2.1. The nD analysis

2D sections in planes parallel to high-symmetry directions through a 6D electron density were calculated. Sections include fivefold axes (5f): [011111] and [100000], threefold axes (3f): [101100] and and twofold axes (2f): [110000] and [001001]. The sections are presented in Fig. 1. All three ODs, already reported by Takakura et al. (2007) in this system, are visible in the 5f section. The red rectangle gives a frame for the 6D unit cell with a lattice constant a_6D = a_r(2^1/2) = 8.045 Å, where a_r is the edge length of the basic rhombohedron. Although perpendicular space can be arbitrarily scaled, it is assumed here that the length of the unit vector in physical space is the same as the length in perpendicular space, and the use of Ångström as a unit in perpendicular space is justified.

Figure 1 High-symmetry sections through the 6D electron density map involving perpendicular space. Three 2D sections are presented, which involve the 5f, 3f and 2f directions. The ODs are marked and labeled in each section. The frame of the 6D unit cell is drawn as a red rectangle. The empty space in the ODB, corresponding to unoccupied positions in the 12-fold subset of the AKNt is marked with a circle a on the 5f section. Two other regions of interest are marked `b' – the overlap between ODB and the ODE forming a phason flip site and `c' – the splitting of atoms along the threefold axis. The distances in parallel space between the ODB and the ODV are given.

OD_B is a high electron density domain, which means that Yb positions are generated by the domain. The center of the domain, marked with an `a' circle, is empty. It corresponds to the nodes of the 12-fold subset of AKNt. With a circle `b', an overlap between OD_B and OD_E is marked, which created an ambiguity in the atomic position. Since two close-distance positions cannot be simultaneously occupied, that is, a split-atom position or otherwise a phason flip site. The electron density is reconstructed based on the reverse Fourier transform; therefore, the smearing of the OD is also caused by the truncation effect akin to the limited resolution of the diffraction data (de Boissieu et al., 1988). The splitting of the electron density marked `c' in the 3f section corresponds to the dodecahedral shell with atoms splitting along the 3f axis. The same was reported in i-CdMgYb (Yamada et al., 2017). The breathing effect can be attributed to the existence of the central atom in the cluster that causes the atoms of the closest shell to relocate [the effect known in the 1/1 approximant of the Bergman QC (Gómez et al., 2008) or face-centered iQC (Buganski et al., 2020c)] or the relaxation due to the disorder in the tetrahedron (Gómez & Lidin, 2003).

The distance between OD_B and OD_V in parallel space is equal to 9.205 Å in the 5f direction and 8.382 Å in the 3f direction. Both numbers are important for understanding the relationship between Tsai and Bergman clusters in the structure, which is explained in Section 2.2.

2.2. Physical space analysis

In this section, the local atomic arrangement reflected by the electron densities calculated in physical space is discussed. The 2D electron density map was calculated in a plane perpendicular to a 2f axis. The matrix (60) from the review paper by Yamamoto (1996) was used. From that map, two regions where the clusters were located were selected. One for the Tsai cluster and the other for the Bergman cluster (Fig. 2). Within each region, a full 3D electron density was calculated to obtain isosurfaces corresponding to atoms in the structure. The level of the isosurface plot was set at ρ_iso = (1/30)ρ_max, where ρ_max is the maximal electron density. That was also the level for later determination of the starting structural model for refinement. In Fig. 2, all shells of both clusters are presented. For both clusters, the center is occupied by an atom. Many clusters with a central atom in the i-CdYb electron density map were found and, therefore, they might be more prevalent than originally expected for the i-CdYb phase. Two clusters presented here are linked to each other along the 5f axis, which is known as the a-linkage. This linkage is not the same as reported in i-ZnMgTm between Bergman clusters (Buganski et al., 2020a). The length of this linkage is equal to 9.205 Å and is longer than the latter by τ. The longer a-linkage will be denoted as a_l-linkage. The common part between the Tsai cluster and Bergman cluster has the shape similar to the rhombic icosahedron but the rhombic faces are truncated and form irregular pentagons. The atomic decoration of this unit is asymmetric along the linkage axis.

Figure 2 Analysis of the local atomic structure in the i-CdYb QC. Two types of clusters are identified, i.e. the Tsai (red) and Bergman (blue) clusters. The shell structure of each cluster is presented as a wireframe skeleton with isosurfaces of the electron density. The isosurface is calculated at the level of ρiso = 1/30ρmax. The Tsai and Bergman clusters are linked along the al-linkage with a length of 9.205 Å which corresponds to the real space distance between the centers of ODV and ODB. The common part between the overlapping RTH outer shell is a rhombic icosahedron with truncated vertices. A Tsai cluster can be pictured as a central cluster surrounded by Bergman clusters with one vertex in the center of the Tsai cluster for each Bergman cluster. Clusters with the 3f symmetry vertex centered are darker blue compared to clusters with the 5f vertex centered.

