Daniel C. Fredrickson

The Samson phase, β-Mg2Al3, revisited

Co-Authors: Michael Feuerbacher, Carsten Thomas, Julien P. A. Makongo, Stefan Hoffmann, Wilder Carrillo-Cabrera, Raul Cardoso, Yuri Grin, Guido Kreiner, Jean-Marc Joubert, Thomas Schenk, Joseph Gastaldi, Henri Nguyen-Thi, Nathalie Mangelinck-Noël, Bernard Billia, Patricia Donnadieu, Aleksandra Czyrska-Filemonowicz, Anna Zielinska-Lipiec, Beata Dubiel, Thomas Weber, Philippe Schaub, Günter Krauss, Volker Gramlich, Jeppe Christensen, Sven Lidin, Daniel Fredrickson, Marek Mihalkovic, Wieslawa Sikora, Janusz Malinowski, Stefan Brühne, Thomas Proffen, Wolf Assmus, Marc de Boissieu, Francoise Bley, Jean-Luis Chemin, Jürgen Schreuer Abstract. The Al−Mg phase diagram has been reinvestigated in the vicinity of the stability range of the Samson phase, β-Mg2Al3 (cF1168). For the composition Mg 38.5 Al 61.5, this cubic phase, space group Fd-3m (no 227), a = 28.242(1) Å, V = 22526(2) Å3, undergoes at 214 °C a first-order phase transition to rhombohedral β′-Mg2Al3(hR293), a = 19.968(1) Å, c = 48.9114(8) Å, V = 16889(2) Å3, (i.e. 22519 Å3 for the equivalent cubic unit cell) space group R3m (no 160), a subgroup of index four of Fd-3m. The structure of the β-phase has been redetermined at ambient temperature as well as in situ at 400 °C. It essentially agrees with Samsons model, even in most of the many partially occupied and split positions. The structure of β′-Mg2Al3is closely related to that of the β-phase. Its atomic sites can be derived from those of the β-phase by group-theoretical considerations. The main difference between the two structures is that all atomic sites are fully occupied in case of the β′-phase. The reciprocal space, Bragg as well as diffuse scattering, has been explored as function of temperature and the β- to β′-phase transition was studied in detail. The microstructures of both phases have been analyzed by electron microscopy and X-ray topography showing them highly defective. Finally, the thermal expansion coefficients and elastic parameters have been determined. Their values are somewhere in between those of Al and Mg.

Perceiving molecular themes in the structures and bonding of intermetallic phases: the role of Hückel theory in an ab initio era

Qualitative molecular orbital theory is central to our understanding of the bonding and reactivity of molecules and materials across chemistry. Advances in computational technology and methodology, however, have made ab initio or density functional theory calculations a simpler alternative, offering reliable results on increasingly large systems in a reasonable time-scale without the need for concerns about the approximations and parameterization of semi-empirical one-electron based methods. In this perspective, we illustrate how the availability of higher-level computational results can augment, rather than supplant, the insights provided by approaches such as the simple and extended Hückel methods. We begin by describing a way to parameterize Hückel-type Hamiltonians against DFT results for intermetallic systems. The potential for chemical understanding embodied by such orbital-based models is then demonstrated with two schemes of bonding analysis that originated in them (but can be extended to DFT results): the μ(3)-acid/base model and the μ(2)-Hückel chemical pressure analysis, which translate the molecular concepts of acidity and electronic/steric competition, respectively, into the context of intermetallic chemistry.

DFT-chemical pressure analysis: visualizing the role of atomic size in shaping the structures of inorganic materials.

