Geometric-acoustics analysis of single-scattering of nonlinearly evolving waves by circular cylinders

Geometric acoustics, or acoustic ray theory, is used to analyze the scattering of high-amplitude acoustic waves incident upon rigid circular cylinders. Theoretical predictions of the nonlinear evolution of the scattered wave field are provided, as well as measures of the importance of accounting for nonlinearity. An analysis of scattering by many cylinders is also provided, though the effects of multiple scattering are not considered. Provided the characteristic nonlinear distortion length is much larger than a cylinder radius, the nonlinear evolution of the incident wave is shown to be of much greater importance to the overall evolution than the nonlinear evolution of the individual scattered waves.

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Inelastic Electron Scattering from Valence Electrons in Aluminum

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010009227x ◽

1976 ◽

Vol 34 ◽

pp. 462-463

Author(s):

P. E. Batson ◽

C. H. Chen ◽

J. Silcox

Keyword(s):

Electron Scattering ◽

Single Scattering ◽

Three Dimensions ◽

Inelastic Electron ◽

Weak Beam ◽

Scattering Spectrum ◽

Valence Electrons ◽

Diffraction Patterns ◽

Electronic Scattering

We wish to report in this paper measurements of the inelastic scattering component due to the collective excitations (plasmons) and single particlehole excitations of the valence electrons in Al. Such scattering contributes to the diffuse electronic scattering seen in electron diffraction patterns and has recently been considered of significance in weak-beam images (see Gai and Howie) . A major problem in the determination of such scattering is the proper correction for multiple scattering. We outline here a procedure which we believe suitably deals with such problems and report the observed single scattering spectrum.In principle, one can use the procedure of Misell and Jones—suitably generalized to three dimensions (qx, qy and #x2206;E)--to derive single scattering profiles. However, such a computation becomes prohibitively large if applied in a brute force fashion since the quasi-elastic scattering (and associated multiple electronic scattering) extends to much larger angles than the multiple electronic scattering on its own.

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Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100133916 ◽

1990 ◽

Vol 48 (2) ◽

pp. 64-65

Author(s):

L. Reimer ◽

R. Oelgeklaus

Keyword(s):

Fourier Transform ◽

Energy Loss ◽

Electron Energy ◽

Rotational Symmetry ◽

Mean Free Path ◽

Single Scattering ◽

Specimen Thickness ◽

Two Dimensional ◽

Image Position ◽

Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

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Background Fitting in the Low-Loss Region of Electron Energy Loss Spectra

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100133874 ◽

1990 ◽

Vol 48 (2) ◽

pp. 56-57

Author(s):

C P Scott ◽

A J Craven ◽

C J Gilmore ◽

A W Bowen

Keyword(s):

Energy Loss ◽

Multiple Scattering ◽

Simple Form ◽

Single Scattering ◽

Low Loss ◽

Characteristic Energy ◽

Normal Method ◽

Fitting In ◽

Electron Energy Loss ◽

Electron Energy Loss Spectra

The normal method of background subtraction in quantitative EELS analysis involves fitting an expression of the form I=AE-r to an energy window preceding the edge of interest; E is energy loss, A and r are fitting parameters. The calculated fit is then extrapolated under the edge, allowing the required signal to be extracted. In the case where the characteristic energy loss is small (E < 100eV), the background does not approximate to this simple form. One cause of this is multiple scattering. Even if the effects of multiple scattering are removed by deconvolution, it is not clear that the background from the recovered single scattering distribution follows this simple form, and, in any case, deconvolution can introduce artefacts.The above difficulties are particularly severe in the case of Al-Li alloys, where the Li K edge at ~52eV overlaps the Al L2,3 edge at ~72eV, and sharp plasmon peaks occur at intervals of ~15eV in the low loss region. An alternative background fitting technique, based on the work of Zanchi et al, has been tested on spectra taken from pure Al films, with a view to extending the analysis to Al-Li alloys.

