scholarly journals Low Energy Resolvent for the Hodge Laplacian: Applications to Riesz Transform, Sobolev Estimates, and Analytic Torsion

2014 ◽  
Vol 2015 (15) ◽  
pp. 6136-6210 ◽  
Author(s):  
Colin Guillarmou ◽  
David A. Sher
Keyword(s):  
Riesz Transform ◽  
Low Energy ◽  
Analytic Torsion ◽  
Hodge Laplacian ◽  
Sobolev Estimates

scholarly journals Sub-Riemannian Limit of the Differential Form Heat Kernels of Contact Manifolds

2020 ◽  
Author(s):  
Pierre Albin ◽  
Hadrian Quan
Keyword(s):  
Blow Up ◽  
Riemannian Metrics ◽  
Contact Manifold ◽  
Heat Kernels ◽  
Analytic Torsion ◽  
Contact Manifolds ◽  
Spectral Invariants ◽  
Eta Invariant ◽  
Hodge Laplacian ◽  
Contact Distribution

Abstract We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the $\eta $-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative $\eta $-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.

Dissociated Σ=27 boundaries in silicon

1988 ◽  
Vol 46 ◽  
pp. 624-625
Author(s):  
A. Garg ◽  
W.A.T. Clark ◽  
J.P. Hirth
Keyword(s):  
Electron Diffraction ◽  
Local Minimum ◽  
Boundary Energy ◽  
Coincidence Site Lattice ◽  
Boundary Plane ◽  
Low Energy ◽  
Theoretical Understanding ◽  
Site Lattice ◽  
Kikuchi Pattern ◽  
Analysis Techniques

In the last twenty years, a significant amount of work has been done in the theoretical understanding of grain boundaries. The various proposed grain boundary models suggest the existence of coincidence site lattice (CSL) boundaries at specific misorientations where a periodic structure representing a local minimum of energy exists between the two crystals. In general, the boundary energy depends not only upon the density of CSL sites but also upon the boundary plane, so that different facets of the same boundary have different energy. Here we describe TEM observations of the dissociation of a Σ=27 boundary in silicon in order to reduce its surface energy and attain a low energy configuration.The boundary was identified as near CSL Σ=27 {255} having a misorientation of (38.7±0.2)°/[011] by standard Kikuchi pattern, electron diffraction and trace analysis techniques. Although the boundary appeared planar, in the TEM it was found to be dissociated in some regions into a Σ=3 {111} and a Σ=9 {122} boundary, as shown in Fig. 1.

Transfer optics for high spatial resolution electron spectroscopy

1988 ◽  
Vol 46 ◽  
pp. 666-667
Author(s):  
G. G. Hembree ◽  
Luo Chuan Hong ◽  
P.A. Bennett ◽  
J.A. Venables
Keyword(s):  
Magnetic Field ◽  
Scanning Transmission Electron Microscope ◽  
Low Energy ◽  
Objective Lens ◽  
State University ◽  
Arizona State University ◽  
Initial Field ◽  
Resolution Electron ◽  
Scanning Transmission ◽  
Low Energy Electrons

A new field emission scanning transmission electron microscope has been constructed for the NSF HREM facility at Arizona State University. The microscope is to be used for studies of surfaces, and incorporates several surface-related features, including provision for analysis of secondary and Auger electrons; these electrons are collected through the objective lens from either side of the sample, using the parallelizing action of the magnetic field. This collimates all the low energy electrons, which spiral in the high magnetic field. Given an initial field Bi∼1T, and a final (parallelizing) field Bf∼0.01T, all electrons emerge into a cone of semi-angle θf≤6°. The main practical problem in the way of using this well collimated beam of low energy (0-2keV) electrons is that it is travelling along the path of the (100keV) probing electron beam. To collect and analyze them, they must be deflected off the beam path with minimal effect on the probe position.

Performance Evaluation of a Novel Type of Low-Energy Electron Microscope

1990 ◽  
Vol 48 (1) ◽  
pp. 354-355
Author(s):  
Bertholdand Senftinger ◽  
Helmut Liebl
Keyword(s):  
Electron Microscope ◽  
High Resolution ◽  
Field Strength ◽  
Numerical Aperture ◽  
Low Energy ◽  
Energy Electron ◽  
High Resolution Imaging ◽  
Lens System ◽  
Resolution Imaging ◽  
Low Energy Electron

During the last few years the investigation of clean and adsorbate-covered solid surfaces as well as thin-film growth and molecular dynamics have given rise to a constant demand for high-resolution imaging microscopy with reflected and diffracted low energy electrons as well as photo-electrons. A recent successful implementation of a UHV low-energy electron microscope by Bauer and Telieps encouraged us to construct such a low energy electron microscope (LEEM) for high-resolution imaging incorporating several novel design features, which is described more detailed elsewhere.The constraint of high field strength at the surface required to keep the aberrations caused by the accelerating field small and high UV photon intensity to get an improved signal-to-noise ratio for photoemission led to the design of a tetrode emission lens system capable of also focusing the UV light at the surface through an integrated Schwarzschild-type objective. Fig. 1 shows an axial section of the emission lens in the LEEM with sample (28) and part of the sample holder (29). The integrated mirror objective (50a, 50b) is used for visual in situ microscopic observation of the sample as well as for UV illumination. The electron optical components and the sample with accelerating field followed by an einzel lens form a tetrode system. In order to keep the field strength high, the sample is separated from the first element of the einzel lens by only 1.6 mm. With a numerical aperture of 0.5 for the Schwarzschild objective the orifice in the first element of the einzel lens has to be about 3.0 mm in diameter. Considering the much smaller distance to the sample one can expect intense distortions of the accelerating field in front of the sample. Because the achievable lateral resolution depends mainly on the quality of the first imaging step, careful investigation of the aberrations caused by the emission lens system had to be done in order to avoid sacrificing high lateral resolution for larger numerical aperture.

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