Multiple scattering in dilute systems

Multiple scattering in the limit in which the constituent scatterers are far removed from each other is studied. It is seen how this situation leads, in the Watson multiple-scattering series, to the dominance of on-shell propagation between the constituents. It also yields a version of the Glauber multiple-scattering theory in which no assumption is made to the effect that the individual projectile-constituent scattering amplitudes are eikonal in form. This approach allows for the immediate extraction of Fresnel corrections to the Glauber result and explains several apparent contradictions as to the relationship between the Watson multiple-scattering series approximated with on-shell propagation only and the Glauber theory.     

Eikonal and Fresnel corrections to the Glauber theory of nuclear multiple scattering

Starting from a Watson-type multiple scattering series we study first-order corrections to the Glauber high-energy collision model. These are obtained by replacing the free-space Green function between successive scatterings by the second-order eikonal propagator. We distinguish between Fresnel corrections and eikonal-type corrections. Due to large cancellation effects the eikonal contribution is very small compared to the Fresnel contribution. First-order corrections to Glauber theory are calculated explicitly for high-energy elastic scattering of protons from light nuclei (⁴He, ¹²C, ¹⁶O).     

Corrections to the Glauber multiple scattering series

The asymptotic accuracy of the Glauber multiple scattering formula is examined in the fixed scatterer approximation in one dimension. If the potentials overlap, the off-shell corrections that dominate are asymptotically of the same order as the corrections to the eikonal approximation results. In the case of nonoverlapping potentials, the corrections to the Glauber formula arise from multiple scattering. The asymptotic behavior of these latter corrections depend essentially on the smoothness of the potentials.     

Multiple scattering of heavy ions,Glauber theory,and optical model

An exact multiple-scattering series for the scattering of two heavy ions is derived and compared to the Glauber approximation. An optical model potential operator is indicated which is an accurate approximation to the multiple-scattering formalism.     

Eikonal theory of electron- and positron-atom collisions

We review recent developments in the application of eikonal methods to the field of electron and positron collisions with atoms. The foundations of the eikonal approximation are first analyzed within the framework of potential scattering, with particular attention to those aspects of the theory which can be generalized to atomic collision processes. We next discuss various many-body applications of the eikonal method, namely: the Glauber approximation, the eikonal-Born series method, optical model theories, the eikonal distorted wave method and the multistate eikonal approximation. We also analyze eikonal exchange amplitudes. Applications of these methods are then considered, first for the case of elastic scattering and then for various inelastic processes.     

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.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

Multiple scattering series in the forward and/or backward direction

A useful formulation of potential scattering is presented in terms of multiple scattering series. This formulation starts from an approximation of the exact Green function in momentum space by the sum of two propagators (forward and backward propagators). This replacement leads naturally to both the forward and backward approximation in the same way. The Fresnel diffraction is incorporated with the to-and-fro motion (reflection) in a potential. High-energy corrections are also presented to the eikonal approximation. This formulation permits a straightforward application to scattering off a composite system, and has a close relationship to the Glauber approximation.     

Multi-step processes in high-energy elastic and inelastic nuclear scattering

Multi-step processes in elastic and inelastic nuclear scattering at intermediate and high energies are investigated using a formulation whereby a finite number of channels are explicitly treated while all the other channels are approximately accounted for through a “second-order potential matrix”. Within the framework of the eikonal approximation the problem reduces to a finite system of first-order coupled integro-differential equations with non-local potentials which depend on the two-body density matrix of the target nucleus. The relationship of the above formulation to the DWIA, the close-coupling method, and the Glauber multiple scattering model is examined. This approach is applied to the elastic and inelastic (2⁺, 4.43 MeV) scattering of 1 GeV nucleons by ¹²C. The corrections to the DWIA are sizeable, and the inelastic scattering appears to be very sensitive to the multi-step contributions and the nuclear structure.     

Correlations in alpha-alpha scattering and semi-classical optical models

We show the equivalence of semi-classical solutions to optical model coupled-channel equations derived from Watson's form of the nucleus-nucleus multiple-scattering series to the Glauber multiple-scattering series. A second-order solution to the semi-classical coupled-channel elastic amplitude is shown to be nearly equivalent to a second-order optical-phase-shift approximation to the Glauber amplitude if the densities of all nuclear excited states are approximated by the ground-state density. Using the Jastrow method to model the two-body density we find an average excited-state density to be of negligible importance in the double-scattering region of alpha-alpha scattering.     

e−-H(2S) elastic scattering in the two-potential eikonal approximation

The differential scattering cross-sections fore ^??H (2S) elastic scattering are calculated at intermediate energies by using the two-potential eikonal approximation. The results are compared with the recent theoretical data and the conventional Glauber cross-sections.     

