Plane Wave Spatial Spectrum

2012 ◽  
pp. 403-414
Keyword(s):  
Plane Wave ◽  
Spatial Spectrum

Characterization of resolution and uniqueness in crosswell direct‐arrival traveltime tomography using the Fourier projection slice theorem

Geophysics ◽  
1994 ◽  
Vol 59 (11) ◽  
pp. 1642-1649 ◽  
Author(s):  
James W. Rector ◽  
John K. Washbourne
Keyword(s):  
Plane Wave ◽  
Oil And Gas ◽  
Fresnel Zone ◽  
Spatial Spectrum ◽  
Limiting Factor ◽  
Angular Aperture ◽  
Data Set ◽  
Traveltime Tomography ◽  
Slice Theorem ◽  
Well Spacing

The process of acquiring a crosswell seismic direct‐arrival traveltime data set can be approximated by a series of truncated plane‐wave projections through an interwell slowness field. Using this approximation, the resolution and uniqueness of crosswell direct‐arrival traveltime tomograms can be characterized by invoking the Fourier projection slice theorem, which states that a plane‐wave projection through an object constitutes a slice of the object’s spatial spectrum. The limited vertical aperture of a crosswell survey introduces a small amount of nonuniqueness into the reconstructed tomogram by truncating the plane‐wave projection. By contrast, the limitations on angular aperture have a significant effect on resolution. The reconstructed tomogram is smeared primarily along the limiting projection angles, with the amount of smearing dependent upon the well spacing and the angular aperture. The amount of smearing was found to be inversely proportional to tan Δϕ, where Δϕ is the angular aperture illuminating a sector of the interwell plane. Consequently, the amount of smearing can be large where the angular aperture becomes small, such as at the top and bottom of the tomogram. For interwell sectors illuminated by large angular apertures, Fresnel zone effects will generally be the limiting factor in crosswell tomogram resolution. However, in some circumstances, angular aperture effects may control the tomogram resolution. The effects of angular aperture and direct‐arrival Fresnel zones produce tomograms with spatial resolution that is dependent upon the well spacing. This study indicates that direct‐arrival traveltime tomography will not usually produce tomograms with substantially greater resolution than surface seismic techniques for normal oil and gas well spacings.

Theory and Errors in Stereo Electron Microscopy

1968 ◽  
Vol 26 ◽  
pp. 336-337
Author(s):  
J. M. Pankratz
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Plane Wave ◽  
Focal Length ◽  
Foil Thickness ◽  
Points Of Interest ◽  
Transmission Electron ◽  
Vertical Spacing ◽  
Illuminating System

It is often desirable in transmission electron microscopy to know the vertical spacing of points of interest within a specimen. However, in order to measure a stereo effect, one must have two pictures of the same area taken from different angles, and one must have also a formula for converting measured differences between corresponding points (parallax) into a height differential.Assume (a) that the impinging beam of electrons can be considered as a plane wave and (b) that the magnification is the same at the top and bottom of the specimen. The first assumption is good when the illuminating system is overfocused. The second assumption (the so-called “perspective error”) is good when the focal length is large (3 x 107Å) in relation to foil thickness (∼103 Å).

A hybrid Gaussian and plane wave density functional scheme

1997 ◽  
Vol 92 (3) ◽  
pp. 477-487 ◽  
Author(s):  
GERALD LIPPERT ◽  
JuRG HUTTER ◽  
MICHELE PARRINELLO
Keyword(s):  
Plane Wave ◽  
Density Functional ◽  
Functional Scheme

Evolution of the electronic structure of metallic multilayers from two to threedimensionality. A Full Potential Linearized Augmented Plane Wave calculation.

2002 ◽  
Vol 727 ◽  
Author(s):  
A. M. Mazzone
Keyword(s):  
Plane Wave ◽  
Density Of States ◽  
Noble Metals ◽  
Full Potential ◽  
Metallic Multilayers ◽  
Augmented Plane Wave ◽  
Narrow Bandwidth ◽  
Epitaxial Multilayers ◽  
Linearized Augmented Plane Wave ◽  
Wave Calculation

AbstractFull Potential Linearized Augmented Plane Wave calculations have been performed for epitaxial multilayers formed by the noble metals Ag and Cu with a thickness n up to 10 layers. The multilayers have a fcc lattice and are pure or compositionally modulated with a structure of the type Agn Cun or (AgCu)n. For n in the range 2,3 the density of states, evaluated at paramagnetic level, exhibits a sharp reduction of the bandwidth which is consistent with the reduced coordination of these structures. For n ≤ 5 the density of states in the central layers converges to the bulk value while the outer layers retain the narrow bandwidth found at n=2. Due to the absence of charge intermixing and hybridization, these features are shared by multilayers of all composition.

Interstitial vs. Substitutional Metal Insertion in V2O5 as Post-Lithium Ion Battery Cathode: A Comparative GGA/GGA+U Study with Localized Bases

2020 ◽  
Author(s):  
Daniel Koch ◽  
Sergei Manzhos
Keyword(s):  
Electronic Structure ◽  
Plane Wave ◽  
Transition Metal Oxides ◽  
Reasonable Agreement ◽  
Correction Term ◽  
Lithium Ion ◽  
Generalized Gradient ◽  
Main Group Metals ◽  
Generalized Gradient Approximation ◽  
Metal Insertion

<p></p><p>The generalized gradient approximation (GGA) often fails to correctly describe the electronic structure and thermochemistry of transition metal oxides and is commonly improved using an inexpensive correction term with a scaling parameter <i>U</i>. We tune <i>U</i> to reproduce experimental vanadium oxide redox energetics with a localized basis and a GGA functional. We find the value for <i>U</i> to be significantly lower than what is generally reported with plane-wave bases, with the uncorrected GGA results being in reasonable agreement with experiments. We use this computational setup to calculate interstitial and substitutional <a>insertion energies of main group metals in vanadium pentoxide</a> and find <a>interstitial doping to be thermodynamically favored</a>.</p><p></p>

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