#
On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

*Dedicated to the memory of Christopher Henley (1955–2015)*.

## Abstract

**:**

## 1. Introduction

^{3}. This model became the canonical lens through which crystal structure and growth was viewed. It seemed to serve crystallography well: the tiles were metaphorical swatches for atomic patterns, while the regularity of their locations suggested crystal growth intuitively (row upon row, layer upon layer). However, the discovery of aperiodic crystals in 1982 [2] showed that this model of growth and form cannot be the whole story. The new models proposed to fill this gap are still debated [3].

_{X}a 72° rotation about a lattice point X. Choose points A and B such that |AB| = r, where r is the minimum distance between points of L. Thus φ

_{A}(B) is a lattice point B′, and φ

^{−1}

_{B}(A) = A′. But then |A′B′| < r, which is impossible (Figure 1).

^{3}. The complexity of these periodic patterns and their close relation to aperiodic crystals is the central theme of this paper.

^{n}and “lift” the structure to a suitable lattice there [11]. The popularity of this formalism owes much to the fact that higher-dimensional analogues of tools developed earlier for periodic crystals can be used and the three-dimensional positions of the crystal pattern recovered by projection. But the process is complicated, and tells us nothing about actual crystal growth. Very recently, the atomic positions in an iQC have been determined directly from X-ray diffraction data, without the high-dimensional formalism [12].

## 2. Nested Clusters

## 3. A Conceptual Breakthrough

^{3}whose points are not only the cube vertices but also their centers. To visualize the bcc packing by equal spheres, recall that the Voronoi cell V of the R

^{3}bcc lattice is a truncated octahedron with six square facets and eight that are regular hexagons. In this lattice, the distance r between the center o of V and the center of a hexagonal facet is one half the minimum interpoint distance, and the distance s from o to the center of a square facet is larger by a factor of approximately 1.15. Thus, spheres of radius r at the bcc lattice points form a packing of R

^{3}in which the spheres are tangent in the directions of three-fold symmetry but with gaps in the four-fold symmetry directions (Figure 5a). Alternatively, if we place spheres of radius s at the lattice points, we get an arrangement in which spheres are tangent in the four-fold direction but overlap along the three-fold axes (Figure 5b).

## 4. Linkages in the Yb-Cd iQC and Its Approximants

## 5. A Newly Identified Cluster Suitable for Growth

## 6. Why Might TDIs Exist in the Melt?

_{6}Sc (a structure identical to that of Cd

_{6}Yb), and how it changed with temperature. This periodic approximant has one Tsai cluster per site in a bcc lattice, as described above for the Yb-Cd system. It was known (their refs [9,10,19]) that there is a critical temperature above which the central tetrahedron appears as disordered. Below that temperature, the crystal transforms to a low-temperature phase where the tetrahedra are ordered). At the even higher temperatures of a melt, the entropy contribution will be even greater, and inner tetrahedron rotation may well be sufficient to stabilize a TDI in the melt.

## 7. Conclusions and Questions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A point lattice in the plane cannot have five-fold rotational symmetry (see the discussion above).

**Figure 2.**Ancient Egyptian tomb ceiling pattern (https://www.pinterest.com/pin/157837161917423903/). Notice that the centers of the stars form a planar point lattice with hexagonal symmetry, although the stars themselves exhibit pentagonal symmetry.

**Figure 3.**Zometools models of Bergman (

**left**) and Mackay (

**right**) clusters. In both models, the vertices of the inner shell are white nodes, the middle shell red, and the outer shell gray. (The inner and outer icosahedra of the two models should, respectively, be congruent but the limited size of Zometool struts precludes that).

**Figure 4.**A Zometools model of the Tsai cluster. In the center is a tetrahedron (green edges) with vertices (white balls) representing Cd atoms. Surrounding it is a shell composed of a dodecahedron (blue edges) with white balls representing Cd atoms at its vertices plus its dual icosahedron with vertices (yellow balls) representing Yb atoms. Outside this is an icosidodecahedron of Cd atoms (red balls each with two short edges to dodecahedron vertices and two longer ones to three-fold rhombic triaconahedron vertices, these four edges forming an X), and outside that is a rhombic triacontahedron with vertices (white balls) and mid-edges (red balls) of Cd atoms. Parts of these two outer shells have been removed so that the inner structure can be more clearly seen.

