Comparison of Plane-Wave Approximations in Potential Scattering

1972 ◽  
Vol 50 (10) ◽  
pp. 1030-1039 ◽  
Author(s):  
John R. Ebden ◽  
J. W. Darewych
Keyword(s):  
Plane Wave ◽  
Cross Sections ◽  
Born Approximation ◽  
Coupling Constants ◽  
Resonance Structure ◽  
Total Cross Sections ◽  
Wave Approximation ◽  
Restricted Form ◽  
Plane Wave Approximation ◽  
Better Than

We compare four different plane-wave approximations for scattering by short-range potentials. These four are the first and second Born approximations, the plane-wave Schwinger approximation and a restricted form of Ford's renormalized Born approximation. For the Yukawa and Gaussian potentials, the differential and total cross sections are calculated in the energy range [Formula: see text] (arbitrary units) and for dimensionless coupling constants ranging between 0.1 and 10. The first Born and renormalized Born are 'first-order' approximations, requiring an order of magnitude less calculational labor than the second Born and Schwinger approximations. The results suggest that the restricted renormalized Born approximation is rarely better than the ordinary first Born approximation. The Schwinger plane-wave approximation is always better than the second Born approximation and since these two involve the evaluation of identical expressions, the Schwinger approximation is always to be preferred over the second Born. The Schwinger plane-wave approximation gives quite consistently the best results of the four methods considered, but even it is not be be trusted for calculating differential cross sections at large angles or for reproducing "resonance structure" in total cross sections.

PLANE-WAVE APPROXIMATION FOR THE SCATTERING AMPLITUDE

1964 ◽  
Vol 42 (6) ◽  
pp. 1017-1029 ◽  
Author(s):  
H. H. Chan ◽  
M. Razavy
Keyword(s):  
Plane Wave ◽  
Born Approximation ◽  
Wave Scattering ◽  
Scattering Amplitude ◽  
Unitarity Condition ◽  
S Wave ◽  
Wave Approximation ◽  
High Energies ◽  
Infinite Sum ◽  
Plane Wave Approximation

Different methods of calculating the S-wave scattering amplitude using the plane-wave approximation are discussed. The resulting scattering amplitudes satisfy the unitarity condition and for high energies approach the Born approximation asymptotically. These methods are applied to the special class of potentials that can be expressed as an infinite sum of exponential potentials, and in particular to some potentials for which the exact solutions are known. It is found that imposing the unitarity condition on the Born approximation, in general, does not improve the result of the calculation.

scholarly journals Hydrodynamic Interactions in Multiple Body Array: A Simple and Fast Approach Coupling Boundary Element Method and Plane Wave Approximation

2013 ◽  
Author(s):  
Jitendra Singh ◽  
Aurélien Babarit
Keyword(s):  
Boundary Element Method ◽  
Plane Wave ◽  
Boundary Element ◽  
Hydrodynamic Interaction ◽  
Computational Time ◽  
Linear Potential ◽  
Wave Approximation ◽  
Interaction Problem ◽  
Plane Wave Approximation ◽  
Element Method

The hydrodynamic forces acting on an isolated body could be considerably different than those when it is considered in an array of multiple bodies, due to wave interactions among them. In this context, we present in this paper a numerical approach based on the linear potential flow theory to solve full hydrodynamic interaction problem in a multiple body array. In contrast to the previous approaches that considered all bodies in an array as a single unit, the present approach relies on solving for an isolated body. The interactions among the bodies are then taken into account via plane wave approximation in an iterative manner. The boundary value problem corresponding to a isolated body is solved by the Boundary Element Method (BEM). The approach is useful when the bodies are sufficiently distant from each other, at-least greater than five times the characteristic dimensions of the body. This is a valid assumption for wave energy converter devices array of point absorber type, which is our target application at a later stage. The main advantage of the proposed approach is that the computational time requirement is significantly less than the commonly used direct BEM. The time savings can be realized for even small arrays consisting of four bodies. Another advantage is that the computer memory requirements are also significantly smaller compared to the direct BEM, allowing us to consider large arrays. The numerical results for hydrodynamic interaction problem in two arrays consisting of 25 cylinders and same number of rectangular flaps are presented to validate the proposed approach.

scholarly journals Post-processing of the plane-wave approximation of Schrödinger equations. Part II: Kohn–Sham models

2020 ◽  
Author(s):  
Geneviève Dusson
Keyword(s):  
Plane Wave ◽  
Ground State ◽  
A Priori ◽  
Processing Method ◽  
Linear Operators ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Post Processing ◽  
Wave Approximation ◽  
Plane Wave Approximation

Abstract In this article, we provide a priori estimates for a perturbation-based post-processing method of the plane-wave approximation of nonlinear Kohn–Sham local density approximation (LDA) models with pseudopotentials, relying on Cancès et al. (2020, Post-processing of the plane-wave approximation of Schrödinger equations. Part I: linear operators. IMA Journal of Numerical Analysis, draa044) for the proofs of such estimates in the case of linear Schrödinger equations. As in Cancès et al. (2016, A perturbation-method-based post-processing for the plane-wave discretization of Kohn–Sham models. J. Comput. Phys., 307, 446–459), where these a priori results were announced and tested numerically, we use a periodic setting and the problem is discretized with plane waves (Fourier series). This post-processing method consists of performing a full computation in a coarse plane-wave basis and then to compute corrections based on the first-order perturbation theory in a fine basis, which numerically only requires the computation of the residuals of the ground-state orbitals in the fine basis. We show that this procedure asymptotically improves the accuracy of two quantities of interest: the ground-state density matrix, i.e. the orthogonal projector on the lowest $N$ eigenvectors, and the ground-state energy.

scholarly journals Limits of the Plane Wave Approximation in the Measurement of Molecular Properties†

2008 ◽  
Vol 112 (39) ◽  
pp. 9439-9447 ◽  
Author(s):  
Zachary B. Walters ◽  
Stefano Tonzani ◽  
Chris H. Greene
Keyword(s):  
Plane Wave ◽  
Molecular Properties ◽  
Wave Approximation ◽  
Plane Wave Approximation

The Scattering of Positrons by Helium: Total Cross Sections up to 270 eV

1975 ◽  
Vol 53 (10) ◽  
pp. 962-967 ◽  
Author(s):  
B. Jaduszliwer ◽  
A. Nakashima ◽  
D. A. L. Paul
Keyword(s):  
Energy Range ◽  
Cross Section ◽  
Cross Sections ◽  
Born Approximation ◽  
Reasonable Agreement ◽  
Experimental Results ◽  
Formation Cross Section ◽  
Sum Rule ◽  
Total Cross Sections ◽  
High Energies

The total cross sections for the scattering of positrons by helium have been measured by the method of transmission in the 16 to 270 eV energy range. The experimental results are higher than those of Canter et al. but are in reasonable agreement with recent results of Griffith et al., and at high energies tend towards Born approximation calculations. The integral of the cross section over positron momentum is smaller than the sum rule estimate made by Bransden et al. A tentative value of (0.034 ± 0.017)πa02 is assigned to the positronium formation cross section at threshold.

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