**2. Crystal Structure of Quasicrystals and Their Approximants**

The discovery of QCs changed the view of the composition of solids and prompted a change in the definition of crystals. QCs contain such types of atomic arrangements whose symmetry does not correspond to the "classical" rules of filling the three-dimensional space with building units. The quasicrystalline arrangement contains a five-fold axial symmetry, which was originally considered to be forbidden in classical crystallography. The original definition, which specified the crystal as a material with a regular periodic arrangement of building units, was changed after the discovery of QCs. At present, the crystal is defined as a material with discrete diffraction peaks or as a material with a point diffraction pattern, respectively [4,6–8,45].

The building blocks of QCs are mostly arranged in clusters. The clusters are composed of several layers that are formed by individual atoms. Three main types of clusters are known: Mackay, Bergman, and Tsai (Figure 2). The clusters differ from one another in the different layering [4–9,46]. Clusters are composed of several shells. In the following lines, three shells will be considered to describe both Mackay and Bergman clusters for the sake of simplicity. The Mackay cluster (Figure 2a) with 54 atoms consists of an inner icosahedron (12 atoms) followed by an icosidodecahedron with 30 atoms. The third shell is an icosahedron with 12 atoms. The Bergman cluster (Figure 2b) differs from the Mackay cluster in the second shell that is formed by a dodecahedron with 20 atoms. Thus, three shells of the Bergman cluster consist of 44 atoms. The Tsai cluster (Figure 2c) has five shells with 158 atoms. The first tetrahedron shell with four atoms sits inside a dodedahedron (20 atoms). The dodecahedron is encased in an icosahedron (12 atoms). The icosahedron sits inside an icosidodecahedron with 30 atoms. The fifth shell is a rhombic triacontahedron (32 atoms + 60 atoms in its edges) [46].

**Figure 2.** Models of Mackay (**a**), Bergman (**b**) and Tsai (**c**) clusters, reproduced from reference [46].

A one-dimensional Fibonacci sequence constitutes the simplest case of quasicrystalline arrangement [47]. In the Fibonacci sequence the sum of the two preceding numbers forms the next number. The ratio of two consecutive numbers is close to <sup>τ</sup> = (1 + √5)/2 = 1.61803398875 and is called the "golden ratio". If two segments are selected, with L representing a longer segment and S a shorter segment, then to form the quasicrystalline arrangement, the shorter segment S is replaced by a longer segment L and the combination SL replaces the longer segment L. This results in a following sequence

$$\text{S} \to \text{L} \to \text{SL} \to \text{LSL} \to \text{SL}\\ \text{LSL} \to \text{LSL}\\ \text{S} \to \text{SL}\\ \text{S} \to \text{SL}\\ \text{S} \to \text{LSL}\\ \text{S} \to \text{LSL}$$

The sequence (1) has features of order, but it is not periodically ordered. There are sections SL and SLL, which alternate but do not repeat with regular periodicity. In fact, there is no periodically repeating segment, a so-called unit cell, in the Fibonacci sequence [6,47]. Based on the Fibonacci sequence, it is possible to draw a one-dimensional quasicrystal graphically (Figure 3). In Figure 3a, letters S and L correspond to shorter and longer segments used in the Fibonacci sequence, respectively. Figure 3b shows the arrangement of S and L segments according to the Fibonacci sequence. If the points dividing the line into segments are atoms, a lattice with the quasicrystalline arrangement in one dimension can be obtained. By adding the second and third dimensions, a simple example of a quasicrystalline lattice can be drawn (Figure 3c).

**Figure 3.** One-dimensional quasicrystalline arrangement: structural segments (**a**), one-dimensional quasiperiodicity (**b**), and simple quasicrystal in three-dimensional space (**c**).

The quasiperiodic arrangement in two-dimensional space can be represented by Penrose tiling [9,45]. This arrangement consists of two tiles that do not repeat periodically in two-dimensional space. The construction is given by two basic units: a wider rhombus with acute angle α = 2π/5 and a narrower rhombus with acute angle α = π/5. These tiles are arranged in shapes with a five-fold axis of symmetry. As with the Fibonacci sequence, the Penrose tiling is not arranged completely randomly. Basic repeating motifs, such as a star with a 5-axis axis of symmetry, are found in the arrangement of tiles, but are not repeated periodically. Such a star in the Penrose tiling may correspond to a cluster in a real quasicrystalline arrangement.