The linkage between Tsai and Bergman clusters in the structure is not rare. In fact, the Tsai cluster can be completely surrounded by Bergman clusters, as shown in Fig. 2 in the bottom right-hand corner. The vertex of the RTH shell of the Bergman cluster is located in the center of the Tsai group. Clusters with the 3f vertex in the center of the Tsai cluster are plotted in a darker color. Various linkages are realized by Bergman clusters around the Tsai cluster. The b-linkage {length b = a_r[4 + 8/(5^1/2)]^1/2 = 15.661 Å} is realized by clusters lying on opposite sides of the Tsai cluster with the 5f vertex located in the center of the Tsai cluster. The adjacent 5f vertex-connected clusters form a short b-linkage (length b_s = a_rτ = 9.205 Å). The adjacent 3f and 5f vertex-connected clusters form the a-linkage (length a = a_r = 5.689 Å). The c-linkage [length c = b(3^1/2)/2 = 13.562 Å] connects 3f with the second neighboring cluster of 5f. Finally, there is also the shortest b-linkage [length b_ss = a(3 − τ)^1/2 = 6.688 Å] between 3f-bound clusters.

The length of the a_l-linkage corresponds to the physical space distance between the center of the OD_V and the OD_B in the 5f-axis direction already given in Section 2.1. If the 6D electron density was shifted along [1 1 1 1 1 1]/2, both ODs would be interchanged. The relationship between Tsai and Bergman clusters is analogous to the shift in 6D space along the diagonal of the 6D unit cell that changes the places of the ODs. Therefore, cluster centers of one type are generated by the vertex-bound OD, and cluster centers of the other type are generated by the center-bound OD. Tsai clusters are associated with the 12-fold subset of the AKNt, but the Bergman clusters are associated with the 12-fold subset of the dual AKNt. Dual AKNt is generated by the RTH OD located not at the origin of the 6D unit cell, but at its center. This is directly related to the definition of a dual tessellation but in 6D space (Mihalkovič et al., 1996; Gailiunas & Sharp, 2005). Both tilings correspond to the uniform shift of 6D space; therefore, the choice of the tiling should not affect the final result of the structural modeling.

3.1. Refinement of Tsai clusters

The refinement converged with R = 11.48% against 424 parameters. This makes the peak-to-parameter ratio equal to ∼12.3. The number of parameters used here is larger than used by Takakura et al. (2007) and is directly related to the number of independent atoms in the asymmetric part of the rhombohedra. In the OD modeling with the nD approach, the number of independent atoms is a hyperparameter that can be arbitrarily chosen. In total, there are 85 refinable atoms in the PR and 60 atoms in the OR. The final composition is Cd_84.9Yb_15.1 with a density of 8.99 g cm⁻³. Regarding the atomic composition, the model reproduces the experimental value (Cd_85.07Yb_14.93). The calculated density is a little larger than reported (8.88 g cm⁻³); however, it was estimated for 1/1 PAC, which is Cd rich with respect to the QC. The density of the QC can be higher. The residual electron density is < 1.6% of the electron density calculated with experimental diffraction amplitudes.

The final model is depicted in Fig. 3. Orientations of units are given by three vectors of the icosahedral lattice, i.e. vectors spanning the PR along the 5f axes. Each shell of the Tsai cluster is plotted separately. The first feature is the presence of the a-linkage at the edge of both rhombohedra with Yb atoms occupying the center of the cluster. Along the same edge, there is a short interatomic distance (∼2 Å) between two Cd atoms, one of which belongs to the outer shell of the RTH and the other to the icosahedral shell of the second cluster in the linkage. It is caused by the existence of the phason flip site exactly in the positions of those atoms. The RTH cluster covering is also possible in the case of i-CdYb, and no division between cluster-belonging atoms and interstitial atoms agglomerated in PR units is necessary, as discussed by Buganski et al. (2020a). All atoms are found to be part of the subsequent shells of Tsai clusters if the a-linkage is allowed. Interestingly, the centers of the Tsai clusters are occupied by Yb atoms, except one cluster which lies on the longer body diagonal with 3f symmetry. The central part of this cluster is occupied by one atom on the body diagonal axis, and one atom slightly shifted from this axis, but is located on the mirror plane. When all symmetries of the asymmetric unit are applied, four atoms with 100% occupancy each are recovered. Later, when the large RD unit is analyzed, it will be evident that this is a tetrahedron. The orientation is the same as in 1/1 PAC of i-ScZn measured at 92 K (Ishimasa et al., 2007).

