Atomic structure and phason modes of the Sc–Zn icosahedral quasicrystal

The detailed atomic structure of the binary icosahedral ScZn7.33 quasicrystal has been investigated by means of high-resolution synchrotron single-crystal X-ray diffraction and absolute scale measurements of diffuse scattering.

Because the hydrodynamic theory is a continuum theory it does not provide a view on a microscopic mechanism on the phason fluctuation. However at the atomic level, the fluctuation of the cut space along Eperp leads to atomic re-arrangements. In case of a 1-D QC, the Fibonacci chain, this corresponds to a LS to SL exchange as shown in Fig. S1(b). These tile reorganizations have been called 'phason flip', in the QC jargon.
A second widely used meaning of the term phason concerns the so-called phason strain, shown in Figure S1c. When the slope of the cut space changes, which is equivalent to a phason strain, this produces systematic atomic re-arrangements, resulting in a breaking of the icosahedral symmetry. On the corresponding diffraction pattern, Bragg peaks shift away from their ideal positions (Socolar et al., 1987). The peak shift (Δ par ) can be expressed as a product between the so-called phason matrix M and the perpendicular reciprocal vector perp of the Bragg reflection. The phason matrix M depends on the symmetry of the system that is a subgroup of the icosahedral symmetry, such as 3 ̅ , 3 ̅ and 5 ̅ . In general the resulting structure remains quasiperiodic. For particular values of the phason matrix M a periodic structure (periodic approximant) is obtained. (Ishii, 1989;Janssen, 1991). When there is a random distribution of phason strain in the sample, the Bragg peaks are no longer delta functions: they have a finite width that scales linearly with = | perp | (Horn, 1986;Lubensky, 1986).
Phason diffuse scattering has been observed for all QC phases to date (see (de Boissieu, 2012) for a recent review). In addition Bragg peak broadening due to the distribution of phason strain is also present. Bragg peak shifts due to linear phason strain are often observed in transmission electron microscopy studies and also in some X-ray diffraction studies. These phenomena are crucial to evaluate the quasicrystal 'quality' i.e. the degree of perfection of the quasiperiodic long-range order of the sample under investigation. There are several conventions for defining the basis vectors for the description of iQCs and therefore, there is no conventional indexing scheme. In the paper we used a definition of the orthonormal set of 6-D vectors proposed by Cahn et al. (1986) for indexing the diffraction pattern of the i-ScZn7.33 QC. On the other hand, one proposed by Yamamoto (1996) was used for the structure analysis, because we used the computer package QUASI07_08 (Yamamoto, 2008) uses this setting. The correspondence between the six indices by Cahn et al. (1986) , n1, …, n6, and that by Yamamoto (1996), h1, …, h6, can be written as following; . (2) The 6-D reciprocal lattice vector has two components and is denoted as = ( par , perp ).
In the indexing scheme by Cahn et al. (1986), the is defined by using a set of orthonormal 6-D vectors , (3) where is the golden mean equals to . (4) In the Yamamoto's scheme the lengths of par and perp are defined respectively by * and * ′ , which correspond to the reciprocal lattice constants in each sub-space. Conventionally, * ′ is set to be the same length to * with the result that the 6-D reciprocal lattice is a 6-D cubic lattice. The lengths of par and perp are defined respectively by and ′ , which correspond to the (direct) lattice constants in each sub-space. In the paper, is denoted as ico , which is an icosahedral lattice constant equals to 6D √2 ⁄ , because the length corresponds to that of icosahedral quasilattice, which is obtained by projecting a subset of the 6-D cubic lattice onto 3-D parallel space. In this paper the structure refinement of the i-ScZn7.33 QC was performed based on the 6-D structure model of the i-YbCd5.7 QC (Takakura et al. 2007) whose occupation domains (ODs) are defined by using the setting of ′ = 1. The position of each OD in the 6-D unit cell is specified by a fractional coordinate as in ordinary crystals.
The detail of the 6-D structure model is described in next section.

