Abstract:
It is shown in this paper that one does not need to use just exponential dumping factor when computing the Rutherford formula within Born approximation. Text, which is very simple, might be of interest for physics students as well as for physics teachers.

Abstract:
Using Fermat's least optical path principle the family of ray-trajectories through a special but common type of a gradient refractive index lens, n(r)=n_0+\Delta n R/r, is solved analytically. The solution, i.e. the ray-equation r(phi), is shown to be closely related to the famous Rutherford scattering and therefore termed photonic Rutherford scattering. It is shown that not only do these classical limits correspond, but also the wave-mechanical pictures coincide: The time-independent Schr\"odingier equation and the inhomogeneous Helmholz equation permit the same mapping between massive particle scattering and diffracted optical scalar waves. Scattering of narrow wave-packets finally recovers the classical trajectories. The analysis suggests that photothermal single particle microscopy infact measures photonic Rutherford scattering in specific limits.

Abstract:
A massless field propagating on spherically symmetric black hole metrics such as the Schwarzschild, Reissner-Nordstr\"{o}m and Reissner-Nordstr\"{o}m-de Sitter backgrounds is considered. In particular, explicit formulae in terms of transcendental functions for the scattering of massless scalar particles off black holes are derived within a Born approximation. It is shown that the conditions on the existence of the Born integral forbid a straightforward extraction of the quasi normal modes using the Born approximation for the scattering amplitude. Such a method has been used in literature. We suggest a novel, well defined method, to extract the large imaginary part of quasinormal modes via the Coulomb-like phase shift. Furthermore, we compare the numerically evaluated exact scattering amplitude with the Born one to find that the approximation is not very useful for the scattering of massless scalar, electromagnetic as well as gravitational waves from black holes.

Abstract:
By considering a cylindrically symmetric generalization of a plane wave, the first Born approximation of screened Coulomb scattering unfolds two new dimensions in the scattering problem: transverse momentum and orbital angular momentum of the incoming beam. In this paper, the elastic Coulomb scattering amplitude is calculated analytically for incoming Bessel beams. This reveals novel features occurring for wide angle scattering when the incoming beam is correctly prepared. The result successfully generalizes the well known Rutherford formula, incorporating transverse and orbital angular momentum into the formalism.

Abstract:
After one century of nuclear physics, the anomalous Rutherford scattering remains a puzzle: its underlying fundamental laws are still missing. The only presently recognized electromagnetic interaction in a nucleus is the so-called Coulomb electric force, in 1/r, only positive thus repulsive in official nuclear physics, explaining the Rutherford scattering at low kinetic energy of the impacting alpha particles. At high kinetic energy the Rutherford scattering formula doesn’t work, thus called “anomalous scattering”. I have discovered that, to solve the problem, it needs only to replace, at high kinetic energy, the Coulomb repulsive electric potential in 1/r, by the also repulsive magnetic Poisson potential in 1/r^{3}. In log-log coordinates, one observes two straight lines of slopes, respectively ？2 and ？6. They correspond with the ？1 and ？3 exponents of the only repulsive electric and magnetic interactions, multiplied by 2 due to the cross-sections. Both Rutherford (normal and anomalous) scattering have been calculated electromagnetically. No attractive force needed.

Abstract:
The first and second Born approximation are studied with the path integral representation for $ {\cal T} $ matrix. The $ {\cal T}$ matrix is calculated for Woods-Saxon potential scattering. To make corresponding integrals solvable analytically, an approximate function for the Woods-Saxon potential is used. Finally it shown that the Born series is converge at high energies and orders higher than two in Born approximation series can be neglected.

Abstract:
We prove the existence of scattering solutions for multidimensional magnetic Schr\"odinger equation which belong to the weighted Sobolev space H^1_s (R^n)(n=2,3) with some s < -1/2. As a consequence of this we formulate the direct Born approximation for the magnetic Schr\"odinger operator. Possible connections with inverse problems (inverse scattering Born approximation) are discussed.

Abstract:
The paper discusses the applicability of WKB and Born (small perturbations) approximations in the problem of the backscattering of quantum particles and classical waves by one-dimensional smooth potentials with amplitudes small compared to the energy of the incident particle (above-barrier scattering). Both deterministic and random potentials are considered. The dependence of the reflection coefficient and localization length on the amplitude and the longitudinal scale of the scattering potential is investigated. It is shown that perturbation and WKB theories are inconsistent in the above-barrier backscattering problem. Not only the solutions but the regions of validity of both methods as well depend strongly on the details of the potential profile, and are individual for each potential. A simple criterion that allows determining the boundary between the applicability domains of WKB and Born approximations is found.

Abstract:
We study the effect of radiation damping on the classical scattering of charged particles. Using a perturbation method based on the Runge-Lenz vector, we calculate radiative corrections to the Rutherford cross section, and the corresponding energy and angular momentum losses.

Abstract:
The hadronic quark structure is investigated in the frame of high energy electron proton scattering. A phenomenological model based on the Born approximation is used to calculate the transition matrix element for the quark system forming the proton target. A potential of electromagnetic nature is assumed for the calculation of the multiple scattering of electron with the constituent valance quarks of the proton. It is found that the first two terms of the Born approximation are sufficient to describe the experimental data of differential cross section for the electron proton system only at low momentum transfer square. On the other hand, a two body scattering amplitude with a relevant form factor may give proper agreement extended to high momentum transfer square region. A harmonic oscillator wave function is used to reproduce the data of the experiment at low momentum transfer. This satisfies quietly the confinement condition of the quarks in the hadron system. However the data of the experiment at relatively high momentum transfer, show that hadronic quarks behave more freely and may be described by just a Coulomb wave function. PACS: 13.40.Fn , 25.40.Cm