Random walks in varying dimensions

We establish recurrence criteria for sums of independent random variables which take values in Euclidean lattices of varying dimension. In particular, we describe transient inhomogeneous random walks in the plane which interlace two symmetric step distributions of bounded support.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research supported in part by National Science Foundation Grant No. DMS 9300191, by a Sloan Foundation Fellowship, and by a Presidential Faculty Fellowship.     

Structural and photoluminescence properties of ZnO nanoparticles on silicon oxide

Isolated, self assembled ZnO nanoparticles are grown in two steps: by the electron beam evaporation of Zn on oxidised silicon wafers, during which isolated Zn nanodots are grown, and a subsequent annealing in oxygen that results in the desired ZnO nanodots. Low temperature PL measurements of the ZnO nanodots show that the near band edge part of the spectra is dominated by a zero phonon line near 3.36 eV which is an overlap of two emitting lines near 3.363 eV and 3.367 eV. Characterization by TEM and EELS shows that the nanoparticles are zinc oxide single crystals grown with their c-axis perpendicular to the substrate; their distribution, size and crystallinity depend on the deposition parameters of zinc and the growth substrate. We discuss the effect of these parameters on the morphology of the resulting material. Our approach demonstrates a simple method for the growth of high purity isolated ZnO nanodots of similar sizes, distributed uniformly on a large surface. PACS 61.46.Df; 81.05.Dz; 81.07.-b     

ZnO homoepitaxy on the O polar face of hydrothermal and melt-grown substrates by pulsed laser deposition

2 cm diameter hydrothermal ZnO crystals were grown and then made into substrates using both mechanical and chemical-mechanical polishing (CMP). CMP polishing showed superior results with an (0002) Ω scan full width half maximum (FWHM) of 67 arcsec and an root mean square (RMS) roughness of 2 Å. In comparison, commercial melt-grown substrates exhibited broader X-ray diffraction (XRD) linewidths with evidence of sub-surface crystal damage due to polishing, including a downward shift of c-lattice parameter. Secondary ion mass spectroscopy revealed strong Li, Fe, Co, Al and Si contamination in the hydrothermal crystals as opposed to the melt-grown substrates, for which glow discharge mass spectroscopy studies had reported high levels of Pb, Fe, Cd and Si. Low temperature photoluminescence (PL) studies indicated that the hydrothermal crystal had high defect and/or impurity concentrations compared with the melt-grown substrate. The dominant bound exciton for the melt-grown substrate was indexed to Al. ZnO films were grown using pulsed laser deposition. The melt-grown substrates gave superior results with XRD (0002) Ω and 2θ/Ω WHM of 124 and 34 arcsec, respectively. Atomic force microscope measurements indicated a low RMS roughness (1.9 nm) as confirmed by fringes in the XRD 2θ/Ω scan. It was suggested that the improvement in XRD response relative to the substrate might be due to “healing” of sub-surface polishing damage due to the elevated T_s used for the growth. Indeed the c-lattice parameter for the homoepitaxial layer on the melt-grown substrate had become that which would be expected for strain-free ZnO. Furthermore, the stability of the PL peak positions relative to bulk ZnO, confirmed that the films appear practically strain free.     

Unimolecular dissociations of C70+ and its noble gas endohedral cations Ne@C70+ and Ar@C70+: cage-binding energies for C2 loss

The energetics and dynamics of unimolecular decompositions of C70+ and its noble gas endohedral cations, Ne@C70+ and Ar@C70+, have been studied using tandem mass spectrometry techniques. The high-resolution mass-analyzed ion kinetic energy (HR-MIKE) spectra for the unimolecular reactions of C70+, Ne@CC70+, and Ar@C70+ were recorded by scanning the electrostatic analyzer and using single-ion counting that was achieved by combination of an electron multiplier, amplifier/discriminator, and multichannel analyzer. These cations dissociate unimolecularly via loss of a C2 unit, and no endohedral atom is observed as fragment. The activation energies for C2 evaporation from Ne@C70+ and Ar@C70+ are lower than those for elimination of the endohedral noble gas atoms. The kinetic energy release distributions (KERDs) for the C2 evaporation have been measured and, by use of the finite heat bath theory (FHBT), the binding energies for the C2 emission have been deduced from the KERDs. The C2 evaporation energies increase in the order DeltaEvap(C70+) < DeltaEvap(Ne@C70+) < DeltaEvap(Ar@C70+), but no big difference in the cage binding was observed for C70+, Ne@C70+, and Ar@C70+, indicating incorporations of the Ne and Ar atoms into C70 contribute a little to the stability of C70 toward C2 loss, which is in good agreement with theoretical calculations but contrasts with the findings in their C60 analogues and in metallofullerenes that the decay energies of the filled fullerenes are much higher than those of the corresponding empty cages.     

The sharp Hausdorff measure condition for length of projections

In a recent paper, Pertti Mattila asked which gauge functions have the property that for any Borel set with Hausdorff measure 0$">, the projection of to almost every line has positive length. We show that finiteness of , which is known to be sufficient for this property, is also necessary for regularly varying . Our proof is based on a random construction adapted to the gauge function.

     

Entangled quantum states as direction indicators

We consider the use of N spin-1/2 particles for indicating a direction in space. If N>2, their optimal state is entangled. For large N, the mean square error decreases as N-2 (rather than N-1 for parallel spins).     

The rotor-router shape is spherical

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.     

Evolving sets, mixing and heat kernel bounds

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |pⁿ(x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.Research supported in part by NSF Grants #DMS-0104073 and #DMS-0244479.