129Xe NMR study of the homogeneity and the size distribution of small sodium metal particles in NaY zeolite

NaY zeolite samples loaded with sodium metal by vapor phase deposition have been investigated using¹²⁹Xe NMR spectroscopy. At low sodium concentration, the¹²⁹Xe NMR spectrum showed three resonance lines which clearly indicate the existence of distinct domains in the zeolite sample. Such an observation suggests that the diffusion of the xenon atoms into each domain only occurs with respect to the NMR time scale (2.9 ms). As the sodium concentration increases, observation of a single broad line indicate a macroscopic homogenization of the system. The shift of this line is explained in part due to a paramagnetic interaction between the xenon atoms and the unpaired electrons of particles containing an odd number of sodium atoms. The linewidth is due to the distribution of the local magnetic fields partially averaged by the rapid motion of the xenon atoms and to the statistical distribution of the sodium particles in the supercage cavities. The paramagnetic interaction vanishes with the oxidation of the sample leading to a narrowing and a shift of the line to higher magnetic fields.     

Investigation of the chemical state of copper in Cu/SiO2 composite films by x-ray photoelectron spectroscopy

The concentration and chemical state of copper in the subsurface region of Cu/SiO₂ composite films obtained by simultaneous magnetron sputtering from two sources (Cu and SiO₂) are determined by x-ray photoelectron spectroscopy (XPS). It is established that copper in the as-grown film is primarily in the form of unoxidized atoms dispersed in a SiO₂ matrix. Annealing of the film results in practically no oxidation, but about 70% of the copper atoms condense into metallic clusters with sizes below 10 Å in the subsurface region and about 50 Å in the bulk of the film. The changes in the binding energy of core electrons, and especially in the energies of Auger electrons, are so large in this situation that photoelectron and Auger spectroscopy are efficient methods for monitoring the chemical state of this composite material.     

Student-Invented Numeration Systems: Pattern-Analysis and Mathematical Understanding

Thirteen fifth graders were given an assignment to invent their own numeration systems, following a unit on bases and a look at early events in the history of numbers. The task presented options that required the students to make decisions (such as whether to use a base, which base to use, design of symbols, etc.), and build a rationale for the elements of their system. Analyses of patterns embedded in their invented systems provided an assessment of student understanding of numeration. The progression of more and less complex thinking related to the student's choice of a base other than 10, consistency of logic throughout the system in words and symbols, rationale for change, and perception of real life examples that would change if the system was adopted. The invention task is presented as another way to make connections.     

On the domain dependence of solutions to the compressible Navier–Stokes equations of a barotropic fluid

We prove a general compactness result for the solution set of the compressible Navier–Stokes equations with respect to the variation of the underlying spatial domain. Among various corollaries, we then prove a general existence theorem for the system in question with no restrictions on smoothness of the spatial domain. Copyright © 2002 John Wiley & Sons, Ltd.     

Some remarks on metaplectic cusp forms and the correspondences of Shimura and Waldspurger

The simple relation between representations of the covering groups of SL₂ and GL₂ makes it possible to fuse and extend the recent metaplectic results of Shimura, Waldspurger, Flicker, and ourselves. By giving a new (purely local andL-function theoretic) treatment of the Waldspurger-Shintani correspondence, we also simplify some of Waldspurger’s original results.