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.     

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.     

The Extraction of Neutron Target Cross Sections from Reactions on Deuterium

The theory of scattering and production reactions on deuterium is developed with a view to the extraction of neutron target amplitudes from analysis of deuterium experiments. Corrections to the spectator model due to the following effects are examined: target particle binding corrections to the Impulse Approximation; (ii) the Pauli principle in charge exchange reactions; (iii) internal or „Fermi”︁ motion of the target particles; (iv) multiple scattering and final state interactions. The closure approximation and the general analysis of single-arm spectrometer experiments are discussed. The connection at high energies and small momentum transfers between the general Watson Theory and the Glauber model is established, and the results of calculations of deuterium corrections for pion and rho photoproduction at high energies are reviewed. “Fresnel” and recoil corrections to Glauber theory are examined. An approximation scheme suitable to the analysis of high energy wide angle coincidence or bubble chamber experiments is developed. A special examination is made of the breakdown of the impulse approximation in the 3-3 resonance region, and of the applicability of lowest order binding and in multiple scattering corrections in this sensitive kinematical region.     

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.     

Glauber amplitude for ionization of helium by electron impact

The high-energy Glauber approximation is used to derive the formula for scattering amplitude for ionization of helium atom by electron impact. The scattering amplitude is expressed as a one dimensional integral involving MeijerG-functions. This may further be expressed in a closed form as simple sums of Meijer functions by writing down the series expansion for Bessel and Meijer functions. Further, the asymptotic behaviour of these amplitudes is examined for both large and small momentum transfers.     

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.     

Noneikonal effects in spin-dependent proton-nucleus interactions

Simple expressions are derived for the non-eikonal corrections to the Glauber diffraction approximation for the proton-nucleus scattering amplitudes, with the spin dependence of the proton-nucleon amplitudes taken into account. As an example, we study the numerical importance of these corrections for elastic p ⁵⁸Ni scattering at 800 MeV.     

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).     

Multiple scattering with a linearized propagator

Scattering by a many-body system is studied within the framework of the “fixed scatterer” approximation and the eikonal approximation formulated in terms of a linearized propagator. If properly treated, the “fixed scatterer” approximation is able to take into account the center-of-mass motion. We specifically study the linearized propagator proposed by Abarbanel and Itzykson. Although for potential scattering the above approximation is essentially equivalent to the Glauber eikonal approximation, its physical implications are quite different when applied to scattering by a composite system. The multiple-scattering series can generally no longer be simply expressed in terms of the individual on-shell scattering amplitudes, and the additivity of phase shifts is shown to break down for overlapping potentials. The implications for phenomenological calculations are discussed. Finally, the above approximation is explicitly applied to high-energy elastic nucleon-deuteron scattering and the results are compared with several variants of the Glauber multiple-scattering formalism.     

HIGH ENERGY NUCLEUS-NUCLEUS ELASTIC SCATTERING

We calculate the differential cross section of α-α elastic scattering with the complete expansion formula of the Glauber amplitude of high energy nucleus-nucleus elastic scattering and the rigid projectile approximation. The results are satisfactory.     

无角度限制的修正的Glauber公式

本文重新推导了Glauber理论中弹性散射振幅公式。如果取直角光学位势作为零级近似,取Woods-Saxon光学位势作为正确的光学位势,发现在原来的散射振幅中,分部积分能够进一步进行下去。最后,得到了修正后的散射振幅公式,而无角度限制。 关键词：     

Off-shell corrections to the glauber theory of nuclear multiple scattering

We study the first-order off-shell corrections to Glauber's multiple scattering model. These consist of a direct and a rescattering term, and are obtained from the eikonal corrections studied in a previous paper by a simple substitution. For an independent particle nuclear model the direct term becomes identical to an expression previously derived by Moniz. We show that the rescattering term is in general not small compared to the direct term. For high incident energy a comparison is made with other corrections to the Glauber model. An optical potential is proposed which consists of the Glauber potential and its off-shell, eikonal, Fresnel and spatial correlation corrections.     

An impact parameter representation for the off-energy-shell scattering amplitude

The unitarized Born approximation for the scattering amplitude, suggested previously, is considered in the short wavelength limit. It is found that its on-energy-shell component is equivalent to the well-known impact parameter representation developed by Glauber. An approximate expression is then derived for the off-energy-shell scattering amplitude on the basis of the unitarized Born approximation which is expected to be as accurate as the Glauber approximation.     

Green function and scattering amplitudes in many-dimensional space

Methods for solving scattering are studied in many-dimensional space. Green function and scattering amplitudes are given in terms of the required asymptotic behaviour of the wave function. The Born approximation and the optical theorem are derived in many-dimensional space. Phase-shift analyses are performed for hypercentral potentials and for non-hypercentral potentials by use of the hyperspherical adiabatic approximation.     

Theory of hadron-deuteron elastic scattering in the GeV region

A multiple scattering formalism for the medium energy hadron-deuteron elastic scattering is developed. This approach provides an extension of the Glauber theory incorporating the complete spin structure and the corrections to the eikonal and fixed scatterer approximations. The systematics of the spin observables and their relation to the hadron-deuteron as well as to hadron-nucleon amplitudes are discussed. An approximate cancellation between few corrections to the double scattering term is found which leaves the non-eikonal correction as the dominant one, and often, very important numerically, e.g., in the deuteron tensor polarization. Comparison of the parameter-free predictions of the theory based on the recent NN phase shift amplitudes with the data on polarization observables in pd elastic scattering is made.     

Scattering of pions by <Superscript>9</Superscript>Ве nuclei in the context of the Glauber diffraction theory

The differential cross section of the elastic scattering of π^±-mesons by the ⁹Ве nucleus at energies of 0.6 and 1.0 GeV are calculated using the Glauber theory of multiple scattering. The wave function of the ⁹Ве nucleus is obtained in the three-cluster 2α-n model. The sensitivity of the differential cross sections to the wave functions of a target nucleus, calculated with different intercluster interaction potentials, and to the contribution from different orders of scattering in Glauber operator Ω is studied.     

Elastic-scattering cross-section for electrons and positrons from atomic hydrogen

The differential cross-sections of atomic hydrogen for elastic scattering of electrons and positrons have been rederived with the help of a method using a single parameter-dependent unitary shift operator for the calculation of the direct contribution. When the parameter approaches zerc the new method leads to the well-known conventional Glauber results. The numerical calculations include polarization effects and the exchange corrections obtained according to alternative approximation methods. Results calculated with Franco’s exchange show a definite improvement over the earlier results for medium energy electrons at large angles of scattering. Total elastic cross-sections have been calculated for 50 and 100eV electrons and positrons.