Two families of non-local scattering models and the weighted curvature approximation

Abstract

There are several nonlocal scattering models available in the literature. Most of them are given with little or no mention of their expected accuracy. Moreover, high- and low-frequency limits are rarely tested. The most important limits are the low-frequency or the small perturbation method (SPM) and the high-frequency Kirchhoff approximation (KA) or the geometric optics (GO). We are interested in providing some insight into two families of non-local scattering models. The first family of models is based on the Meecham–Lysanov ansatz (MLA). This ansatz includes the non-local small slope approximation (NLSSA) by Voronovich and the operator expansion method by Milder (OEM). A quick review of this first family of models is given along with a novel derivation of a series of kernels which extend the existing models to include some more fundamental properties and limits. The second family is derived from formal iterations of geometric optics which we call the ray tracing ansatz (RTA). For this family we consider two possible kernels. The first is obtained from iteration of the high-frequency Kirchhoff approximation, while the second is an iteration of the weighted curvature approximation (WCA). In the latter case we find that most of the required limits and fundamental conditions are fulfilled, including tilt invariance and reciprocity. A study of scattering from Dirichlet sinusoidal gratings is then provided to further illustrate the performance of the models considered.     

The weighted curvature approximation in scattering from sea surfaces

A family of unified models in scattering from rough surfaces is based on local corrections of the tangent plane approximation through higher-order derivatives of the surface. We revisit these methods in a common framework when the correction is limited to the curvature, that is essentially the second-order derivative. The resulting expression is formally identical to the weighted curvature approximation, with several admissible kernels, however. For sea surfaces under the Gaussian assumption, we show that the weighted curvature approximation reduces to a universal and simple expression for the off-specular normalized radar cross-section (NRCS), regardless of the chosen kernel. The formula involves merely the sum of the NRCS in the classical Kirchhoff approximation and the NRCS in the small perturbation method, except that the Bragg kernel in the latter has to be replaced by the difference of a Bragg and a Kirchhoff kernel. This result is consistently compared with the resonant curvature approximation. Some numerical comparisons with the method of moments and other classical approximate methods are performed at various bands and sea states. For the copolarized components, the weighted curvature approximation is found numerically very close to the cut-off invariant two-scale model, while bringing substantial improvement to both the Kirchhoff and small-slope approximation. However, the model is unable to predict cross-polarization in the plane of incidence. The simplicity of the formulation opens new perspectives in sea state inversion from remote sensing data.     

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract

We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.     

Two comments on the structure function approximation for deep-inelastic scattering

The mechanism of radiative return to resonance can be effectively used to describe radiative corrections in terms of the Drell-Yan process. The iteration procedure is proposed. It is shown that the y→1 kinematic region can be described in terms of modified structure functions and the Sudakov formfactor, which significantly changes the result obtained both in the lowest order and with allowance for all leading orders of perturbation theory.     

Multiple scattering theory for non-local and multichannel potentials

Methodological advances in multiple scattering theory (MST) in both wave and Green's function versions are reported for the calculation of electronic ground and excited state properties of condensed matter systems with an emphasis on core-level photoemission and absorption spectra. Full-potential MST is reviewed and extended to non-local potentials. Multichannel MST is reformulated in terms of the multichannel density matrix whereby strong electron correlation of atomic multiplet type can be accounted for in both ground and excited states.     

Formal tilt invariance of the local curvature approximation

Abstract

Tilt invariance is a stringent but necessary condition that a second-order wave scattering model must satisfy in order to qualify for a broad range of applications. This invariance expresses the fact that the scattering model is unchanged whether the tilting of the scattering surface is implemented before or after its reduction to the limit of the small-perturbation method (SPM). Our scattering model is based on a second-order kernel which is quadratic in its lowest order with respect to successive derivatives of the rough surface. Hence, it is termed the local curvature approximation (LCA). We have previously demonstrated that the LCA is approximately tilt invariant in the quasi-specular and quasi-backscattering geometries. In this contribution, LCA is made formally tilt invariant up to first order in the tilting vector. It will be shown that this formal tilt invariance is achieved mainly through inclusion of polarization mixing due to out-of-plane tilt. Even though the LCA formally reduces to the SPM and Kirchhoff limits in addition to tilt invariance, its curvature kernel stays reasonably concise and practical to implement in both analytical and numerical evaluations. This curvature kernel may also be used in the other two formulations of our model, namely the non-local curvature approximation and the weighted curvature approximation.     

Sensitivity of inelastic pion scattering to non-local interactions

We show that the excitation of nuclear electric dipole states by small angle inelastic scattering of pions can be used to detect the degree of non-locality of pion nucleus scattering matrix.     

A non-local approximation to the exchange energy of the non-homogeneous electron gas

A non-local approximation to the exchange energy in an inhomogeneous electron gas is proposed which preserves the main characteristics of the correct Fermi hole. Tested for the atoms He and Ne, it gives a substantial improvement over the local approximation in the calculation of the exchange energy and the exchange energy density and reproduces closely the Hartree-Fock results. The results show that this approximation should be useful in solids.     

Coulomb scattering in the eikonal approximation

We derive a new expression for the eikonal approximation of the nuclear Coulomb scattering amplitude. Our expression should be particularly useful for the treatment of Coulomb scattering in hadron-nucleus and nucleus-nucleus scattering at high and intermediate energies.     

Electron-molecule and atom-molecule scattering within the few-body approximation

A quantum theory of few-body scattering based on the Faddeev-Yakubovsky equations is applied to the calculation of cross sections of electron and atom scattering by diatomic molecules in specified excited rovibrational states. The results of the calculations are compared with the available experimental data and other calculations.     

Validity of the impulse approximation in meson-nucleus scattering

An equation for the meson—bound-nucleon amplitude is derived which allows us to examine corrections to the impulse approximation. We also comment on an approximation where only the nuclear ground-state energy is kept in the propagators. In a model calculation the corrections to these approximations are found to be important at low energy.     

Two-photon exchange contributions to elastic ep scattering in the non-local field formalism

We construct a non-local gauge-invariant Lagrangian to model the electromagnetic interaction of the proton. The Lagrangian includes all allowed operators with dimension up to five. We compute the two-photon exchange contribution to elastic electron–proton scattering using this effective non-local Lagrangian. The one-loop calculation in this model includes the standard box and cross box diagram with the standard on-shell form of the hadron electromagnetic vertices. Besides this we find an extra contribution, which depends on an unknown constant. We use experimentally extracted form factors for our calculation. We find that the correction to the reduced cross section is slightly non-linear as a function of the photon longitudinal polarization ε. The non-linearity seen is within the experimental error bars of the Rosenbluth data. The final result completely explains the difference between the form factor ratio G_E/G_M extracted by Rosenbluth separation technique at SLAC and polarization transfer technique at JLAB.