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Local thickness determination by Electron Energy Loss Spectroscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100172450 ◽

1994 ◽

Vol 52 ◽

pp. 944-945

Author(s):

Suichu Luo ◽

John R. Dunlap ◽

Richard W. Williams ◽

David C. Joy

Keyword(s):

Energy Loss ◽

Analytical Electron Microscopy ◽

Mean Free Path ◽

Single Scattering ◽

Specimen Thickness ◽

Three Dimension ◽

Angular Correction ◽

Electron Intensity ◽

Image Position ◽

Inelastic Mean Free Path

In analytical electron microscopy, it is often important to know the local thickness of a sample. The conventional method used for measuring specimen thickness by EELS is:where t is the specimen thickness, λi is the total inelastic mean free path, IT is the total intensity in an EEL spectrum, and I0 is the zero loss peak intensity. This is rigorouslycorrect only if the electrons are collected over all scattering angles and all energy losses. However, in most experiments only a fraction of the scattered electrons are collected due to a limited collection semi-angle. To overcome this problem we present a method based on three-dimension Poisson statistics, which takes into account both the inelastic and elastic mixed angular correction.The three-dimension Poisson formula is given by:where I is the unscattered electron intensity; t is the sample thickness; λi and λe are the inelastic and elastic scattering mean free paths; Si (θ) and Se(θ) are normalized single inelastic and elastic angular scattering distributions respectively ; F(E) is the single scattering normalized energy loss distribution; D(E,θ) is the plural scattering distribution,

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Monte Carlo Modelling of Secondary Electron Yield from Rough Surfaces

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100180860 ◽

1990 ◽

Vol 48 (1) ◽

pp. 422-423

Author(s):

John C. Russ

Keyword(s):

Monte Carlo ◽

Energy Loss ◽

Secondary Electron ◽

Secondary Electron Emission ◽

Analytical Calculation ◽

Mean Free Path ◽

Single Scattering ◽

High Energy ◽

Secondary Electron Yield ◽

Electron Yield

Monte-Carlo programs are well recognized for their ability to model electron beam interactions with samples, and to incorporate boundary conditions such as compositional or surface variations which are difficult to handle analytically. This success has been especially powerful for modelling X-ray emission and the backscattering of high energy electrons. Secondary electron emission has proven to be somewhat more difficult, since the diffusion of the generated secondaries to the surface is strongly geometry dependent, and requires analytical calculations as well as material parameters. Modelling of secondary electron yield within a Monte-Carlo framework has been done using multiple scattering programs, but is not readily adapted to the moderately complex geometries associated with samples such as microelectronic devices, etc.This paper reports results using a different approach in which simplifying assumptions are made to permit direct and easy estimation of the secondary electron signal from samples of arbitrary complexity. The single-scattering program which performs the basic Monte-Carlo simulation (and is also used for backscattered electron and EBIC simulation) allows multiple regions to be defined within the sample, each with boundaries formed by a polygon of any number of sides. Each region may be given any elemental composition in atomic percent. In addition to the regions comprising the primary structure of the sample, a series of thin regions are defined along the surface(s) in which the total energy loss of the primary electrons is summed. This energy loss is assumed to be proportional to the generated secondary electron signal which would be emitted from the sample. The only adjustable variable is the thickness of the region, which plays the same role as the mean free path of the secondary electrons in an analytical calculation. This is treated as an empirical factor, similar in many respects to the λ and ε parameters in the Joy model.

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Scattering of Electrons at High - Pressure and Restoration of Image Contrast in High Pressure Scanning Electron Microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100180677 ◽

1990 ◽

Vol 48 (1) ◽

pp. 384-385

Author(s):

J. S. Shah ◽

R. Durkin ◽

A. N. Farley

Keyword(s):

Electron Microscopy ◽

Scanning Electron Microscopy ◽

High Pressure ◽

Cross Sections ◽

Single Scattering ◽

Image Contrast ◽

Experimental Scheme ◽

Medium Resolution ◽

Scattering Approximation ◽

Scanning Electron

It is now possible to perform High Pressure Scanning Electron Microscopy (HPSEM) in the range 10 to 2000 Pa. Here the effect of scattering on resolution has been evaluated by calculating the profile of the beam in high pressure and assessing its effect on the image contrast . An experimental scheme is presented to show that the effect of the primary beam ionization is to reduce image contrast but this effect can be eliminated by a novel use of specimen current detection in the presence of an electric field. The mechanism of image enhancement is discussed in terms of collection of additional carriers generated by the emissive components.High Pressure SEM (HPSEM) instrumentation is establishing itself as commercially viable. There are now a number of manufacturers, such as JEOL, ABT, ESCAN, DEBEN RESEARCH, selling microscopes and accessories for HPSEM. This is because high pressure techniques have begun to yield high quality micrographs at medium resolution.To study the effect of scattering on the incident electron beam, its profile - in a high pressure environment - was evaluated by calculating the elastic and inelastic scattering cross sections for nitrogen in the energy range 5-25 keV. To assess the effect of the scattered beam on the image contrast, the modification of a sharp step contrast function due to scattering was calculated by single scattering approximation and experimentally confirmed for a 20kV accelerated beam.

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