Focusing of high-energy particles in the electrostatic field of a homogeneously charged sphere and the effective momentum approximation

The impact of the strongly attractive electromagnetic field of heavy nuclei on electrons in quasi-elastic (e, e') scattering is often accounted for by the effective momentum approximation. This method is a plane wave Born approximation which takes the twofold effect of the attractive nucleus on initial- and final-state electrons into account, namely the modification of the electron momentum in the vicinity of the nucleus, and the focusing of electrons towards the nuclear region leading to an enhancement of the corresponding wave function amplitudes. The focusing effect due to the attractive Coulomb field of a homogeneously charged sphere on a classical ensemble of charged particles incident on the field is calculated in the highly relativistic limit and compared to results obtained from exact solutions of the Dirac equation. The result is relevant for the theoretical foundation of the effective momentum approximation and describes the high-energy behavior of the amplitude of continuum Dirac waves in the potential of a homogeneously charged sphere. Our findings indicate that the effective momentum approximation is a useful approximation for the calculation of Coulomb corrections in (e, e') scattering off heavy nuclei for sufficiently high electron energies and momentum transfer.     

Eikonal analysis of Coulomb distortion in quasi-elastic electron scattering

An eikonal expansion is used to provide systematic corrections to the eikonal approximation through order 1/k ², where k is the wave number. Electron wave functions are obtained for the Dirac equation with a Coulomb potential. They are used to investigate distorted-wave matrix elements for quasi-elastic electron scattering from a nucleus. A form of effective-momentum approximation is obtained using trajectory-dependent eikonal phases and focusing factors. Fixing the Coulomb distortion effects at the center of the nucleus, the often-used ema approximation is recovered. Comparisons of these approximations are made with full calculations using the electron eikonal wave functions. The ema results are found to agree well with the full calculations.     

Long-range Coulomb forces in DIS: missed radiative corrections?

The Born approximation, one photon exchange, used for DIS (deep inelastic scattering) is subject to virtual radiative corrections which are related to the long-range Coulomb forces. They may be sizeable for heavy nuclei since Zα is not a small parameter. So far, these corrections are known only for two processes, elastic scattering and bremsstrahlung on the Coulomb field of a point-like target. While the former amplitude acquires only a phase, in the latter case also the cross-section is modified. Although the problem of Coulomb corrections for DIS on nuclei is extremely difficult, it should be challenged rather than “swept under the carpet”. The importance of these radiative corrections is questioned in the present paper. We show that, in the simplest case of a constant hadronic current, the Coulomb corrections provide a phase to the Born amplitude, therefore the cross-section remains the same. Inclusion of more realistic hadronic dynamics changes this conclusion. The example of coherent production of vector mesons off nuclei reveals large effects. So far a little progress has been made deriving lepton wave functions in the Coulomb field of an extended target. Employing available results based on the first-order approximation in Zα, we conclude that the Coulomb corrections are still important for heavy nuclei. We also consider an alternative approach for extended nuclear targets, the eikonal approximation, which we demonstrate to reproduce the known exact results for Coulomb corrections. Calculating electroproduction of vector mesons, we again arrive at a large deviation from the Born approximation. We conclude that one should accept with caution the experimental results for nuclear effects in DIS based on analyses done in the Born approximation. Received: 16 May 2001 / Accepted: 4 July 2001     

Approximate scattering solutions of the dirac equation for electron-nucleus processes in light nuclei

Scattering solutions of the second-order Dirac equation for the case of the Coulomb potential and which are correct to first order in the coupling constantZe ²/hc are investigated and found to describe pure Coulomb scattering equally well as the Sommerfeld-Maue wave functions. Errors introduced by the use of these solutions are studied in a numerical calculation of cross sections for nuclear electric-quadrupole excitation by high-energy electrons. The use of these wave functions is suggested for simplified calculations of lowest-order Coulomb corrections to Born approximation results for various electron-nucleus processes.     

Coulomb corrections for coherent electroproduction of vector mesons: Eikonal approximation

Virtual radiative corrections due to the long-range Coulomb forces of heavy nuclei with charge Z may lead to sizeable corrections to the Born cross-section usually used for lepton-nucleus scattering processes. An introduction and presentation of the most important issues of the eikonal approximation is given. We present calculations for forward electroproduction of rho-mesons in a framework suggested by the VDM (vector dominance model), using the eikonal approximation. It turns out that Coulomb corrections may become relatively large. Some minor errors in the literature are corrected.Received: 3 October 2003, Revised: 2 December 2003, Published online: 6 July 2004PACS: 25.30.-c Lepton-induced reactions - 25.30.Rw Electroproduction reactions - 13.40.-f Electromagnetic processes and properties - 25.30.Bf Elastic electron scattering     

Electron scattering on two-neutron halo nuclei: The case of He

The formalism to describe electron scattering reactions on two-neutron halo nuclei is developed. The halo nucleus is described as a three-body system (core +n + n), and the wave function is obtained by solving the Faddeev equations in coordinate space. We discuss elastic and quasielastic scattering using the impulse approximation to describe the reaction mechanism. We apply the method to investigate the case of electron scattering on ⁶He. Spectral functions, response functions, and differential cross sections are calculated for both neutron knockout and knockout by the electron.     

