A stannaethene with a planar environment of the tricoordinated tin and carbon atoms

Addition of the diarylstannylene R″₂Sn (R″ = 2−^tBu−4,5,6-Me₃C₆H) to the cryptodiborylcarbene (Me₃Si)₂C(B^tBu)₂C furnishes the stannaethene (Me₃Si)₂C(B^tBu)₂C=SnR″₂ (10). The X-ray structure analysis of 10 reveals a strictly planar environment of the tricoordinated tin and carbon atoms and a slight twisting of the Sn=C double bond.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

An oblique-incidence optical reflectivity difference and LEED study of rare-gas growth on a lattice-mismatched metal substrate

We studied the growth of Xe on Nb(110) from 33 K to 100 K using a combination of low-energy electron diffraction and an in situ oblique-incidence optical reflectivity difference technique. We found that a hexagonal close-packed Xe film grows after a transition layer of three monoatomic layers thick is formed. The first two monolayers, influenced by both the interaction with the Nb substrate and the Xe–Xe interaction, lack long-range order. The third monolayer forms a bulk-like hexagonal close-packed structure. Subsequently a bulk-phase Xe(111) film grows in step-flow mode from 54 K down to 40 K. At 40 K, we observed a brief crossover to a layer-by-layer mode. At 33 K the growth proceeds in a kinetically limited multilayer or a three-dimensional island mode. PACS 68.35.Fx; 68.35.Ja; 82.20.-w; 61.16.Ch     

Spatially resolved dynamics of the TiO2(1 1 0) surface reconstruction

We use low-energy electron microscopy to image the reversible transformation of the TiO₂(1 1 0) surface between a high-temperature 1 × 1 structure and a low-temperature 1 × 2 structure. The reconstruction dynamics are novel: 1 × 2 bands nucleated during cooling at the steps of the starting 1 × 1 surface and then grew laterally from the steps. The transformation kinetics are dominated by mass flow from the surface to the bulk, a process that facilitates converting the high-density 1 × 1 phase to the lower-density 1 × 2 phase. We have also imaged how the 1 × 1 surface reconstructs to 1 × 2 phase after sufficient oxygen is removed from the crystal’s bulk during vacuum annealing. 1 × 2 bands also nucleated and grew laterally from the initial 1 × 1-surface’s steps. However, because this isothermal 1 × 1-to-1 × 2 transition occurs largely by mass redistribution on the surface, the steps of the initial 1 × 1 surface and final 1 × 2 surface are offset. We propose models of mass redistribution during the 1 × 1/1 × 2 phase transition to explain this effect. We conclude that the phase transition is first-order because it always occurred by the nucleation and growth of discrete phases. Finally, we show that quenching can roughen TiO₂’s surface by forming pits and that changing temperature causes step motion on 1 × 2 surfaces.     

Smoothness of subdivision surfaces at extraordinary points

A stationary subdivision scheme such as Catmull and Clark's is described by a matrix iteration around an extraordinary point. We show how higher order smoothness of a limiting surface obtained by a stationary subdivision algorithm for tri- or quadrilateral nets depends on the spectral properties of the matrix and give necessary and sufficient conditions. The results are also useful to construct subdivision algorithms for surfaces of any smoothness order at extraordinary points. This revised version was published online in August 2006 with corrections to the Cover Date.