In three-dimensional space, two types of rhombohedra form the three-dimensional quasicrystalline arrangement containing elements with the icosahedral symmetry. This arrangement is called icosahedral. In the structure of QCs, no unit cell repeating regularly can be found. Thus, its size is theoretically infinite. In fact, the size of the whole grown crystal is the size of the cell. These large unit cells are in contrast to other metallic materials, whose unit cells are built from small number of atoms [48].

Two-dimensional QCs containing a quasicrystalline arrangement along two axes (in one plane) can be further divided based on crystallographic rules of symmetry, which are based on their diffraction patterns. There are octagonal QCs (O-type) with eightfold rotational symmetry, decagonal (D-type) with ten-fold rotational symmetry, and dodecagonal (DD-type) with twelve-fold rotational symmetry [12]. The three-dimensional QC, also called icosahedral QC (i-QC), is quasiperiodic along all three axes [49]. The clusters usually comprise one of the icosahedral–shaped layers, or the entire clusters can be arranged in the icosahedral shape. The presence of five-fold axes of symmetry in the icosahedral structure is related to the point diffraction spectrum of the i-QC showing ten-fold symmetry.

In addition to QCs, there are also arrangements with many atoms in the lattice along with the presence of a cluster-based structure. The arrangements are called quasicrystalline approximants. The QCs and the quasicrystalline approximants (schematic representation given in Figure 4, [50,51]) may consist of equally formed clusters. While QCs have the clusters arranged quasiperiodically in space (i.e., non-periodically), the quasicrystalline approximants have clusters arranged with regular periodicity. Therefore, a unit cell is present in the structure of the quasicrystalline approximant, which is periodically repeated in three-dimensional space. However, a cluster in a quasicrystalline approximant structure may also comprise icosahedron-shaped layers or other shapes with the presence of five- or ten-fold axis of symmetry, and thus their structure may exhibit a diffraction pattern with a hint of a ten-fold axis of symmetry despite the regular arrangement.

**Figure 4.** Schematic structure of quasicrystal (**a**) and quasicrystalline approximant (**b**) with denoted star-shape corresponding to clusters in real structure. Quasicrystalline arrangement is represented by Penrose tiling.

In Figure 5, the comparison of electron diffraction patterns of both QC and quasicrystalline approximant is shown. The electron diffraction pattern of the QC (Figure 5a) has a perfect five-fold symmetry. The electron diffraction pattern of a quasicrystalline approximant (Figure 5b) with an orthorhombic unit cell has either two-fold or four–fold axis of symmetry, but there are indications of five-fold symmetry related to the icosahedral arrangement of atoms in clusters [10,11,52–59].

**Figure 5.** Comparison of electron diffraction patterns in [010] zone axis: decagonal quasicrystal in Al–Co-Cu system (**a**), ε<sup>16</sup> decagonal quasicrystalline approximant in Al–Pd–Co system (**b**).

The surface structure of SCIPs is significantly less understood compared to bulk [60,61]. Preliminary results show that the adsorption of small, covalently bonding molecules on icosahedral quasicrystals is very similar to that of pure Al substrate. This is consistent with other studies, which indicate that the surface termination of most SCIPs is Al-rich [60]. A scanning tunneling microscopy (STM) has been utilized to study the surfaces of Al– Pd–Mn quasicrystals [62–64]. The STM permits a visualization of the local atomistic surface structure. Specific planes of the bulk structure have been observed as surface terminations [63]. The termination planes are characterized by high atomic density and include elements with the lowest surface energy. Nevertheless, the interpretation of individual STM images is challenging and often needs to be accompanied by theoretical models of the surface [61]. Therefore, ab initio density functional theory (DFT) calculations have been utilized to model quasicrystalline surfaces [65,66]. To perform the calculations, a Vienna ab initio simulation package (VASP) has been used [66]. The atomic structure model of the five-fold Al–Pd–Mn surface is derived from the icosahedral approximant model. In the model, the surface was cut perpendicular to one of its pseudo-five-fold axes. The cleavage position was selected to create high density surface layers consistent with experimental findings. The resulting surface structure is characterized by Penrose tiling [65]. Most tiling vertices coincide with the center of Bergman clusters.
