A novel route to aligned nanotubes and nanofibres using laser-patterned catalytic substrates

We describe the generation of aligned carbon nanotube bundles and films by pyrolysis of solid organic precursors (for example 2-amino-4,6-dichloro-s-triazine, s-triaminotriazine) at 950-1050 °C over laser-patterned thin metal (Fe, Co, Ni) films, deposited on silica substrates. Scanning electron microscopy (SEM) and atomic force microscopy (AFM) studies reveal that surface roughness of the laser-etched catalytic substrates plays a key role in achieving control of nanotube growth. We believe that, during the etching process, the energised (ablated) metal clusters condense and recrystallise evenly, possibly as the metal oxide, within the edges or surface of the eroded regions. During pyrolysis these catalytic particles, embedded in the silica substrates, are responsible for carbon agglomeration and subsequent tube axial growth, suggesting that nanotube alignment strongly depends upon the etching conditions (for example laser power, pulse duration, and focal distance). The pyrolysed products (usually nanotubes or nanofibres) were characterised by SEM, high-resolution transmission electron microscopy (HRTEM), electron energy loss spectroscopy (EELS) and energy dispersive X-ray spectroscopy (EDX). Samples containing only small amounts of amorphous carbon and other carbonaceous particles are notably absent. We observe that the degree of graphitisation is dependent upon the catalyst and the organic precursor. Interestingly, a nitrogen content З% was detected within the nanofibres, which exhibit corrugated graphite-like morphologies. This pyrolytic method may be used to advantage in generating aligned heteroatomic nanostructures such as B_xC_yN_z systems.     

Differentiability of convex functions and the convex point of continuity property in Banach spaces

We show that ifX is a separable Banach space, then every continuous, convex, Gateaux differentiable function onX is Fréchet differentiable on a dense set if and only ifX* has theweak*-Convex Point of Continuity Property (C*PCP). Research completed while a visitor at the University of Alberta. Research supported in part by an H. R. MacMillan Fellowship from the University of British Columbia. Research partially supported by NSERC (Canada).     

Singularity of orbits in classical Lie algebras

We determine the maximum k such that the k-fold sum of some nontrivial, adjoint orbit in the Lie algebra of a classical, compact Lie group has measure zero. The orbits of minimal dimension are seen to be the extreme examples. We show that for this choice of k there is a central, continuous measure on the group such that ^k is singular to L ¹. For Lie groups other than type B _n or C ₃ this result is sharp.     

Precise measurement of the positive muon anomalous magnetic moment

A precise measurement of the anomalous g value, a(mu) = (g-2)/2, for the positive muon has been made at the Brookhaven Alternating Gradient Synchrotron. The result a(mu+) = 11 659 202(14) (6) x 10(-10) (1.3 ppm) is in good agreement with previous measurements and has an error one third that of the combined previous data. The current theoretical value from the standard model is a(mu)(SM) = 11 659 159.6(6.7) x 10(-10) (0.57 ppm) and a(mu)(exp) - a(mu)(SM) = 43(16) x 10(-10) in which a(mu)(exp) is the world average experimental value.     

On the equality of the grundy and ochromatic numbers of a graph

It is proved in this note that the Grundy number, Γ(G), and the ochromatic number, χ⁰(G), are the same for any graph G.