Ba6Ge25: low-temperature Ge-Ge bond breaking during temperature-induced structure transformation

In order to find the optimal conditions for sample preparation of the binary germanide Ba₆Ge₂₅, the germanium-rich part of the Ba-Ge phase diagram was redetermined by means of metallography, X-ray powder diffraction and differential thermal analysis. The temperature behavior of cubic Ba₆Ge₂₅ was investigated both on polycrystalline samples and single crystals. The temperature dependence of the lattice parameter exhibits two anomalies at about 180 and 230 K, respectively, which are caused by a structure transformation in two steps with hysteresis. Powder (T=10-295 K) and single-crystal (T=95-295 K) X-ray diffraction studies confirm that the symmetry of Ba₆Ge₂₅ (space group P4₁32) remains unchanged within the entire temperature range. A reconstructive behavior of the structural transformation is observed, involving Ge-Ge bond breaking and barium cation displacements. Some Ge4 type atoms (∼28%) are so significantly displaced during cooling that Ge4-Ge6 bonds break and new three-bonded (3b)Ge⁻ species (electron acceptors) are formed. Consequently, the number of charge carriers is reduced, affecting the physical properties. The reversible bond breaking involved in this process is a typical characteristic of a solid-state chemical reaction.     

The axiom of choice holds iff maximal closed filters exist

It is shown that in ZF set theory the axiom of choice holds iff every non empty topological space has a maximal closed filter.     

Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization

It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.