Atomic size effects have long played a role in our empirical understanding of inorganic crystal structures. At the level of electronic structure calculations, however, the contribution of atomic size remains difficult to analyze, both alone and relative to other influences. In this paper, we extend the concepts outlined in a recent communication to develop a theoretical method for revealing the impact of the space requirements of atoms: the density functional theory-chemical pressure (DFT-CP) analysis. The influence of atomic size is most pronounced when the optimization of bonding contacts is impeded by steric repulsion at other contacts, resulting in nonideal interatomic distances. Such contacts are associated with chemical pressures (CPs) acting upon the atoms involved. The DFT-CP analysis allows for the calculation and interpretation of the CP distributions within crystal structures using DFT results. The method is demonstrated using the stability of the Ca(2)Ag(7) structure over the simpler CaCu(5)-type alternative adopted by its Sr-analogue, SrAg(5). A hypothetical CaCu(5)-type CaAg(5) phase is found to exhibit large negative pressures on each Ca atom, which are concentrated in two symmetry-related interstitial spaces on opposite sides of the Ca nucleus. In moving to the Ca(2)Ag(7) structure, relief comes to each Ca atom as a defect plane is introduced into one of these two negative-pressure regions, breaking the symmetry equivalence of the two sides and yielding a more compact Ca coordination environment. These results illustrate how the DFT-CP analysis can visually and intuitively portray how atomic size interacts with electronics in determining structure, and bridge theoretical and experimental approaches toward understanding the structural chemistry of inorganic materials.

Generality of the 18-n Rule: Intermetallic Structural Chemistry Explained through Isolobal Analogies to Transition Metal Complexes

Intermetallic phases exhibit a vast structural diversity in which electron count is known to be one controlling factor. However, chemical bonding theory has yet to establish how electron counts and structure are interrelated for the majority of these compounds. Recently, a simple bonding picture for transition metal (T)-main group (E) intermetallics has begun to take shape based on isolobal analogies to molecular T complexes. This bonding picture is summarized in the 18-n rule: each T atom in a T-E intermetallic phase will need 18-n electrons to achieve a closed-shell 18-electron configuration, where n is the number of electron pairs it shares with other T atoms in multicenter interactions isolobal to T-T bonds. In this Article, we illustrate the generality of this rule with a survey over a wide range of T-E phases. First, we illustrate how three structural progressions with changing electron counts can be accounted for, both geometrically and electronically, with the 18-n rule: (1) the transition between the fluorite and complex β-FeSi2 types for TSi2 phases; (2) the sequence from the marcasite type to the arsenopyrite type and back to the marcasite type for TSb2 compounds; and (3) the evolution from the AuCu3 type to the ZrAl3 and TiAl3 types for TAl3 phases. We then turn to a broader survey of the applicability of the 18-n rule through a study of the following 34 binary structure types: PtHg4, CaF2 (fluorite), Fe3C, CoGa3, Co2Al5, Ru2B3, β-FeSi2, NiAs, Ni2Al3, Rh4Si5, CrSi2, Ir3Ga5, Mo3Al8, MnP, TiSi2, Ru2Sn3, TiAl3, MoSi2, CoSn, ZrAl3, CsCl, FeSi, AuCu3, ZrSi2, Mn2Hg5, FeS2 (oP6, marcasite), CoAs3 (skutterudite), PdSn2, CoSb2, Ir3Ge7, CuAl2, Re3Ge7, CrP2, and Mg2Ni. Through these analyses, the 18-n rule is established as a framework for interpreting the stability of 341 intermetallic phases and anticipating their properties.

Orbital origins of helices and magic electron counts in the Nowotny chimney ladders: the 18 - n rule and a path to incommensurability.

Valence electron count is one of the key factors influencing the stability and structure of metals and alloys. However, unlike in molecular compounds, the origins of the preferred electron counts of many metallic phases remain largely mysterious. Perhaps the clearest-cut of such electron counting rules is exhibited by the Nowotny chimney ladder (NCL) phases, compounds remarkable for their helical structural motifs in which transition metal (T) helices serve as channels for a second set of helices formed from main group (E) elements. These phases exhibit density of states pseudogaps or band gaps, and thus special stability and useful physical properties, when their valence electron count corresponds to 14 electrons per T atom. In this Article, we illustrate, using DFT-calibrated Hückel calculations and the reversed approximation Molecular Orbital analysis, that the 14-electron rule of the NCLs is, in fact, a specific instance of an 18 - n rule emerging for T-E intermetallics, where n is the number of E-supported T-T bonds per T atom. The structural flexibility of the NCL series arises from the role of the E atoms as supports for these T-T bonds, which simply requires the E atoms to be as uniformly distributed within the T sublattice as possible. This picture offers a strategy for identifying other intermetallic structures that may be amenable to incommensurability between T and E sublattices.

