The first stable binary icosahedral quasicrystals (iQCs) were found in Cd-Yb and Cd-Ca alloy systems [1, 2], which was followed by finding of nine isostructural iQCs in Cd-R (R = Y, Gd-Tm) [3] and Zn-Sc systems [4]. The structural type of these iQC is called Tsai-type, and it has been extended to ternary or quaternary alloys by atomic substitutions.

Higher-dimensional structure analysis of the Cd-Yb iQC by single-crystal X-ray diffraction revealed that the atomic structure consists of two main building blocks, rhombic triacontahedron (RT) and acute rhombohedron (AR) units [5]. In ternary iQCs, the structure analysis becomes more difficult because occupational disorder has to be taken into account in the 6D structure model. Furthermore, recent studies have shown that some sites are preferentialy occupied by the substituting elements [6,7], which indicates that the higher-dimensional structure model must be optimized to ternary iQCs. To build such model, knowledge of atomic structures in ternary quasicrystal approximants (APs) is quite important.

In the first part of my talk, I will present the superstructure and basic structure of ternary Yb-Cd-Mg 1/1APs with the compositions Yb_12.9Cd_78.4Mg_8.8 and Yb_13.3Cd_64.2Mg_22.5 [8]. The former was determined to have a face-centred packing structure comprising two distinguishable RT units (space group Fd3, a = 31.377(1) Å), while the latter was found to have a body-centred packing structure made of identical RT units (space group Im3, a = 15.7596(4) Å). The distinction between the two types of RT units in the superstructure is based on the positional disorder of the first tetrahedron shell and the relative Cd/Mg occupancy at sites (48h) in the fourth icosidodecahedron shell.

In the second part, I will introduce a Python package (PyQCstrc) for building the higher-dimensional models of iQCs [9] and present a modification of six-dimensional structural model for the primitive Tsai-type iQCs so as to incorporate the selective Cd/Mg occupation found in the Cd-Mg-Yb 1/1 APs [7].

[1] A.P. Tsai, J.Q. Guo, E. Abe, H. Takakura, and T.J. Sato, (2000), Nature, 408, 537–538.
[2] Guo, J. Q., Abe, E., Tsai, A. P. Phys. Rev. B. (2000), 62, R14605−R14608.
[3] Goldman, A. I., Kong, T., Kreyssig, A., Jesche, A., Ramazanoglu, M., Dennis, K. W., Bud’ko, S. L., Canfield, P. C., (2013), Nat. Mater, 12, 714−718.
[4] Canfield, P. C., Caudle, M. L., Ho, C. S., Kreyssig, A., Nandi, S., Kim, M. G., ... & Goldman, A. I. (2010), Phys. Rev. B, 81(2), 020201.
[5] Takakura, H., Pay Gómez, C., Yamamoto, A., de Boissieu, M., and Tsai, A.P., (2007), Nat. Mater. 6, 58–63.
[6] Pay Gómez, C. & Tsai, A. P. (2013). Comptes Rendus Physique, 15(1), 1–10.
[7] Yamada, T., Takakura, H., de Boissieu, M. and Tsai, A.-P., (2017), Acta Cryst. B73, 1125-1141.
[8] Yamada, T, (2021), Phil. Mag., 101(3), 257-275.
[9] Yamada, T, J. Appl. Cryst., in press.

Keywords: Quasicrystal, Approximant

This work was supported by JSPS KAKENHI grants (numbers JP18K13987, JP19H05818).

It has been known for some time now that normally a crucial element in QC formation, at least in soft matter, is the presence of two prominent wave numbers in the linear response behaviour to periodic modulations of the particle density distribution. This is equivalent to having two prominent peaks in the static structure factor or in the dispersion relation. In the first half of the talk, we demonstrate how the crucial pair of prominent wave numbers are connected to the length and energy scales present in the pair potentials. Whilst the ratio between the two length scales is important, we show here that for thermodynamically stable soft matter quasicrystals, the ratio of these wave numbers should be close to certain special values. We identify features in the particle pair interaction potentials which can suppress or encourage density modes with wave numbers associated with one of the regular crystalline orderings that compete with quasicrystals, enabling either the enhancement or suppression of quasicrystals. In the second half of the talk we look how to compute phase diagrams for a given interaction potential in an efficient manner. In order to do this, we focus on the representation of the density distribution in soft matter systems. The form of the average (probability) density distribution in solids is often represented as a sum of Gaussian peaks (or similar functions) centred on lattice sites or via a Fourier sum. Here, we argue that representing instead the logarithm of the density distribution via a Fourier sum is better. The advantage of this representation is that it excels both deep in the crystalline region of the phase diagram and also close to melting. Additionally, we show how a strongly nonlinear theory (SNLT) enables efficient computation of the phase diagram for a threedimensional quasicrystal-forming system using an accurate nonlocal density functional theory.

The atomic structure of the decagonal Al-Cu-Rh quasicrystal with a space group is refined based on five X-ray diffraction datasets, collected at 293 K, 1013 K, 1083 K, 1153 K and 1223 K with the use of a synchrotron radiation [1]. The real-space structure solution with the tiling-and-decoration approach based on the moment series expansion [2] is executed.

