Test of correlation expansions for inelastic scattering of protons from 16O and 40Ca in Glauber theory

Two different versions of a correlation expansion for the A-body nuclear transition density required to evaluate the Glauber amplitude for inelastic proton-nucleus scattering are tested. Antisymmetrized oscillator wave functions, containing only Pauli correlations, are used to calculate the “exact” amplitude as well as various terms in the correlation expansions for the excitation of the 3^? (6.13 MeV) state of ¹⁶O and the 5^? (4.49 MeV) state of ⁴⁰Ca. The leading term in both expansions, which ignores all correlations and corresponds to the Glauber theory version of the DWIA, leads to errors which are larger than present experimental errors, especially at large momentum transfers. In one version of the correlation expansion, due to Alkhazov et al., the first-order correction contains both elastic and inelastic two-body correlations and leads to satisfactory results. In the other version, used by Abgrall et al., the first-order correction contains only inelastic two-body correlations. In this case the first- and second-order corrections are needed to obtain accuracy comparable to that of the latest experiments.     

Elastic scattering of high-energy protons from nuclei

The elastic scattering of high-energy protons from zero-spin nuclei is considered within the framework of Glauber theory. The sensitivity of the cross sections to the nuclear structure and to the proton-nucleon amplitude is studied. Results of calculations demonstrating the investigated effects are presented for the case of 1 GeV proton scattering from ⁴He, ¹⁶O and ⁵⁸Ni.     

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).     

Elastic scattering of the carbon isotopes from light nuclei (Z = 7 to 16) in optical model,strong-absorption,and folding approaches

Recently obtained data for elastic scattering of ^12,13C from ¹⁴N, ¹⁶O, ²⁸Si and ³²S are analyzed by a variety of methods, including the conventional six-parameter optical model, Ericson's parameterization of the strong-absorption S-matrix, the Glauber formalism, and a simple single-folding model. It is shown that earlier analyses of similar data, in which the Glauber approach was used exclusively, have a dubious validity.     

Proton total cross sections on 1H, 2H, 4He, 9Be,C and O in the energy range 180 to 560 MeV

Proton total cross sections have been measured for the nuclei ¹H, ²H, ⁴He, ⁹Be, C and O from 180 to 560 MeV (610 to 1170 MeV/c). The standard transmission technique was used with a resulting total error of 1 % to 2 %. Statistical errors were small (< 1 %) and the major contribution to the final error comes from uncertainties in applying the correction for Coulomb-nuclear interference in elastic scattering at small angles. For ⁴He, ⁹Be, ¹²C. and ¹⁶O this experiment also gives new information on the real part of the spin-independent forward scattering amplitude for proton-nucleus elastic scattering. Total cross sections have been calculated using a Glauber model approach and poor agreement with the data is obtained, even for deuterium.     

Structure of 6Li from high-energy proton scattering

The elastic and inelastic (2.189 MeV level) scattering of 1 GeV protons from ⁶Li is predicted using Glauber multiple-scattering theory in conjunction with an α-d cluster model capable of fitting quantitatively a large number of measured form factors, in particular the most recent high-momentum-transfer electron scattering data. The predictions, especially in the inelastic channel, are quite sensitive to the parameters of the model. A valuable opportunity is thus provided via experiments well within the capabilities of LAMPF for comparison of structure information obtained from proton scattering with that from other sources in the case of a “test” nucleus whose properties are relatively well determined and unambiguous. Moreover, proton scattering goes beyond electron scattering in possibly providing information about the effective real part of the proton-deuteron cluster amplitude, and thus indirectly about D-state effects in the deuteron cluster.     