Isolobal Analogies in Intermetallics: The Reversed Approximation MO Approach and Applications to CrGa4- and Ir3Ge7-Type Phases

Intermetallic phases offer a wealth of unique and unexplained structural features, which pose exciting challenges for the development of new bonding concepts. In this article, we present a straightforward approach to rapidly building bonding descriptions of such compounds: the reversed approximation Molecular Orbital (raMO) method. In this approach, we reverse the usual technique of using linear combinations of simple functions to approximate true wave functions and employ the fully occupied crystal orbitals of a compound as a basis set for the determination of the eigenfunctions of a simple, chemically transparent model Hamiltonian. The solutions fall into two sets: (1) a series of functions representing the best-possible approximations to the model systems eigenstates constructible from the occupied crystal orbitals and (2) a second series of functions that are orthogonal to the bonding picture represented by the model Hamiltonian. The electronic structure of a compound is thus quickly resolved into a series of orthogonal bonding subsystems. We first demonstrate the raMO analysis on a familiar molecule, 1,3-butadiene, and then move to illustrating its use in discovering new bonding phenomena through applications to three intermetallic phases: the PtHg4-type CrGa4 and the Ir3Ge7-type compounds Os3Sn7 and Ir3Sn7. For CrGa4, a density of states (DOS) minimum coinciding with its Fermi energy is traced to 18-electron configurations on the Cr atoms. For Os3Sn7 and Ir3Sn7, 18-electron configurations also underlie DOS pseudogaps. This time, however, the 18-electron counts involve multicenter interactions isolobal with classical Ir-Ir or Os-Os covalent bonds, as well as Sn-Sn single bonds serving as electron reservoirs. Our results are based on DFT-calibrated Hückel calculations, but in principle the raMO analysis can be implemented in any method employing one-electron wave functions.

The modulated structure of Co3Al4Si2: incommensurability and Co-Co interactions in search of filled octadecets.

Incommensurate modulations are increasingly being recognized as a common phenomenon in solid-state compounds ranging from inorganic materials to molecular crystals. The origins of such modulations are often mysterious, but appear to be as diverse as the compounds in which they arise. In this Article, we describe the crystal structure and bonding of Co3Al4Si2, the δ phase of the Co-Si-Al system, whose modulated structure can be traced to a central concept of inorganic chemistry: the 18 electron rule. The structure is monoclinic, conforming to the 3 + 1D superspace group C/2m(0β0)s0. The basis of the crystal structure is a rod packing of columns of the fluorite (CaF2) type, a theme that is shared by the recently determined structure of Fe8Al(17.4)Si(7.6). The columns are arranged into sheets, within which the fluorite structures primitive cubic network of Si/Al atoms continues uninterrupted from column to column. Between the sheets, layers of interstitial Si/Al atoms occur, some of which are arranged with a periodicity incommensurate with that of the fluorite-type columns. Strong modulations in the interstitial layers result. Electronic structure calculations, using a DFT-calibrated Hückel model on a commensurate approximate structure, reveal that the complex pattern of atoms within these interstitial layers serves to distribute Si/Al atoms around the Co atoms in order to reach 18 electron counts (filled octadecets). Central to this bonding scheme is the covalent sharing of electron pairs between Co atoms. The shared electron pairs occupy orbitals that are isolobal to classical Co-Co σ and π bonds, but whose stability is tied to multicenter character involving bridging Si/Al atoms. Through these features, Co3Al4Si2 expands the structural and electronic manifestations of the 18 electron rule in solid-state inorganic compounds.

