Vecteur unitaire | EPFL Graph Search

vignette|Deux vecteurs unitaires dans un espace vectoriel normé. Dans un espace vectoriel normé (réel ou complexe) E, un vecteur unitaire est un vecteur dont la norme est égale à 1.

Si le corps des scalaires est R, deux vecteurs unitaires v et w sont colinéaires si et seulement si v = w ou v = –w.
Si le corps des scalaires est C, et si v est un vecteur unitaire de E, alors les vecteurs unitaires colinéaires à v sont αv où α est un complexe de module 1.

Les vecteurs unitaires permettent de définir la direction et le sens d'un vecteur non nul de E. Tout vecteur non nul v est la multiplication du vecteur unitaire u = v/║v║ par un nombre réel strictement positif, à savoir la norme ║v║ de v. :v = ║v║u. Pour tout vecteur ayant un sens opposé à v, on a :v = -║v║u. En physique, pour dénoter les vecteurs unitaires, il est usuel d'utiliser un accent circonflexe : \hat{x}, \hat{y}, \hat{z} . En mécanique quantique, les états sont des vecteurs unitaires d'espaces de Hilbert. En

Metal Clusters on Surfaces

Michael Hugentobler

The morphology, stability and reactivity of nanometre-sized particles (Au, Pt) is investigated on three different supports. Gold nanoparticles with a diameter comprised between 4 and 6nm are stabilized in nanosized pits of well defined depth in highly oriented pyrolytic graphite (HOPG). These pits are produced by creation of artificial defects, followed by etching under a controlled oxygen atmosphere. At low Au coverage, clusters are found on the edges of the hexagonal pits maximizing the contact to dangling bonds on graphite multisteps. Larger coverage results in Au beads of well defined shape and with a constant bead density per unit length. Most remarkable is the stability of these nanostructures under ambient conditions. Temperatures as high as 650 K do not alter the morphology of the gold clusters. Above 700 K, the gold clusters catalyze the etching of graphite layers when heated under atmospheric conditions. The depth of the etching channel is defined by the depth of the pit and the channel width can be controlled by the cluster size. Hydrogen, oxygen and water are dissociated on the Au clusters at low temperatures (> 400 K). CO and CO2 is produced above 700 K which confirms the etching of the multilayer graphene sheets. The so-called water-gas shift reaction has been observed where oxygen and water dissociate on the Au clusters at low temperatures. Thermal desorption spectroscopy (TDS) measurements on the Pt=YSZ system have been performed and the desorption peaks of CO and O2 have been evidenced. Carbon monoxide desorbs from the surface at a temperature of 535 K and oxygen at 705 K, in good agreement with values reported in literature. Catalytic measurements have confirmed the known strong catalytic activity of Pt. The CO oxidation takes place at temperatures between 450 K and 700 K. The adsorption and desorption temperature of the two educts have been confirmed during the catalysis measurements. Size-selected Aun clusters (n = 5, 7) are deposited from the gas phase at room temperature with well-controlled kinetic energy on rutile TiO2 . Two different surface reconstructions have been used as support: TiO2(110) – (1 × 1) and TiO2(110) – (1 × 2). Systematic studies of morphology, stability and adsorption sites have been performed using scanning tunnelling microscopy (STM). The mean size of the clusters is maintained during deposition at a kinetic energy of 7.1eV per atom, indicating that there is no fragmentation. Au clusters are not stable neither on the TiO2(110) – (1 × 1) nor on the TiO2(110) – (1 × 2) reconstruction during annealing of the substrate to 800 K. A progressive transition into a pronounced 3-d formation is observed. The size distribution of the particles is small and the growth mode (Ostwald ripening) independent of the surface reconstruction. The higher corrugation of the TiO2(110) – (1 × 2) reconstruction results in a pronounced stabilization of the clusters on terraces, in contrast to the TiO2(110) – (1 × 1) reconstruction where 90% of the clusters are found on step edges at elevated temperatures. The catalytic activity of the Au clusters sets in during the 2d - 3d transition. These measurements have to be confirmed and a full understanding of the process should be found.

EPFL2010

Effects of Motion and Configural Complexity on Color Transparency Perception

Peggy Gerardin, Sabine Süsstrunk

We tested whether motion and configural complexity affect perceived transparency. A series of five coherent chromatic transformations in color space was applied across a figure: translation, convergence, shear, divergence and rotation. The stimuli consisted of a bipartite or a checkerboard configuration (10x10 deg), with a central static or moving overlay (5x5 deg). Three different luminance conditions (the plane of chromatic transformation oriented toward higher, lower or equal luminances) were also tested for each of three modulation depths. For each stimulus, the observer judged whether the overlay appeared transparent or not. The main results indicated an interaction between the type of chromatic transformation and stimulus motion and complexity. For example, convergences are judged to appear transparent significantly more often when motion is added for bipartite configurations, or when they are generated in a checkerboard configuration. Surprisingly, shears which have been reported to appear opaque, are more frequently reported to appear transparent with short vector lengths and when combined with motion. Other transformations are also affected by motion, although the effectiveness of figural complexity on transparency seems to depend on both the type of color shifts and the presence of motion. The results indicate that adding motion and stimulus complexity are not necessarily neutral with respect to the chromatic shifts evoking transparency. Thus, studies that have used motion to enhance transparency may yield different results about the color shifts supporting transparency perception from those that did not. The same might be supposed for stimulus complexity under some conditions.

2006

Morphology and stability of Au nanoclusters in HOPG nanopits of well-defined depth

Simon Bonanni, Wolfgang Harbich, Michael Hugentobler

Gold nanoparticles with a diameter comprised between 4 and 6 nm are stabilized in nanosized pits of well defined depth in highly oriented pyrolytic graphite (HOPG). These pits are produced by creation of artificial defects, followed by etching under a controlled oxygen atmosphere. At low Au coverage, clusters are found on the edges of the hexagonal pits maximizing the contact to dangling bonds on graphite multisteps. Larger coverage results in Au beads of surprisingly well defined shape and with a constant bead density per unit length. Most remarkable is the stability of these nanostructures under ambient conditions. Temperatures as high as 650 K do not alter the morphology of the gold clusters. Higher temperatures do not lead to a change of the cluster morphology but to catalytically driven etching of the HOPG substrate.

2011

Metal Clusters on Surfaces

Michael Hugentobler

EPFL2010