Figure 3 Atomic decoration of rhombohedral units in the i-CdYb refined model with Tsai clusters. The orientation of each unit is given by three vectors of icosahedral setting, directed toward the vertices by a common face of the icosahedron. All shells of the Tsai cluster are given individually for clarity. The shells are dodecahedron (#1), icosahedron (#2), icosidodecahedron (#3) and rhombic triacontahedron (#4). In most cases, there is a Yb atom at the center of the clusters.

To better understand the local atomic configurations in the model, we have drawn the RD with two PRs and two ORs with all atomic positions restored by using symmetry operations. The RD with all Yb atoms and Cd atoms contained within six icosahedra is given in Fig. 4(a). The six icosahedra together with two others located along the 2f axis and connected via a short b-linkage form a hexagonal bipyramid marked with blue. The edge length of the hexagonal bipyramid aligned with the threefold direction is equal to the length of the c-linkage. The edge-bound icosahedra provide a grid for the remaining Yb atoms but are shaded with a light color; otherwise, the image would be incomprehensible. We have not plotted all Cd atoms either for the sake of clarity.

Figure 4 The local atomic arrangement in i-CdYb is based on the agglomeration of PR and OR in the RD unit (colors of the atoms as in Fig. 3). In (a), the RD with Yb atoms is plotted with a hexagonal bipyramid (blue) formed by eight icosahedral shells. Atoms of Cd, located on the dodecahedral shells within marked icosahedra, are plotted. With light shading, all edge-bound icosahedra are plotted to form a grid for the remaining Yb atoms. In (b), the magnified picture of the two hexagonal types of icosahedra is presented. The tetrahedron-centered one is generated by the PR unit, while the other is part of the OR. The tetrahedron has one of its faces perpendicular to the 3f axis, directed along the longer body diagonal of the PR. In (c), the magnified image of short b-linked triacontahedra is presented. The common part has the shape of a double-capped RD presented in (d). The inner structure is formed by a hexagonal prism of Cd and an octahedron with two Yb atoms in apices. Sites `a' are generated by an icosidodecahedral shell, while sites `b' are formed by dodecahedra. The sites `b' are split. The correlation plot (e) between calculated Fcalc and observed Fobs. The diffraction amplitudes show good agreement; however, a classical broadening for low-intensity peaks is observed. The orientations of each unit in space are given by three perpendicular vectors directed along the 2f axes.

The tetrahedron can be found in four icosahedra that were generated by clusters in PR units only. Tetrahedra along the 3f axis (body diagonal) of the PR point toward each other like arrows. The icosahedra within the OR have centers occupied by Yb with a significant disorder of Cd atoms in the dodecahedral shell. The magnified picture of both icosahedron types with and without a tetrahedron inside is presented in Fig. 4(b). The face of the tetrahedron is perpendicular to the 3f axis of the PR with the apex atom on the axis. The tetrahedron is not centered within the dodecahedron. However, no distortion was created to the dodecahedron, as reported in i-ScZn (Yamada et al., 2016a). The distortion in the dodecahedron occurs in a pseudo-Tsai cluster (Gebresenbut et al., 2020) with an Yb atom inside. With the label `e', the split atomic position is marked at the 3f vertex of the dodecahedron. There is also a Cd atom that protrudes through the icosahedral shell.

The inner clusters in the RD form a short b-linkage presented by a `d' motif in Fig. 4(a) and magnified in Fig. 4(c). What is most clearly visible here is a disorder within the outer shell of the RTH. Atoms that seem to float above the RTH shells are Cd atoms linked to the 3f vertices of the RTH. Split atoms along the 3f direction were mentioned in Section 2.1. Here, it was not modeled as a split atom, but the residue of that disorder is a shift from the original position.

The common part of RTH clusters here forms a double-capped RD which is τ³ deflated with respect to the one in Fig. 4(a). This is a manifestation of the self-similarity (Takakura et al., 2007) of QCs . The inner structure of the small RD is complex but fully explained in Fig. 4(d). Two Yb atoms (labelled c) occupying the apex positions of the octahedron are shared by two overlapping icosahedra visible in a motif `d'. Within the square basis of an octahedron, there are four atoms which belong to the dodecahedral (b) and icosidodecahedral (a) shells. Atoms that belong to dodecahedra are split with half-occupancy between dodecahedral shells of linked clusters. Atoms `a' are all generated by an icosidodecahedron, also in a hexagonal prism.