Eikonal expansion

An eikonal expansion of the potential scattering T matrix is evaluated, without approximation, through third order in the inverse momentum. Based on the results, their correspondence with the WKB approximation and a new statement of the unitarity constraint, we propose a sequence of four approximations to the exact impact parameter (Fourier-Bessel) representation of the scattering matrix. The sequence consists of the Glauber approximation and three systematic corrections to the Glauber approximation. The corrections are analytic functions of the impact parameter for Yukawa and Gaussian potentials; they vanish for a Coulomb potential.The sequence of eikonal amplitudes is convergent at high energy and is clearly established for small momentum transfer. Validity for all momentum transfer is conjectured based on systematic cancellation, explicitly verified through third order in the expansion, of momentum transfer dependence in the eikonal impact parameter representation. Such cancellation is shown to occur in the explicit construction of the eikonal expansion of the second Born amplitude for a Yukawa potential.Numerical tests of the sequence of eikonal amplitudes show systematic increase of the angular range of validity by comparison with partial wave results for continuous potentials; the theory is not convergent for discontinuous potentials.The WKB phase shift formula is shown to produce a systematic connection with eikonal expansion results. From this we deduce a generating function for the eikonal phase corrections of arbitrary order and also conjecture a sum of the eikonal expansion valid in the limit of high energy and arbitrary potential strength.     

Nuclear isospin mixing and elastic parity-violating electron scattering

The influence of nuclear isospin mixing on parity-violating elastic electron scattering is studied for the even–even, N=Z nuclei ¹²C, ²⁴Mg, ²⁸Si, and ³²S. Their ground-state wave functions have been obtained using a self-consistent axially-symmetric mean-field approximation with density-dependent effective two-body Skyrme interactions. Some differences from previous shell-model calculations appear for the isovector Coulomb form factors which play a role in determining the parity-violating asymmetry. To gain an understanding of how these differences arise, the results have been expanded in a spherical harmonic oscillator basis. Results are obtained not only within the plane-wave Born approximation, but also using the distorted-wave Born approximation for comparison with potential future experimental studies of parity-violating electron scattering. To this end, for each nucleus the focus is placed on kinematic ranges where the signal (isospin-mixing effects on the parity-violating asymmetry) and the experimental figure-of-merit are maximized. Strangeness contributions to the asymmetry are also briefly discussed, since they and the isospin mixing contributions may play comparable roles for the nuclei being studied at the low momentum transfers of interest in the present work.     

Hulthén势散射态的解析解

在任意l波的离心项1/r²用λ²e^-λr/(1-e^-λr)²近似表达的条件下,对Hulthén势的径向Schrdinger方程作自变量指数变换,使此转化为超几何微分方程,获得了Hulthén势s波散射态的精确解和非s波散射态的近似解析解.给出了相移的解析表达式和按“k/2π标度”归一化的用超几何函数表示的径向波函数.讨论了解析解的意义.     

Note on relativistic coulomb wave functions

Some results on the wave functions of a Dirac particle in a point charge Coulomb field are presented. Angular momentum and parity eigenfunctions are tabulated and used to construct solutions representing asymptotically a plane wave plus incoming or outgoing spherical waves. The solutions forr → 0 are exhibited in a form closely similar to the corresponding free particle spinors and are used to construct generalizations of the Casimir projection operators for positive or negative energy particles. The various wave functions presented are useful in calculations such as nuclear beta decay in which it is necessary to take into account final (or initial) state Coulomb interactions. Because of the similarity of the wave functions to those of a free particle, calculations including rigorously all Coulomb corrections for such processes as allowed beta decay can be performed with little more effort than is involved in a calculation using only plane waves.     

A systematic comparison between born approximation and distorted wave Born approximation for inelastic electron scattering

The correction factors for Coulomb effects in inelastic electron scattering are given for electric quadrupole transitions. The cross sections in Born approximation and distorted wave Born approximation are calculated in the liquid drop model for electron energies between 20 and 80 MeV and nuclei up toZ=26.     