A RELATIVISTIC CORRECTION TO THE GLAUBER THEORY AND THE HIGH ENERGY BEHAVIOR OF THE SCATTERING AMPLITUDE

In this paper, the basis of the Glauber theory is changed from the schrodinger equation to a relativistic equation, which is then solved using eikonal approximation to investigate the high energy behavior of the scattering amplitude and its relationship with phenomenological nuclear forces. A correction to the Glauber theory including the modification of transverse and longitudinal momentum transfer is also proposed.     

Non-eikonal corrections in potential scattering on two centres

Scattering of a spinless projectile from a system of two (over-lapping or non-overlapping) local potentials is studied in the first-order eikonal expansion. Corrections to the Glauber formula are found to come mainly from the non-eikonal propagation in the “direct” double scattering, and not from multiple scattering (“reflections”).A comparison is made between the exact (to the leading order) and two approximate results: (i) based on the theory of scattering from non-overlapping potentials, and (ii) neglecting the reflection terms and adopting the local “Born-like” off-shell extrapolation of the two-body amplitudes. The second approximation is found to be more accurate and technically simpler in calculation of multiple scattering from many-body systems.     

Four-body multiple-scattering expansion of the total transition amplitude—MST

The elastic scattering differential cross section is calculated for proton scattering from ⁶He at 717 MeV, using single-scattering terms of the multiple-scattering expansion of the total transition amplitude. We analyze the effects of different scattering frameworks, specifically the factorized impulse approximation and the fixed scatterer (adiabatic) approximation. The text was submitted by the authors in English.     

Comparison of eikonal-Born series and modified Gluaber approach for the study of elastic electron-hydrogen scattering

Elastic electron-hydrogen scattering at medium and high energies has been analysed in eikonal-Born series and modified Glauber approach forE _i ≥ 50 eV. We have used closed form expressions for evaluating the second Born term in the fixed scatterer approximation and for higher order terms of Glauber-eikonal series. The exchange effect has also been included via Glauber-Bonham-Ochkur type procedure. The fixed scatterer approximation calculations are compared with results obtained using other approximations for the second Born term available in the literature. The results compare fairly well with experimental data.     

Second-order eikonal corrections for

The first-order eikonal approximation is frequently adopted in interpreting the results of A(e,e^′p) measurements. Glauber calculations, for example, typically adopt the first-order eikonal approximation. We present an extension of the relativistic eikonal approach to A(e,e^′p) which accounts for second-order eikonal corrections. The numerical calculations are performed within the relativistic optical model eikonal approximation. The nuclear transparency results indicate that the effect of the second-order eikonal corrections is rather modest, even at Q²≈0.2 (GeV/c)². The same applies to polarization observables, left–right asymmetries, and differential cross sections at low missing momenta. At high missing momenta, however, the second-order eikonal corrections are significant and bring the calculations in closer agreement with the data and/or the exact results from models adopting partial-wave expansions.     

Lepton scattering off few-nucleon systems at medium and high energies

The interpretation of recent JLab experimental data on the exclusive process A(e,e′p)B off few-nucleon systems is analyzed in terms of realistic nuclear wave functions and the Glauber multiple scattering theory, both in its original form and within a generalized eikonal approximation. The relevance of the exclusive process ⁴He(e,e′p)³H for possible investigations of QCD effects is illustrated.     

The fixed scatterer approximation and nucleon deuteron scattering

A method of solving the Schroedinger Equation for the scattering from two fixed local potentials is presented. The solutions are used within the framework of the fixed scatterer approximation to perform model calculations of N-D scattering using both effective range theory potentials and a “semi-realistic” potential with a strong repulsive core. For smooth potentials approximate solutions to the fixed scatterer problem are proposed and found to be quite accurate.Other procedures for calculating elastic scattering were compared with the exact fixed scatterer approximation. The results show that the neglect of longitudinal momentum transfer in the Glauber multiple diffraction theory is a severe effect except in the forward direction. For small angle scattering the Glauber prediction for the double scattering amplitude is quite accurate, and does not depend strongly upon either the extent of potential overlap, or the ratio $\text{V}\text{E}$. Comparisons with the Agassi and Gal results for nonoverlapping potentials indicate that the effects of potential overlap are important, at least for the lowest partial waves. Conclusions about the overall importance of off-energy shell effects in nucleon-deuteron scattering, and about the interference of these effects with the determination of the correlation structure of nuclei are not free from ambiguities.     

Pion scattering on medium-weight nuclei and distortion effects in double charge exchange reactions

An eikonal version of the DWIA is used to compute absorption and dispersion effects on the cross section for double charge exchange to its double analogue. Lacking information on the π?⁵¹V optical potential, recent data on π?⁴⁰ Ca elastic scattering have been analyzed in the Glauber approximation and the extracted elastic phase function has been scaled up to correspond to ⁵¹V. The result is used in a computation of the DWIA amplitude. Large reduction factors are obtained in agreement with experimental upper bounds. Discrepancies with previous estimates are discussed.