**Figure 5.**Two-dimensional cross-sections, perpendicular to a face diagonal, of bcc packings by equal spheres. (

**a**) A bcc packing of spheres of radius r; (

**b**) A bcc packing of spheres of radius s.

**Figure 6.**(

**Left**): A rhombic triacontahedron viewed along a three-fold rotation axis; (

**Right**): The six vertices ringing a vertex on the three-fold axis, together with that vertex, are seven of the eight vertices of an obtuse golden rhombohedron (OR).

**Figure 7.**(

**a**) Left: Two rhombic triacontahedra sharing a rhombus (b-linkage). The main diagonal of the shared rhombus is identified by a blue strut; (

**b**) Right: Two rhombic triacontahedra overlapping in an obtuse rhombohedron (c-linkage). A yellow strut marks its short diagonal. Observe that the common vertices that are three-fold for one rhombic triacontahedron are five-fold for the other.

**Figure 8.**(

**a**) The OR complex. Observe from its shadow that the OR looks like a regular hexagon when looking down the yellow strut; (

**b**) The AR (acute golden rhombohedron) complex.

**Figure 9.**Zometools models of the tetrahedron-dodecahedron-icosahedron (TDI) cluster and its surroundings. (

**a**) A single TDI. The white balls represent Cd atoms at the vertices of a dodecahedron, while blue struts represent the dodecahedron edges. The yellow balls represent Yb atoms at the vertices of an icosahedron. This picture shows all five of the red bonds from a yellow ball to vertices of the dodecahedron. In the center are four blue balls, also representing Cd atoms, at the corners of a tetrahedron; (

**b**) The essential structure of Cd

_{6}Yb around each of its TDIs: yellow struts indicate c-linkages and long blue struts indicate the longer diagonal of the shared rhombus for each b-linkage. The blue struts are on the faces of a cube and the yellow struts point into the cube’s corners. This combination of linkages occurs for each TDI in Cd

_{6}Yb, and always in the same orientation.

**Figure 10.**Development of a two-TDI cluster. (

**a**) The initial bond between two TDIs. Only one of the five red bonds around each yellow Yb is shown for the right TDI; they are otherwise identical and have the same orientation; (

**b**) Three red atoms from the icosidodecahedral shell have gelled into position on the right TDI at vertex V2; (

**c**) Three red atoms have likewise gelled into position on V2, and a red bond and red ball have also been added to each of the three added in (

**b**) to V2; (

**d**) The entire OR complex has assembled around the yellow strut.

**Figure 11.**Development of a three-TDI structure. (

**a**) Initial bond to the third TDI. V1′ and V2′ are the white balls at the ends of the yellow strut on the right. One additional red ball has gelled into place between the second and third TDIs. (

**b**) A close-up of the enforced b-linkage of a shared rhombus between the developing Tsai clusters of the second and third TDIs (marked by a long blue strut, seen almost end-on in the reflection-symmetry plane between those two TDIs).

**Figure 12.**A schematic illustration of a two-dimensional section of the three-unit cluster, with dotted circles denoting the TDIs and solid circles the Tsai clusters. The single point of contact marked as a b-linkage in the figure becomes a shared face.

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**MDPI and ACS Style**

Taylor, J.E.; Teich, E.G.; Damasceno, P.F.; Kallus, Y.; Senechal, M.
On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters. *Symmetry* **2017**, *9*, 188.
https://doi.org/10.3390/sym9090188

**AMA Style**

Taylor JE, Teich EG, Damasceno PF, Kallus Y, Senechal M.
On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters. *Symmetry*. 2017; 9(9):188.
https://doi.org/10.3390/sym9090188

**Chicago/Turabian Style**

Taylor, Jean E., Erin G. Teich, Pablo F. Damasceno, Yoav Kallus, and Marjorie Senechal.
2017. "On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters" *Symmetry* 9, no. 9: 188.
https://doi.org/10.3390/sym9090188