An analytical expression of the amplitude of stripping reaction in the coulomb approximation

A method is derived for calculating the matrix element of stripping reaction (d,p) in the socalled distorted-wave Born approximation; only Coulomb scattering of incident and outgoing particles is taken into consideration. Coulomb wave functions are expressed by means of integral representation permitting integration over angular variables and, when the radial wave function of the trapped neutron is specially chosen, also calculation of the radial integral for non-zero orbital momentum of the trapped neutron. Relations are given for calculating the differential cross-section of reaction and the polarization of products. The special case, when the orbital momentum of the captured neutron is equal to one or two, is discussed in detail.     

Electron scattering on two-neutron halo nuclei: The case of 6He

The formalism to describe electron scattering reactions on two-neutron halo nuclei is developed. The halo nucleus is described as a three-body system (core +n + n), and the wave function is obtained by solving the Faddeev equations in coordinate space. We discuss elastic and quasielastic scattering using the impulse approximation to describe the reaction mechanism. We apply the method to investigate the case of electron scattering on ⁶He. Spectral functions, response functions, and differential cross sections are calculated for both neutron knockout and α knockout by the electron.     

Effects of the Dirac sea on the Coulomb sum value for electron scattering

Polarization effects of the Dirac sea on the Coulomb sum values for electron scattering are investigated within the framework of the random-phase approximation (RPA) in the relativistic σ-ω model. Divergence due to NN̄ excitations in the correlation functions is renormalized using a method of dimensional regularization. Particle-hole correlations reduce the sum values of the relativistic Hartree approximation by 15≈20%, while effects of the Dirac sea reduce them further by about 10≈20%. The quenching phenomena of the Coulomb sum values observed in medium heavy nuclei are rather well explained in the renormalized relativistic RPA.     

Charged lepton production from iron induced by atmospheric neutrinos

The charged current lepton production induced by neutrinos in ⁵⁶Fe nuclei has been studied. The calculations have been done for the quasielastic as well as the inelastic reactions assuming Δ-dominance and take into account the effect of Pauli blocking, Fermi motion and the renormalization of weak transition strengths in the nuclear medium. The quasielastic production cross-sections for lepton production are found to be strongly reduced due to nuclear effects, while there is about 10% reduction in the inelastic cross-sections in the absence of the final-state interactions of the pions. The numerical results for the momentum and angular distributions of the leptons averaged over the various atmospheric-neutrino spectra at the Soudan and Gran Sasso sites have been presented. The effect of nuclear-model dependence and the atmospheric-flux dependence on the relative yield of μ to e has been studied and discussed.     

Calculation by plane wave Born approximations of electron-impact ionization of silver and copper

The plane wave Born approximation is used to calculate total electron impact ionization cross section of silver and copper. Wavefunctions of the target and residual ions were modeled by non orthogonal Hartree-Fock and Dirac-Fock orbitals. The wave functions of the atom and residual ion are calculated with allowance for relaxation effects. The one-electron wavefunction of the continuous spectrum for the ejected electron is obtained using single-configuration Hartree-Fock and Dirac-Fock method. The orthogonalization of the ejected electron wave functions to all occupied orbitals of the target atom is performed. Results of calculations are compared to available experimental measurements and theoretical calculations performed by non relativistic one-electron PWBA, where the ejected electrons is modeled by the hydrogenic Coulomb wave function.     

Elastic scattering, muon transfer, bound states and resonances in the three-body mesic molecular systems in the reduced adiabatic hyperspherical approach

The uniform method of numerical investigation of bound states and scattering processes 2→ 2 (including resonance states) in the Coulomb three-body (CTB) systems is developed. It is based on the adiabatic hyperspherical approach (AHSA) and includes the numerical realization and applications to the three-body mesic atomic systems. The results of calculations of bound states of these systems (including the local characteristics of the wave functions) and the scattering processes 2→ 2 (including the characteristics of the resonance states) are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Eikonal expansion in electron scattering: I. Elastic scattering

A systematic eikonal expansion for the scattering of high-energy electrons from nuclei is derived which starts from the iterated Dirac equation. The resulting scattering amplitude is written in an impact parameter representation depending on eikonal phases which are proportional to inverse powers of the energy. The first two correction terms to the leading Glauber-Baker amplitude are calculated. For a Coulomb potential they agree with a sinθ-expansion of the relativistic Coulomb scattering amplitude. In the case of scattering from an extended charge distribution at sufficiently high energies numerical partial wave calculations are accurately reproduced.