All the crystallographic – factors are ranging from 5.9% to 6.4% for the datasets of common 1460 symmetry-inequivalent peaks. What is the most intriguing is the correlation (Pearson correlation equal to 0.85) between lattice parameters (edge-length of rhombus and the interatomic layer distance) and the maximum of the residual electron density. The identical temperature dependence presented in figure 1 for the parameters implies the phase transformation. The residual density is agglomerated in the origin of the 4D unit cell what implies the phase transformation is related to the General Penrose Tiling (GPT). Additionally, we can observe a local minimum around the 1083-1153 K of the moments values being directly related to phasons. This is the temperature the structure is the most stable around. The existence of the local minimum in all the plots proves the phason disorder is related to the structure stability what was previously questioned due to insufficient quality of the refinement [1].

We modified the moment series approach to accommodate the existence of the 5^th atomic surface arising for the random-tiling model of the decagonal quasicrystal (figure 2). After the structural refinement with the updated model we obtained much better results in terms of the R-factors. Even more, up to the uncertainty estimated with Hesse matrix, we could prove the 5^th atomic surface existence is not only the artefact of the electron density calculation but the crucial feature of the structure in the 1083-1153 K. The calculations prove the random-tiling hypothesis of the structural stability is true for the decagonal quasicrystals and the structure is stabilized by phasons.

Beyond Golay-Rudin-Shapiro S. I. Ben-Abrahamshelomo.benabraham@gmail.com

I briefly recapitulate the necessary background about the original pseudorandom Golay-Rudin-Shapiro sequence (GRS) and its known generalizations [1-5]. The standard method to make the one-sided GRS based on a two-letter alphabet A₂ = {a, b} two-sided is by constructing a proto-GRS structure based on a four-letter alphabet A₄ = {a, b, c, d} and then reduce it to A₂. In order to generalize to higher dimensions one proceeds analogically. Here I extend GRS to eight symbols (alias letters, digits or colors). I also refine the terminology introducing the designation dD GRSn for a GRS structure based on n symbols and supported by Z^d.

The most natural support for 3D GRS8, that is a structure is based on A₈ = {0, 1, 2, 3, 4, 5, 6, 7} is Z³. The respective substitution is

(1)

The bottom matrix refers to 2D GRS8. The bottom line, in turn, refers to 1D GRS4, while the alphabet A₈ splits into two disjoint A₄'s. Thus 1D necessitates special treatment. As in the case of GRS4, the substitution must be applied twice.

Fig.1 shows an isometric projection of the hull of the second generation of 3D GRS8.

Figure 1. Aspect of hull of 3D GRS8 generation 2.The Fourier spectrum of all GRS structures is absolutely continuous [7, 8].

[1] Golay, M. J. E. (1949) J. Opt. Soc. Amer. 39 437-444.

[2] Rudin, W. (1959) Proc. Amer. Math. Soc. 10 855-859.

[3] Shapiro, H. S. (1951) Extremal problems for polynomials and power series, Master's thesis (MIT, Cambridge MA).

[4] Queffélec, M. (1995) Substitution dynamical systems – spectral analysis, LNM 1294, 2nd. ed. (Springer Verlag, Berlin).

[5] Ben-Abraham, S. I. and David, A. (2020) J. Phys.: Conf. Ser. (in press).

[6] Allouche J.-P. and Shallit J. (2003) Automatic Sequences: Theory, Applications, Generalizations, (Cambridge University Press.

[7] Baake, M. and Grimm, U. (2013) Aperiodic Order. Volume 1: A Mathematical Invitation, (Cambridge University Press).

[8] Barbé, A. and von Haeseler, F. (2003) J. Phys. A: Math. Gen. 38 2599-2622.

Keywords: Golay-Rudin-Shapiro structures

Developing realistic three-dimensional growth models for quasicrystals is a fundamental requirement. Uchida found a general principle for building crystal structures (the Uchida stacking motif) in complex alloys such as the μ-Al₄Mn phase [1]. Here, we investigated the Uchida stacking motif using molecular dynamics (MD) simulations to search for clues to the origins of the atomic arrangements in quasicrystals. We used the LAMMPS code for the MD simulations. Our MD simulation results well reproduce the Uchida stacking motif seen in the μ-Al₄Mn phase. The simulations also reveal the formation of a deformed icosahedron. Our results provide new insights into the growth mechanism and origin of complex alloys and quasicrystals.

[1] Uchida, M. & Matsui. Y. (2000). Acta Cryst. B56, 654.

The atomic-resolution holography (ARH) technique [1,2] offers the possibility to experimentally determine the local atomic-scale structure of quasicrystals. This method can selectively investigate specific elements and their 3-dimensional local atomic environment in a range of up to around 2 nm, without the need of a priori information on the structure. Therefore, it can provide a novel perspective for the visualization of the structure of aperiodic systems.

Recently, we have described the results of the ARH reconstruction for the Penrose lattice, which can be regarded as a reference system for decagonal quasicrystals. The resulting pattern of atomic images can be interpreted a projection of the average structure.[3] Using this framework, we can now describe how the experimental results for decagonal Al-Co-Ni quasicrystals compare with the projection of the average structure from a computational model.[4]

An example is shown in the Figure below, with exemplary data of an experimental hologram of an Al-Co-Ni quasicrystal (a). The intense lines in the hologram are the so-called X-ray standing wave lines, which indicate the 10-fold symmetry of the system. The reconstruction of the environment around the Co atoms from the holograms is illustrated in (b), and is compared with the corresponding projection from the computational model (c). Shown here is the quasi-periodic plane that includes the emitter atom at the origin. The atomic images at the vertices of the dashed polygons can be identified with transition metal atoms, while Al atoms are mainly distributed along the polygon edges.

We will also demonstrate the differences of the quasiperiodic structure versus a crystalline approximant and illustrate the ARH results for icosahedral structures.