The correlation plot between the calculated and observed diffraction amplitudes is given in Fig. 4(e). High-intensity peaks are aligned along the center-of-mass line (red); however, single peaks deviate from it even above 10⁻². Apart from that, there is a natural broadening, mostly symmetrical, for small peaks. For QCs, small-amplitude diffraction peaks very often deviate toward smaller F_calc values appearing as a tail, which is attributed to a multiple-scattering effect (Fan et al., 2011) or phasonic disorder (Buganski et al., 2016, 2019). The phasonic ADP was estimated as . To compare this value with the results given by Yamada et al. (2016a) for i-ScZn or Yamada et al. (2017) for i-Cd-Mg-Yb we need to recalculate the values in those papers. The conversion constant is 8π². The general Debye–Waller formula for phasons used in those papers is , where b_ph is a phasonic ADP parameter and is the length of the vector of perpendicular subspace of reciprocal space. The formula used here is . After conversion, the result for i-ScZn is and for i-Cd-Mg-Yb it ranges from to depending on the Mg concentration (Yamada et al., 2017). Our result is two to three times smaller, indicating a very small phasonic disorder.

3.2. Refinement of Bergman clusters

The refinement converged with R = 11.69% against 432 parameters. It makes the peak-to-parameter ratio equal to ∼12. In total, there are 81 refinable atoms in PR and 57 atoms in OR. These are atoms that are not affined to the tetrahedron. Because the tetrahedron is now located in the vertices of rhombohedral units, we needed to treat the tetrahedron as a partially occupied icosahedron. In the asymmetric part of rhombohedron there are three vertices, which convert to 36 additional atoms each with one-third occupation probability. The refinement of each atom independently would be inefficient; therefore, for all the icosahedra, we assumed two parameters: the scale parameter and the phononic ADP, the same for all atoms in the icosahedra. The scale parameter is responsible for increasing/decreasing the size of the icosahedron as a whole object. Initially, the icosahedron was oriented with one of the faces being perpendicular to the 3f axis and centered in the vertex. Only the size of the icosahedron changes during refinement, not its orientation. The final composition is Cd_84.97Yb_15.03 with a density of 8.89 g cm⁻³. Also, in this model, both calculated atomic composition and the density reproduce the experimental values very well. The residual electron density was also < 1.6%. The number of atoms in the model is larger than that for the Tsai cluster refinement, which is akin to the treatment of the tetrahedron in the structure.

In contrast to the model with Tsai clusters, one site in the OR is treated as a mixed-occupied site. It is 70/30% occupied by Cd/Yb. The atom can be seen in Fig. 5(b), located at the mid-edge position on the RTH outer shell

Figure 5 Local atomic arrangement in i-CdYb based on the agglomeration of PR and OR in the RD unit (colors of the atoms as in Fig. 5). In (a), the RD with Yb atoms is plotted with hexagonal bipyramid (blue) formed by eight RTH shells. Atoms of Cd, located in the vertices of icosahedral #1 shells, are marked with the addition of partially occupied sites associated with the tetrahedron. With light shading, all edge-bound RTH units are plotted to form a grid for the remaining Yb atoms. In (b), the magnified image of half of the Tsai cluster with a tetrahedron is depicted. The tetrahedron is assumed to be a partially occupied icosahedron with 0.3 occupancy probability. In (c), the magnified image of short b-linked triacontahedra is presented. The common part has the shape of a double-capped RD (d). The internal structure is made up of a hexagonal bipyramid and Cd atoms belonging to polyhedra (e) and (f). Atoms `b' and `c' create short distances. Atom `b' comes from an icosahedral shell that forms the b-linkage and is located at the phason flip site. Atom `c' is generated by an external icosahedral shell. We believe that this is not a consistent feature because the symmetry-related side from the RD is free of this atom. The orientations of each unit in space are given by three perpendicular vectors directed along the 2f axes. In (g), the correlation plot between calculated Fcalc and observed Fobs diffraction amplitudes are shown.

In Fig. 6 the atomic decoration of the asymmetric rhombohedral units with Bergman clusters is depicted. Analogously to the previous structural model, all atomic positions can be described as packings of RTH clusters. The positions of RTH clusters are the same as in the previous refinement (see Fig. 3 for comparison), but this time the shell structure is Bergman type.