S3. 6-D model of i-YbCd5.7 QC
In the superspace description of iQCs their atomic arrangement in the parallel space, Epar, is obtained as a result of an irrational cut of corresponding 6-D periodic structure. The 6-D periodic structure is described as a decorated 6-D lattice with 3-D objects, called occupation domains (ODs) or atomic surfaces, lying in the perpendicular space, Eperp.
The OD that generates the vertices of this 3-D AKN tiling is obtained as a projection of the 6-D unit cell, which has the shape of triacontahedron with an edge length of unity, onto Eperp. The overall shape of this OD is shown in Fig. S2(a). This OD, a measure of the size of any ODs, can be defined using a set of three vectors specifying its asymmetric part.  (1992) is used. The shape of the whole OD is obtained from the asymmetric part by the symmetry operations of the icosahedral group. Accordingly, the diameters of the OD along five-, three-and two-fold directions are 2 , (2 + 2 ) √2 + ⁄ and 2√3 √(2 + ) ⁄ , respectively. The volume is 20(3 − ) −3/2 , which results in the point density: These values are equivalent to those found in a literature (Elser, 1986). They are the subset vertices of the 3-D Amman-Kramer-Neri (AKN) tilling. Fig. S2(b) shows the overall shape of the OD that generates the twelve-fold packing sites. This OD is obtained by modifying −2 times smaller triacontahedral OD in Fig. S2(a). In the 6-D model of the i-YbCd5.7 QC this OD is configured to express the characteristic Tsai-type RTH clusters (Takakura et al. 2007). obtained by placing the archetype ODs suitably, so as to generate the atoms located at the vertices of the polyhedral shells of the RTH cluster shown in Fig. S2(e). The OD at E generates the atoms located at the mid-edges of a RTH and the DFP shown in Fig. S2(f). Its dimpled shape is due to the shape of the archetype ODs as a castingmold. No unreasonably short atomic distances appear in the physical space atomic structure except for the disordered tetrahedral sites located on the inner most shell of the RTH cluster. The structure refinements were carried out using the asymmetric parts of the ODs as shown in Fig. S2(d) by taking into account different local environments for the atomic sites.
The atomic structure in Epar can be described with two building blocks, namely, the RTH cluster and DFP, as shown in Fig. S2(e) and ( Similarly to periodic crystal, there is some arbitrariness in defining the origin of the 6-D unit cell. In particular the origin can be shifted by the vector (1,1,1,1,1,1)/2. Applying this shift to the current 6-D model of i-YbCd5.7 QC the OD that generates clusters centres is found at the vertex position, (0,0,0,0,0,0). Fig. S3 displays the two-, three-and five-fold sections of the 6-D electron density as shown in Fig. 5 but with the new origin shifted by the vector (1,1,1,1,1,1 -Quiquandon et al., 1991;Boudard et al., 1992;Gratias et al,. 2000;Yamamoto et al., 2003;Quiquandon & Gratias, 2006), and the Bergman-type QCs such as i-Al-Li-Cu (de Boissieu et al., 1991;Yamamoto, 1992) so that the cluster generation can be directly compared. Indeed, the i-YbCd5.7 QC contains 6-D 'ingredients' that are seen in both the 6-D structure model for the Mackay and Bergman-type QCs. On the Fig. S3 one can see that for instance the Sc12 icosahedron is generated by the OD at V in a way similar for the larger icosahedron in Mackay-type iQC. On the other hand the Zn20 dodecahedron is generated by the OD at B similarly to the dodecahedron in Bergman-type iQC. The Zn32 icosidodecahedon is generated by the OD at V as for the icosidodecahedron in Mackay-type iQC. Finally, the large RTH is generated by the OD at B, and the small OD at E, now in the position (0,1,1,1,1,1)/2, generates a mid-edge decoration of the large RTH, that is characteristic to the Tsai-type iQC.