Abstract

Abstract

Abstract

Higher-dimensional crystals have been studied for the last thirty years. However, most practicing chemists, materials scientists, and crystallographers continue to eschew the use of higher-dimensional crystallography in their work. Yet it has become increasingly clear in recent years that the number of higher-dimensional systems continues to grow from hundreds to as many as a thousand different compounds. Part of the problem has to do with the somewhat opaque language that has developed over the past decades to describe higher-dimensional systems. This language, while well-suited to the specialist, is too sophisticated for the neophyte wishing to enter the field, and as such can be an impediment. This Focus Review hopes to address this issue. The goal of this article is to show the regular chemist or materials scientist that knowledge of regular 3D crystallography is all that is really necessary to understand 4D crystal systems. To this end, we have couched higher-dimensional composite structures in the language of ordinary 3D crystals. In particular, we developed the principle of complementarity, which allows one to identify correctly 4D space groups solely from examination of the two 3D components that make up a typical 4D composite structure.

We present a new geometric description of Mg₄₄Rh₇, a compound with 408 atoms in its cubic unit cell. Using both experimental site preferences and LDA-DFT-calibrated extended Hückel (eH) calculations as guides, we highlight the structural units within Mg₄₄Rh₇ that reflect the electron-richness or electron-poorness of each crystallographic site. The units that best account for these site preferences and electron populations are 34- and 25-atom fragments of the Ti₂Ni structure, rather than the variety of clusters often used to describe complicated intermetallic and ionic structures. These Ti₂Ni pieces, located using a systematic search algorithm, fit together in a beautifully intricate network. An examination of this network reveals some surprising geometric features of Mg₄₄Rh₇, including a fractal-like arrangement of similar atomic formations on different length scales, geometrically connected to an approximate fivefold symmetry.

The search for meaningful clusters of atoms in the unit cell of a complicated intermetallic crystal is reminiscent of the search for constellations of stars in the sky. In their Full Paper on page 7852 ff., R. Hoffmann, S. Lee and R. F. Berger describe a quantum mechanically guided search for clusters of atoms in Mg₄₄Rh₇.

In den 1960er Jahren bestimmte Samson die Strukturen einiger der komplexesten intermetallischen Phasen, darunter NaCd₂, Mg₂Al₃ und Cu₃Cd₄, die jeweils über 1000 Atome pro Elementarzelle aufweisen. Basierend auf den bemerkenswerten Arbeiten von Samson und Andersson verwenden wir quantenmechanische Rechnungen zur Beschreibung dieser Strukturen. Der formale Aufbau der Struktur auf der Grundlage der elektronischen Verhältnisse beginnt mit der relativ einfachen Mg₁₇Al₁₂-Struktur und setzt sich bis zu Samsons NaCd₂-Struktur fort, wobei die Aufteilung in elektronenreiche und elektronenarme Gitterplätze (bezüglich der durchschnittlichen Elektronenzahl) bei beiden Strukturen das Vorliegen von Fragmenten des MgCu₂-Typs aufdeckt. Die äußeren und inneren Bereiche eines solchen Fragments weisen unterschiedliche Bindungsverhältnisse auf: Der innere Bereich ist eher polar, während die Grenzflächen relativ unpolar sind. Dieser Sachverhalt kann mit der Geometrie der Grenzflächenplätze begründet werden, die zugleich an elektronenreichen wie auch elektronenarmen Gittern beteiligt sind. Die polaren und unpolaren Gitterplätze in NaCd₂ sind durch eine Minimalfläche, die D-Fläche, voneinander getrennt. Die Argumente, die für eine solche Struktur sprechen, beruhen auf elektronischen Überlegungen. Ein entscheidendes Merkmal besteht darin, dass sich polare und unpolare Bindungstypen gegenseitig durchdringen. Diese Methode lässt sich auch auf andere Strukturen anwenden.

In the 1960s, Samson solved the structures of some of the most complicated intermetallic phases known, including those of NaCd₂, Mg₂Al₃, and Cu₃Cd₄ (each with over 1000 atoms per unit cell). Following remarkable earlier constructions by Samson and by Andersson, we use quantum-mechanical calculations as a guide to describing and understanding these structures. Our electronic Aufbau begins with the relatively simple Mg₁₇Al₁₂ structure and works up to Samson's NaCd₂ structure. In both structures, a division of the sites into electron-rich and electron-poor (with respect to an average electron count) reveals MgCu₂-type fragments. Between the interiors and exteriors of these fragments, a change in bonding character takes place—the interiors are more polar, the interfaces relatively nonpolar. This electronic situation is traced to the geometry of the interface sites; they lie simultaneously on electron-rich and electron-poor networks. The resulting polar and nonpolar sites in NaCd₂ are separated by a minimal surface, the D surface. The driving force for assuming this structure is electronic: NaCd₂ features an interpenetration of polar and nonpolar bonding regions. This sort of thinking can be applied to other structures.

We present single-crystal studies of Pd_0.213Cd_0.787 and Pd_0.235Cd_0.765, synchrotron powder studies of Pd_1−xCd_x, 0.755≥x≥0.800, and LDA-DFT and extended Hückel (eH) calculations on these or related phases. The two single-crystal structures have a, b, and c axis lengths of 9.9013(7), 14.0033(10), 37.063(24) and 9.9251(3), 14.0212(7), 60.181(3) Å, respectively and they crystallize in the space groups Ccme and F2mm, respectively (solved as (3+1)-dimensional crystals their most convenient superspace group is Xmmm(00γ)s00). The structures have two different structural components each with their own separate axis parameters. Powder data shows that the ratio of these separate axes (S/L) varies from 1.615 to 1.64, values near the golden mean (1.618). For Pd_0.213Cd_0.787, different Pd and Cd site occupancies lead to variation in the R factor from 2.6–3.6 %. The site occupancy pattern with the lowest R factor (among the 26 820 variants studied) is the exact site occupancy pattern predicted by LDA-DFT parameterized eH Mulliken charge populations. The phases can be understood through a chemical twinning principle found in γ-brass, the parent structure for the above phases (a relation with the MgCu₂ Laves phase is also noted). This twinning principle can be used to account for Cd and Pd site preferences. At the same time there is a clean separation among the Cd and Pd atoms for the two separate chain types at height b=0 and 1/2. These results indicate that Cd:Pd stoichiometry plays a role in phase stability.

The total electronic energies of the six electrons per atom (e⁻ per atom) alloys W, TaRe, HfOs, and YIr and the seven electrons per atom alloys Re, WOs, TaIr, HfPt, and YAu have been calculated in the local density approximation of density functional theory. When one considers common alloy structures such as atomically ordered variants of the body-centered cubic, face-centered cubic, or hexagonally closest packed structures and plots the total electronic energy as a function of the unit cell parameter, one finds for both the six and seven electrons per atom series energetic isosbestic points. An energetic isosbestic point corresponds to a critical value of the size parameter for which all members of the 6 or 7 e⁻ per atom series of compounds have nearly identical total electronic energy. Just as in spectroscopy, where the existence of such isosbestic points is the hallmark of two compounds present in the mixture, an energy isosbestic point1, 2 implies there are just two separate energy curves. For both series it is found that the total electronic energy can be viewed as the weighted sum of a purely covalent term and a purely ionic term. Two semi-quantitative models are proposed to account for these two separate energies. In the first model the total energy is viewed as the sum of the elemental structural energy plus an ionic energy based on the Born–Mayer ionic model. In the second model one considers within the confines of μ₂-Hückel theory the evolution of the total electronic energy as the Coulombic H_ii integrals change in value.