Figure 6 Atomic decoration of rhombohedral units in the i-CdYb refined model with the dual AKNt and Bergman clusters. The orientation of each unit is given by three vectors of icosahedral setting, directed toward the vertices a common face of the icosahedron. All shells of the Bergman cluster are given individually for clarity. The shells are icosahedron (#1), dodecahedron (#2), icosahedron (#3) (shells #2 and #3 form a rhombic triacontahedron), soccerball polyhedron (#4) and rhombic triacontahedron (#5). In most cases, there is a Cd atom in the center of the cluster. The disordered tetrahedron treated as a partially occupied icosahedron can be seen in the vertices of PR and OR.

The atoms in the vertices of the RTH outer shell are Yb, with an occasional Cd atom or a disordered tetrahedron. The density of RE elements within this shell is higher than in i-ZnMgTm, which is a natural consequence of the crystal composition. i-ZnMgTm has a concentration of RE elements almost twice lower that of i-CdYb. There is a Cd atom on the edge of the asymmetric unit of the PR, very close to the vertex-bound tetrahedron. It is generated by the vertex of the dodecahedral shell. This atom is very close to an empty site next to the Yb atom lying on the edge of the PR. The empty site belongs to the #3 icosahedron and is empty due to the vicinity of the Yb atom, exactly like in the AgInYb PAC structure described by Li et al. (2008). The centers of Bergman clusters are occupied by Cd atoms.

By analogy to the refinement with Tsai clusters, the local atomic arrangement is considered within the RD. The visualization of the most interesting features is presented in Fig. 5. The RD unit is plotted in Fig. 5(a), where Yb atoms are presented with specially selected Cd atoms. Cd atoms are only plotted within eight icosahedral #1 shells centered within RTH clusters and for the disordered tetrahedron. Eight RTH clusters form a hexagonal bipyramid of the same size and orientation as in the Tsai cluster refinement. The grid for Yb atoms is formed by the RTH outer shell.

The short b-linkage formed by the apex positions in the hexagonal bipyramid is depicted in Fig. 5(c) with the analysis of the internal motifs in Figs. 5(d)–5(f). Once again, a double-capped small RD unit is formed, but with Yb atoms in the vertices. One disordered tetrahedron, generated by the internal vertex of the RD, replaces the Yb atom. Most Cd atoms belong to the intersection of two soccerball polyhedra and occupy sites at rhombic faces of the small RD. It is the local structural feature reported in the Bergman type i-ZnMgHo QC by Takakura et al. (2006). Eight Cd atoms inside the RD form a small hexagonal bipyramid that is rotated by 90° with respect to the large hexagonal bipyramid. The rotation is along the 2f axis, going through the vertices in the caps of RD. The hexagonal bipyramid is a motif found in the Bergman QC related and used to constitute a model of QC in i-AlZnMg (Henley & Elser, 1986) or i-AlCuLi (Yamamoto, 1992). In these structures, the bipyramid had a non-rotated orientation. Atoms in the hexagonal bipyramid are generated by a small icosahedron and a small RTH shell that is a combination of shells #2 and #3 [Fig. 6(e)]. In the double-capped RD, there are two sites that create small distances. The first one, `b', is generated by the icosahedral #3 shell. This atom belongs to the phason flip site. Therefore, it can occupy two positions with the same probability marked with a dotted line circle. The position `c' is close to the atoms in the soccerball polyhedron, making those atoms split. These atoms are generated by an icosahedral shell #3 that is outside the short b-linkage. On the opposite, symmetry-related side of RD the atom does not appear.

The tetrahedron is located at the vertices of rhombohedral units. This was expected since the tetrahedron in non-shifted electron density was found along the 3f axis of PR in positions corresponding to body-centered positions of the 6D unit cell. After the shift, the body-centered positions become vertex-centered sites. All shells surrounding the tetrahedron are plotted in Fig. 6(b) and constitute a Tsai cluster. The Tsai cluster was found at the central node vertex of the large RD.

The correlation plot between the calculated and observed diffraction amplitudes is given in Fig. 6(g). High-intensity peaks are aligned along the center-of-mass line (red) with an occasional deviation above 10⁻² in amplitude. A similar effect was observed in the refinement of the Tsai cluster. The broadening effect is also present here, being mostly symmetric. The value of the phasonic ADP is . It is larger than in the previous refinement, but still smaller than reported for other Tsai-type quasicrystals or even CdYb modeled by Takakura et al. (2007).