HRTEM image and its Fourier transform (inset in left image) presented in log scale. The inverse Fourier transform for the co-linear peaks of the reciprocal space, encircled with a red continuous line in the inset (right) magnified Fourier transform, was calculated in order to show the Ammann lines. (Right) Image shows wavy lines indicating the phonon strain. A few instances of dark patches are marked with red dotted circles. The line pattern is continuous on both sides of the black area hence there is no jag that can be ascribed to linear phason strain.

Figure 5

Relations between different tilings and clusters found in decagonite. The atomic decoration of the basic building blocks of the structure, the thick and thin rhombi of the 16.79 Å RPT, are found based on the relation to the Hiraga cluster.

Figure 6

Initial atomic decoration of two rhombi of the RPT at each atomic layer. Color coding is as follows: red – TM, blue – Al, blue/red – mixed atom, blue half-sphere – partially occupied site by Al. Atom 23 (see the supporting information for details) is indicated. The atom potentially causes the point density of the model to be too high. The distributions of electron density (ρ_max) in the positions of local maxima are plotted for thick (top) and thin (bottom) rhombi. The electron density is normalized by the maximal electron density within each rhombus. Two emerging sectors correspond to two atomic species likely to be present in the structure.

Figure 7

Refined atomic decoration for each type of rhombi in the RPT and every atomic layer. The color coding is the same as in Fig. 6 ▸ with an addition of a red half-sphere defining the partially occupied TM site. The basic 2.45 Å RPT is plotted within every rhombus confirming the positions of atoms follow the specific sites of rhombi. The Gummelt cluster is plotted for guidance. The correlation plot F _calc versus F _obs shows good agreement of the model with experimental data. The distribution of diffraction peak errors shows a two-domain behavior. The value of the R factor (standard and weighted) decreases with the decreasing number of weak reflections taken for calculations.

Figure 8

100 × 100 Å sections through the refined structure of decagonite. The RPT and PPT are plotted over the structure for guidance. Two exemplary Hiraga clusters with the subdivision into Deloudi clusters are additionally presented. The formation of pentagonal bipyramids is marked with green pentagons.

Figure 9

Scaling property of the DQC for decagonite. Decagonal clusters at five different length scales are presented. The characteristic motif of five decagons with a star-like cluster in the center is also shown. This motif is often used to define the atomic decoration of clusters for the cluster-embedding approach.

Figure 10

Decoration of the refined units of the RPT with Gummelt clusters. The overlap rules for Gummelt clusters are shown. The m symmetry of the decorated Gummelt clusters by atomic species is broken which means that our model cannot be explicitly explained on the basis of Gummelts cluster.

Figure 11

Mean displacement parameter and atomic shift after the refinement for each atom in the asymmetric part of the model. The distribution of the parameters is plotted below with red assigned to the thin rhombus and blue for the thick rhombus. The correlation between the magnitude of the atomic shift and the phononic mean displacement parmeter is presented. For both units a weak positive correlation is found.

Figure 12

Vector and magnitude of the atomic shift after the refinement from the initial position. The colors of the arrows refer to the atomic species (following the same scheme as in Fig. 2 ▸).

Figure 13

Recovered atomic surfaces for the decagonite structure. Red –TM atoms, blue – Al atoms and green – mixed atoms. The partially occupied sites are not differentiated in this image. The phason flip domain in the atomic surfaces is highlighted by the red circle. The domain corresponds to Al atoms located at the edges of rhombi.

Figure 14

2D section through the electron density map recovered on the phases of the peaks coming from the refinement. The experimental amplitudes, coming from the model and their differences, were used to recover density maps. The residual electron density is lower than the potential atom, therefore we can conclude that our model does not miss any atom.