Three-dimensional gallium sulphide open framework

The synthesis and crystal structure of four gallium sulphide open frameworks, built from supertetrahedral clusters, are described. The structures of [C4NH12]6[Ga10S18][C4NH12]6[Ga10S18](1) and [C4NH12]12[Ga20S35.5(S3)0.5O](2) contain supertetrahedral T3 clusters, while in the isostructural compounds [C4NH12]16[Ga10S18M4Ga16S33][C4NH12]16[Ga10S18M4Ga16S33] (M=CoM=Co(3), Zn (4)), T3 and T4 clusters alternate. These materials exhibit three-dimensional frameworks, with topologies consisting of two interpenetrating diamond lattices, and contain over 50% of solvent accessible void space. UV–Vis diffuse reflectance measurements indicate that these compounds are semiconducting, with band gaps over the range 3.4–4.1 eV.

Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

We consider the billiard dynamics in a non-compact set of ℝ d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called quenched random Lorentz tube. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.