V2Cu3Ga8, Mo2Cu3Ga8 and W2Cu3Ga8—New compounds with a novel order variant of a bcc packing and motifs of self-similarity

Single crystals of the new compounds TM₂Cu₃Ga₈ (TM=V, Mo, W) were synthesised from the elements. Structure determinations of the isotypic compounds (cI104, space group , Z=8; Mo₂Cu₃Ga₈: a=11.9171(10) Å, 613 refl., 23 param., R₁(F)=0.022, wR₂(F²)=0.047; W₂Cu₃Ga₈: 11.9248(8) Å, 346 refl., 23 param., R₁(F)=0.048, wR₂(F²)=0.086; V₂Cu₃Ga₈: 11.7861(14) Å, 374 refl., 24 param., R₁(F)=0.033, wR₂(F²)=0.081) showed a new cubic structure type which can be classified as an ordered defect variant of a bcc packing with a^′=4a: [(TM)₂(Cu)₃(□)₃][Ga₈]. The coordination polyhedra of the transition metals consist of Ga₈-cubes with 3 sides capped by Cu leading to coordination number 11. The arrangement of the TMGa₈Cu₃-polyhedra is in a way they form itself a 3-fold capped cube. All compositions were confirmed by EDX measurements.     

Synthesis and crystal structure of MgB12

Single crystals of MgB₁₂ were synthesized from the elements in a Mg/Cu melt at 1600 °C. MgB₁₂ crystallizes orthorhombic in space group Pnma with , and . The crystal structure (Z=30, 5796 reflections, 510 variables, R₁(F)=0.049, wR₂(I)=0.134) consists of a three dimensional net of B₁₂ icosahedra and B₂₁ units in a ratio 2:1. The B₂₁ units are observed for the first time in a solid compound. Mg is on positions with partial occupation. The summation reveals the composition MgB_12.35 or Mg_0.97B₁₂ , respectively. This is in good agreement with the value of MgB_11.25 as expected by electronic reasons to stabilize the boron polyhedra and .     

Ca14Si19 – a Zintl Phase with a Novel Twodimensional Silicon Framework

Ca₁₄Si₁₉ is an overlooked binary phase in the Ca/Si system with a novel type of twodimensional silicon framework (R3 c, a = 867.85(6), c = 6852.8(8) pm, Z = 6). The basic building units are 3,3,3-barrelanes Si₁₁ which are linked by Si₃ bridges to form a twodimensional silicon framework leaving space for interstitial calcium atoms. The thickness of the silicon layers is about 800 pm. The compound is a semiconductor with a band gap of about E_G = 0.1 eV and a diamagnetic moment of χ_mole = ?9 · 10^?4 cm³mol^?1. According to the relatively high linking of silicon atoms the reaction with air and moisture is fairly slow.     

Eigenvalues of the difference and product of projections

There is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary (i.e. zero-one) matrices. This isomorphism allows many problems concerning nonbinary Boolean matrices to the referred to the binary ease. However, there are some features of the general (i.e. nonbinary) case that have not been mentioned, although they differ somewhat from the binary case. We exhibit characterizations of the linear operators that preserve several invariants of matrices over finite Boolean algebras to illustrate the differences and similarities of the general vs. the binary cases. We employ a canonical form that is useful in applying the isomorphism.     

Boron: Elementary Challenge for Experimenters and Theoreticians

Many of the fundamental questions regarding the solid‐state chemistry of boron are still unsolved, more than 200 years after its discovery. Recently, theoretical work on the existence and stability of known and new modifications of the element combined with high‐pressure and high‐temperature experiments have revealed new aspects. A lot has also happened over the last few years in the field of reactions between boron and main group elements. Binary compounds such as B₆O, MgB₂, LiB_1?x, Na₃B₂₀, and CaB₆ have caused much excitement, but the electron‐precise, colorless boride carbides Li₂B₁₂C₂, LiB₁₃C₂, and MgB₁₂C₂ as well as the graphite analogue BeB₂C₂ also deserve special attention. Physical properties such as hardness, superconductivity, neutron scattering length, and thermoelectricity have also made boron‐rich compounds attractive to materials research and for applications. The greatest challenges to boron chemistry, however, are still the synthesis of monophasic products in macroscopic quantities and in the form of single crystals, the unequivocal identification and determination of crystal structures, and a thorough understanding of their electronic situation. Linked polyhedra are the dominating structural elements of the boron‐rich compounds of the main group elements. In many cases, their structures can be derived from those that have been assigned to modifications of the element. Again, even these require a critical revision and discussion.     

Compact High-Order Finite Differences for Interface Problems in One Dimension

This paper is concerned with the construction and analysis ofcompact finite difference approximations to the model linearsource problem –(pu')' + qu = f where the functions p,q, and f can have jump discontinuities at a finite number ofpoints. Explicit formulae that give O(h²) O(h³) and O(h⁴) accuracyare derived, and a procedure for computing three-point schemesof any prescribed order of accuracy is presented. A rigoroustruncation and discretization error analysis is offered. Numericalresults are also given.