Coulomb scattering amplitude

We now generalise to the case where the potential V r) has the Coulomb form at long range. We must add the Coulomb scattering amplitude (SchifF, 1955)... 

At high energies the Coulomb scattering is not the simple Rutherford cross-section from a point charge. The finite extension of the charge density of the nucleus must be taken into account. In Born approximation the Coulomb scattering amplitude from a finite sized charged distribution becomes... 

The proton-nucleus Coulomb scattering amplitude is obtained by solving the Schrodinger equation with relativistic kinematics (eq. 3.30) with the Coulomb potential from eq. (3.59). The p 4 Coulomb amplitude, to order a, is given by... 

To summarize, the relativistic kinematic prescription is to replace COM system momenta by their relativistic values, to replace the reduced mass by the reduced total energy (eq. 3.29), and to use the M0ller factor, eq. (3.37). This prescription yields the correct, relativistic proton-nucleus Coulomb scattering amplitude to order a. The usual proton-nucleus Coulomb potential must also be multiplied by the relativistic correction factor rj (eq. 3.56). 

Asymptotic Condition.—In Section 11.1, we exhibited the equivalence of the formulation of quantum electrodynamics in the Coulomb and Lorentz gauges in so far as observable quantities were concerned (t.e., scattering amplitudes). We also noted that both of these formulations, when based on a hamiltonian not containing mass renormalization counter terms, suffered from the difficulty that the... 

The first-order Bom transition amplitude for the Coulomb scattering of a Volkov electron reads ... 

To compute relativistic operators Hle, one considers scattering amplitudes both in NRQED and in full QED treating Coulomb interaction as a perturbation. By matching these amplitudes, one determines Hie. Therefore, the question of how Hiei should be computed is related to the question of how full QED amplitudes can be effectively calculated in threshold kinematics. 

The data show that the differential cross section is dominated by the direct scattering amplitude / the modulus for the spin-flip amplitude g is in general an order of magnitude smaller. Thus the spin—orbit interaction has only a small influence on the cross section, which is mainly influenced by the Coulomb interaction, exchange, and charge-cloud polarisation. 

The conclusion is that also in the higher energies level the Bom approximation finely combines with the Yukawa potential to correctly model the photonic scatterings on Coulombic manifested interaction. Worth therefore to further compute, from the scattering amplitude, the dififerential and total cross section induced by the Yukawa (and then Coulombic) potentials - a matter in the next approached. 

The above reduces to the famous Rutherford formula when the Coulomb amplitude is all that contributes to the scattering amplitude... 

Recently Elster, Liu and Thaler (ELT) [El 91] proposed a novel method for dealing with the momentum space Coulomb problem, which is, in principle, exact and may be less prone to numerical difficulties than the VP method. Their approach is based on the separation of the optical potential in eq. (3.63) and employs the two-potential formula [Ro 67] to express the full scattering amplitude as a sum of the point Coulomb amplitude and the point Coulomb distorted nuclear amplitude. The latter is obtained by numerically solving an integral equation represented in terms of Coulomb wave function basis states rather than the usual plane wave states. 

Consider the amplitude for the creation of secondary electrons upon atom excitation by electron impact. As a result of the Coulomb interaction with the atom, the incident electron loses a part of its energy and goes into an inelastically scattered, state and the atom goes into an excited state characterized by a core hole and a secondary electron. In the context of the single-electron approach, the initial state of the system is characterized by i) = w, a) and the final states are characterized by I/) = Ip, ) where u)) and h) are single-electron wave functions of the incident and inelastically scattered electrons, and p) and a) are singleelectron wave functions of the secondary election and the core level electron, respectively. Then the amplitude for creation of the secondary electron is defined by the matrix element... 

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