Electronic Packing Frustration in Complex Intermetallic Structures: The Role of Chemical Pressure in Ca2Ag7

The assignment of distinct roles to electronics and sterics has a long history in our rationalization of chemical phenomena. Exploratory synthesis in the field of intermetallic compounds challenges this dichotomy with a growing list of phases whose structural chemistry points to an interplay between atomic size effects and orbital interactions. In this paper, we begin with a simple model for how this interdependence may arise in the dense atomic packing of intermetallics: correlations between interatomic distances lead to the inability of a phase to optimize bonds without simultaneously shortening electronically under-supported contacts, a conflict we term electronic packing frustration (EPF). An anticipated consequence of this frustration is the emergence of chemical pressures (CPs) acting on the affected atoms. We develop a theoretical method based on DFT-calibrated μ(2)-Hückel calculations for probing these CP effects. Applying this method to the Ca(2)Ag(7) structure, a variant of the CaCu(5) type with defect planes, reveals its formation is EPF-driven. The defect planes resolve severe CPs surrounding the Ca atoms in a hypothetical CaCu(5)-type CaAg(5) phase. CP analysis also points to a rationale for these results in terms of a CP analogue of the pressure-distance paradox and predicts that the impetus for defect plane insertion is tunable via variations in the electron count.

First-Principles Elucidation of Atomic Size Effects Using DFT-Chemical Pressure Analysis: Origins of Ca36Sn23’s Long-Period Superstructure

The space requirements of atoms are empirically known to play key roles in determining structure and reactivity across compounds ranging from simple molecules to extended solid state phases. Despite the importance of this concept, the effects of atomic size on stability remain difficult to extract from quantum mechanical calculations. Recently, we outlined a quantitative yet visual and intuitive approach to the theoretical analysis of atomic size in periodic structures: the DFT-Chemical Pressure (DFT-CP) analysis. In this Article, we describe the methodological details of this DFT-CP procedure, with a particular emphasis on refinements of the method to make it useful for a wider variety of systems. A central improvement is a new integration scheme with broader applicability than our earlier Voronoi cell method: contact volume space-partitioning. In this approach, we make explicit our assumption that the pressure at each voxel is most strongly influenced by its two closest atoms. The unit cell is divided into regions corresponding to individual interatomic contacts, with each region containing all points that share the same two closest atoms. The voxel pressures within each contact region are then averaged, resulting in effective interatomic pressures. The method is illustrated through the verification of the role of Ca-Ca repulsion (deduced earlier from empirical considerations by Corbett and co-workers) in the long-period superstructure of the W5Si3 type exhibited by Ca36Sn23.

New roles for icosahedral clusters in intermetallic phases: micelle-like segregation of Ca-Cd and Cu-Cd interactions in Ca10Cd27Cu2.

Despite significant progress in the structural characterization of the quasicrystalline state, the chemical origins of long- and short-range icosahedral order remain mysterious and a subject of debate. In this Article, we present the crystal structure of a new complex intermetallic phase, Ca10Cd27Cu2 (mC234.24), whose geometrical features offer clues to the driving forces underlying the icosahedral clusters that occur in Bergman-type quasicrystals. Ca10Cd27Cu2 adopts a C-centered monoclinic superstructure of the 1/1 Bergman approximant structure, in which [110] layers of Bergman clusters in the 1/1 structure are separated through the insertion of additional atoms (accompanied by substantial positional disorder). An examination of the coordination environments of Ca and Cu (in the ordered regions) reveals that the structure can be viewed as a combination of coordination polyhedra present in the nearest binary phases in the Ca-Cd-Cu compositional space. A notable feature is the separation of Ca-Cd and Cu-Cd interactions, with Bergman clusters emerging as Ca-Cd Friauf polyhedra (derived from the MgZn2-type CaCd2 phase) encapsulate a Cu-Cd icosahedron similar to those appearing in Cu2Cd5. DFT chemical pressure calculations on nearby binary phases point to the importance of this segregation of Ca-Cd and Cu-Cd interactions. The mismatch in atomic size between Cu and Cd leads to an inability to satisfy Ca-Cu and Ca-Cd interactions simultaneously in the Friauf polyhedra of the nearby Laves phase CaCd2. The relegation of the Cu atoms to icosahedra prevents this frustration while nucleating the formation of Bergman clusters.