Superconductivity in domains with corners

We study the two-dimensional Ginzburg-Landau functional in a domain with corners for exterior magnetic field strengths near the critical field where the transition from the superconducting to the normal state occurs. We discuss and clarify the definition of this field and obtain a complete asymptotic expansion for it in the large $\kappa$ regime. Furthermore, we discuss nucleation of superconductivity at the boundary

Weyl asymptotics for magnetic Schr\"odinger operators and de Gennes' boundary condition

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof relies on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [A. Kachmar, J. Math. Phys. Vol. 47 (7) 072106 (2006)].Comment: 28 pages (revised version). to appear in Rev Math Phy

Microscopic work function anisotropy and surface chemistry of 316L stainless steel using photoelectron emission microscopy

International audienceWe have studied the variation in the work function of the surface of sputtered cleaned 316L stainless steel with only a very thin residual oxide surface layer as a function of grain orientation using X-ray photo-electron emission microscopy (XPEEM) and Electron Backscattering Diffraction. The grains are mainly oriented [1 1 1] and [1 0 1]. Four distinct work function values spanning a 150 meV energy window are measured. Grains oriented [1 1 1] have a higher work function than those oriented [1 0 1]. From core level XPEEM we deduce that all grain surfaces are Cr enriched and Ni depleted whereas the Cr/Fe ratio is similar for all grains. The [1 1 1] oriented grains show evidence for a Cr 2 O 3 surface oxide and a higher concentration of defective oxygen sites

Simulation of resonant tunneling heterostructures: numerical comparison of a complete Schr{ö}dinger-Poisson system and a reduced nonlinear model

Two different models are compared for the simulation of the transverse electronic transport through an heterostructure: a $1D$ self-consistent Schr{ö}dinger-Poisson model with a numerically heavy treatment of resonant states and a reduced model derived from an accurate asymptotic nonlinear analysis. After checking the agreement at the qualitative and quantitative level on quite well understood bifurcation diagrams, the reduced model is used to tune double well configurations for which nonlinearly interacting resonant states actually occur in the complete self-consistent model

On the third critical field in Ginzburg-Landau theory

Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, $H_{C_3}$, describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near $H_{C_3}$ only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables

Fracture of M23_{23}C6_6 carbides / austenitic matrix interfaces in austenitic stainless steels

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Asymptotic behavior of u-capacities and singular perturbations for the Dirichlet-Laplacian

In this paper we study the asymptotic behavior of u-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets ω and ω of R 2, containing the origin. First, if ϵ is close to 0 and if u is a function defined on ω, we compute an asymptotic expansion of the u-capacity Capω(ϵω¯,u) as ϵ → 0. As a byproduct, we compute an asymptotic expansion for the Nth eigenvalues of the Dirichlet-Laplacian in the perforated set ω(ϵω¯) for ϵ close to 0. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near 0 and on the shape ω of the hole

Ion irradiation behaviour and chemical durability of simple borosilicate glasses

International audienceSimple alkali alumino-borosilicate glasses were doped with iron and/or molybdenum oxides. The chemical composition was then fully characterized before irradiation at room temperature with 255 keV Xe ions to fluence ranging from 10(15) Up to 10(17) ionS/cm(2). After irradiation, glass samples were leached in deionised water at 90 degrees C for 7 days. Scanning electron microscopy, electron microprobe analysis and ion beam analysis have been used to characterize the investigated samples. Irradiation induces a strong anti-correlated migration of boron and sodium. Alternatively enriched/impoverished zones are formed in the centre (irradiated) as well as at the rim (unirradiated) of the glass samples. Silicon, calcium, iron and molybdenum do not exhibit any perturbations in both surface and volume distributions. Boron and sodium mobility is mainly due to the coupling of thermal, electrical and stress gradients during ion irradiation. Glasses pre-irradiated before leaching exhibit stronger Na and B release than unirradiated glasses. A higher aluminium and iron surface enrichment is observed; it is probably due to the highest reactivity of the irradiated glasses having lost part of B and Na from the damaged network. (C) 2008 Elsevier B.V. All rights reserved

Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

For each pair (Formula presented.) of positive parameters, we define a perforated domain (Formula presented.) by making a small hole of size (Formula presented.) in an open regular subset (Formula presented.) of (Formula presented.) ((Formula presented.)). The hole is situated at distance (Formula presented.) from the outer boundary (Formula presented.) of the domain. Thus, when (Formula presented.) both the size of the hole and its distance from (Formula presented.) tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain (Formula presented.) and we denote its solution by (Formula presented.) Our aim is to represent the map that takes (Formula presented.) to (Formula presented.) in terms of real analytic functions of (Formula presented.) defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of (Formula presented.) to suitable subsets of (Formula presented.) we prove a global representation formula that holds on the whole of (Formula presented.) Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct