DOI: 10.1002/chem.201203758 Pseudo-Fivefold Diffraction Symmetries in Tetrahedral Packing Stephen Lee,* Ryan Henderson, Corey Kaminsky, Zachary Nelson, Jeffers Nguyen, Nick F. Settje, JoshuaTeal Schmidt, and Ji Feng [a] Introduction Tetrahedral packing in metal crystals is ubiquitous; it is also complex. [1–4] While tetrahedra combined with octahedra fill space in the well-known face-centered-cubic and hexagonal- closest-packed structures, tetrahedral packing leads to the Frank–Kasper phases, [5–8] structures such as the Laves phases, [9, 10] Al 6 Mg 11 Zn 11 , [11] and Cd 3 Cu 4 , [12–14] with unit cells extending from tens to hundreds or even thousands of atoms. [15–20] Worse yet for their understanding, perfect regular tetrahe- dra do not fill space. While tetrahedra plus octahedra assem- ble into clusters of easily discerned octahedral or hexagonal symmetry, symmetries which naturally extend into the crys- tal symmetry as a whole, twenty tetrahedra come together to make a filled icosahedron, a cluster whose fivefold rota- tions bear little relation to crystal symmetries. In this paper we will discuss the most regular of tetrahe- dral cluster packing, packing in which tetrahedra come to- gether to make filled icosahedra. We will show how, in these cases, pure tetrahededral packing may not lead to true crys- tal symmetries, but in general leads to a variety of pseudo fi- vefold rotational symmetries. We will find that the bonds at the center of these tetrahedral clusters form the pseudo-five- fold rotation axes, rotation axes which can be shown to dominate the single crystal x-ray diffraction patterns. [8, 12, 21, 22] We will see that tetrahedrally-packed icosahedral and deca- gonal quasicrystals are just two of four limiting cases. We will further show the symmetries of the strongest dif- fraction peaks are not only controlled by the central bonds of the tetrahedral clusters, but that, viewed as vectors, the directions defined by these strongest diffraction peaks ac- tually are the directions of the central bonds. We will find these strong diffraction peaks viewed as vectors form recip- rocal space clusters of unusual shape and pseudosymmetry. Examining these diffraction peaks, we can apply the Jones theory of intermetallic stability, [23–25] a quantum theory which connects strong diffraction to optimal numbers of va- lence electrons, and which, as we further discuss in this paper, can be beneficially related to tight-binding and ex- tended Hückel theories. [26–28] We therefore relate tetrahedral packing to possible atomic stoichiometries and connect the Abstract: We review the way in which atomic tetrahedra composed of metal- lic elements pack naturally into fused icosahedra. Orthorhombic, hexagonal, and cubic intermetallic crystals based on this packing are all shown to be united in having pseudo-fivefold rota- tional diffraction symmetry. A unified geometric model involving the 600-cell is presented: the model accounts for the observed pseudo-fivefold symme- tries among the different Bravais lat- tice types. The model accounts for vertex-, edge-, polygon-, and cell-cen- tered fused-icosahedral clusters. Vertex-centered and edge-centered types correspond to the well-known pseudo-fivefold symmetries in I h and D 5h quasicrystalline approximants. The concept of a tetrahedrally-packed re- ciprocal space cluster is introduced, the vectors between sites in this cluster cor- responding to the principal diffraction peaks of fused-icosahedrally-packed crystals. This reciprocal-space cluster is a direct result of the pseudosymmetry and, just as the real-space clusters, can be rationalized by the 600-cell. The re- ciprocal space cluster provides insights for the Jones model of metal stability. For tetrahedrally-packed crystals, Jones zone faces prove to be pseudosymmet- ric with one another. Lower and upper electron per atom bounds calculated for this pseudosymmetry-based Jones model are shown to accord with the observed electron counts for a variety of Group 10–12 tetrahedrally-packed structures, among which are the four known Cu/Cd intermetallic com- pounds: CdCu 2 , Cd 3 Cu 4 , Cu 5 Cd 8 , and Cu 3 Cd 10 . The rationale behind the Jones lower and upper bounds is re- viewed. The crystal structure of Zn 11 Au 15 Cd 23 , an example of a 1:1 MacKay cubic quasicrystalline approx- imant based solely on Groups 10–12 el- ements is presented. This compound crystallizes in Im3 ¯ (space group no. 204) with a = 13.842(2) . The struc- ture was solved with R 1 = 3.53 %, I > 2s ; = 5.33 %, all data with 1282/0/38 data/restraints/parameters. Keywords: diffraction · intermetal- lic phases · noble metals · quasicrys- tals · solid state structures [a] Prof. S. Lee, Dr. R. Henderson, C. Kaminsky, Z. Nelson, J. Nguyen, N.F. Settje, Dr. J.T. Schmidt, Dr. J. Feng Department of Chemistry and Chemical Biology Cornell University, Ithaca, NY 14853-1301 (USA) E-mail : sl137@cornell.edu Supporting information for this article is available on the WWW under http://dx.doi.org/10.1002/chem.201203758. Chem. Eur. J. 2013, 00,0–0  2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim These are not the final page numbers! ÞÞ &1& FULL PAPER