Research Article: The atomic structure of the Bergman-type icosahedral quasicrystal based on the Ammann–Kramer–Neri tiling

Date Published: March 01, 2020

Publisher: International Union of Crystallography

Author(s): Ireneusz Buganski, Janusz Wolny, Hiroyuki Takakura.


The article discusses the atomic structure modelling based on the Ammann–Kramer–Neri tiling of the ternary Bergman quasicrystal in the 3D real space.

Partial Text

Single grains of the ZnMgTm iQC were grown by a solution growth technique (Canfield & Fisk, 1992 ▸). High-purity elements Zn (Nilaco, 99.99%), Mg (Nilaco, 99.99%) and Tm (Rare Metallic Co. Ltd, 99.9%) with the nominal composition of Zn62.8Mg33.6Tm3.6 were put in an alumina crucible and sealed inside a silica ampoule together with 0.333 atm (1 atm = 101.325 kPa) of argon. The elements in the crucible were melted at 1023 K for 5 h and cooled to 863 K using a muffle furnace with a cooling rate of 2 K h−1. After holding at this temperature for about 50 h, the silica ampoule was quenched in water. The grains of the Zn–Mg–Tm iQC, which have a shiny fracture surface, were embedded in a solidified solution. The chemical composition of the iQC was determined to be Zn69.5Mg20.9Tm9.6 by wavelength-dispersive X-ray spectroscopy (WDX) on an electron probe microanalyser (EPMA: Jeol, JXA-8530F).

The derivation of the initial model for the refinement of aperiodic crystals, especially QCs lacking the average, periodic structure, has become much more convenient since the algorithm for the phase retrieval of diffraction amplitudes was invented. There are two iterative algorithms serving the purpose: LDEM (Takakura et al., 2001 ▸) implemented in the QUASI07_08 package (Yamamoto, 2008 ▸) and the charge-flipping algorithm (Oszlányi & Sütő, 2004 ▸) used in the Superflip software (Palatinus, 2004 ▸). The ab initio phase retrieval allows one to obtain a value of the crystallographic R factor of around 14–18% in most cases, which helps immensely to construct an atomic model of the structure. The phase retrieval could not be so accurate without the availability of high-quality diffraction data. Fortunately, modern detectors allow the collection of hundreds of thousands of diffraction peaks with a high precision.

The infinite structure model of the Bergman ZnMgTm iQC was constructed by ascribing the atomic decoration to two golden rhombohedra in the AKNt. Even though the AKNt is indirectly used in all the models of iQCs, i.e. it provides the 12-fold vertex environment and defines the atomic decoration of the interstitial part, it has never been utilized as a quasilattice for which the prototiles are building blocks of the structure. The main reason why it has never been used is that the unique decoration of rhombohedra cannot be determined in real structures (Janssen et al., 2018 ▸; Qiu et al., 1996 ▸; Qiu & Jarić, 1995 ▸). To our knowledge, the latter statement has never been tested, nor proven for the inflated tiling. PI QCs obey the scaling rule (Ogawa, 1985 ▸). Therefore, the edge length of the golden rhombohedra can be inflated from a regular 5.13 Å to 21.7 Å. We faithfully decided to try this approach and carry out the structure modelling. Of course, we worked with a strong assumption that the structure is well modelled by the decorated AKNt, which is inflated.

The structure solution and refinement were based on the real-space modelling of atomic positions. The atomic elements were assigned to specific positions in the rhombohedra and later, by assembling rhombohedral units to form the AKNt, a whole structure in physical space was recreated. The real-space structure refinement, based on the average unit cell (AUC) approach, which is known to work exclusively in the physical space, was previously used for decagonal structures (Kuczera et al., 2011 ▸, 2012 ▸). Its main principle is the construction of the atomic distribution function (Wolny, 1998 ▸; Wolny et al., 2018 ▸; Buczek & Wolny, 2006 ▸). The distribution arises from projection of all positions on the periodic, reference lattice. In practical applications, only the reference vertex of the rhombohedra must be projected because positions of atoms are related to that chosen site by vector translation. The AUC approach has never been used for the structure refinement of iQCs; therefore, a whole methodology including the code for the structure refinement had to be developed from scratch.

The structure was refined within the scheme of the atomic decoration of golden rhombohedra forming the AKNt. The standard method of modelling the iQC and approximants is the cluster-based approach where the local atomic arrangement is expressed by three known so far families of clusters. It is interesting to see how our refined structure corresponds to the cluster model.

The traditional approach, with , involves modelling the ODs located in the internal space, being orthogonal to the physical space. Our model, despite being based on the real space, can still be lifted to the 6D space to compare our result with previous attempts on the Bergman-type QC. In order to lift the structure to 6D a large portion (>500 000) of the atomic positions were generated. All the positions were represented as 6D vectors, where the physical space coordinates were derived from the structure itself and the internal space coordinates were assigned 0. It is due to the principle of the section method where the physical space coordinate is generated as an intersection of the physical space with the OD. That always occurs for the internal space coordinates equal to 0. After multiplying the 6D vector of each atom by the inverse projection matrix , the coordinates in the 6D space were found. The coordinates were then reduced to one 6D unit cell by the modulo 1 operation. Every position of the generated structure was then assigned to the corresponding OD. The assignment is not deterministic because of the phason flip sites that allow for an atom to be ascribed to two different ODs with equal probability. The recreated ODs are plotted in Fig. 11 ▸ and compared with their equivalents coming from the simple decoration model (Elser & Henley, 1985 ▸). The inner part of the ODB generates Tm atoms in the structure. Tm is only located in that OD which corresponds well with the fivefold section plotted in Fig. 2 ▸. In the ODV the inner part is empty which also corresponds very well to the plot in Fig. 2 ▸ obtained from the ab initio phasing procedure. Those positions are related to the cluster centres in the body-diagonal of the AR. The ODE is located at the low-symmetry site () which is also recreated in our model. The recreated ODs resemble those for the simple decoration model in terms of size and shape, but the details are different. For instance, we can see unoccupied sites along the threefold direction in the ODV and ODB. That fine structure corresponds to fine-tuning carried out by Yamamoto to solve the structure of the AlCuLi iQC (Yamamoto, 1992 ▸). He additionally modified the simple decoration model by removing the atom from the 12-fold vertices (empty centre in ODB), and also removing some atoms inside the rhombic dodecahedron lying on the mid-edge positions and putting atoms in off-edge-centre positions. However, the fine details of the ODs are different, e.g. we observe an aggregation of Mg atoms nearby fivefold vertices in the ODV, which does not occur in Yamamoto’s model, possibly due to its idealized decoration of off-centre mid-edge positions. In addition, our model is significantly more ordered chemically.

The structure model for the PI ZnMgTm QC is proposed based on the atomic decoration of the two rhombohedra in the Ammann–Kramer–Neri tiling. The rhombohedra used in the model are inflated in comparison with standard lattice size. The inflation allows us to avoid the problem of an ambiguous decoration of rhombohedra in the interstitial part of the structure which exists in a cluster-based model. The ambiguity was the main obstacle in the construction of the occupation domains for the structure modelling in a 6D embedding. For the structure refinement, we rejected the higher-dimensional approach in favour of the real-space approach. Since the structure modelling was based on the real-space atomic decoration of rhombohedra, it is redundant to lift the structure to 6D just for the refinement.




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