Application of the diffraction theory to proton scattering by the 15C nucleus in inverse kinematics

The calculation of the amplitude of proton elastic scattering by unstable neutron-rich nucleus ¹⁵C within the Glauber diffraction theory with the ¹⁵C nucleus wave function in the multiparticle shell model is presented. In optical limit approximation (single scattering), the differential cross section of proton elastic scattering is calculated at energies of 0.2, 0.6, and 1 GeV/nucleon. For an energy of 0.2 GeV/nucleon, the contribution of double scattering, determining the differential cross section at angles exceeding 20° is taken into account. Original Russian Text ? E.T. Ibraeva, A.M. Zhusupov, O. Imambekov, G. Nurbakova, 2009, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2009, Vol. 73, No. 6, pp. 892–897.     

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.     

Short-range correlations in high-energy hadron collisions

High-energy collisions of hadrons on nuclei are studied by means of Glauber multiple scattering theory, using Jastrow correlated wave functions. To this end a cluster expansion is derived, and its convergence properties are studied. Calculations have been carried out for ⁴He and ¹⁶O nuclei, and they are compared with experimental data.     

Quasi-elastic pion scattering on 16O: A comparison between the Δ-hole approach and a standard first-order calculation

We present calculations of quasi-elastic pion scattering on ¹⁶O in both the Δ- hole model and a standard first-order theory. The basic approach is a modified “DWIA”, where the same in-medium πN t-operator is used for the transition and the optical potential generating the distorted pion wave functions. In order to allow a meaningful comparison with the recent SIN data, all technical complications such as background interactions, non-static vertex-corrections, 1/A effects, are taken into account, and the DWIA amplitude is evaluated without further numerical simplification like, e.g., factorization. In contrast to the closure approximation, the Δ-hole model predicts correctly the doubly differential cross sections for (π, π') on ¹⁶O over a wide range of angles and energy-losses.     

Interpretation of airy minima in 16O+16O and 16O+12C elastic scattering in terms of a barrier-wave/internal-wave decomposition

The observation of refractive effects in ¹⁶O+¹⁶O and ¹⁶O+¹²C elastic scattering data has definitively established the fact that the optical potential for some light heavy-ion systems is relatively transparent and that its real part is deep. Most of the interpretations of the rainbow features of these data rely on the so-called nearside-farside decomposition of the scattering amplitude. Starting from recent optical model analyses of ¹⁶O+¹⁶O and ¹⁶O+¹²C elastic scattering around 100 MeV incident energy as an example, we present an alternative interpretation based on the barrier-wave/internal-wave decomposition first proposed by Brink and Takigawa. This method, which complements the nearside-farside approach, demonstrates clearly the exceptional transparency of the ¹⁶O+¹⁶O, and to a lesser extent ¹⁶O+¹²C, interactions at the investigated energies and makes possible the extraction of the two contributions whose interference explains the Airy oscillations seen in the farside amplitude.     

Inelastic sum rule for high-energy hadron scattering and nucleon-nucleon correlations

We have considered the influence of short-range dynamical and statistical correlations on the elastic and summed inelastic scattering of high-energy hadrons from light nuclei. Our calculations are based on the complete multiple scattering series of Glauber. The correlation effects are discussed on the example of the inelastic sum rule for p-¹⁶O scattering.     

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.     

Elastic scattering of polarised deuterons from 16O at 200, 400 and 700 MeV

Angular distributions of cross section, and A_y and A_yy analysing powers were measured for polarised deuteron elastic scattering from ¹⁶O at 200, 400 and 700 MeV. The data at 200 MeV bear evidence of the nuclear rainbow phenomenon while those at 400 and 700 MeV are reminiscent of the proton scattering results at equivalent energies. The data were analysed in terms of the optical model. The real central potential shape changes from an attractive Woods-Saxon form at 200 MeV to a wine-bottle-bottom form with a repulsive interior at 700 MeV. The total reaction cross sections deduced display a clear nuclear transparency effect in the present energy domain in agreement with predictions from the Glauber theory optical limit. Comparison with previous results for ⁴⁰Ca and ⁵⁸Ni targets is made.     

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.     

Elastic scattering of heavy ions — The simplest interpretation?

Elastic scattering of heavy ions with incident energies well above the Coulomb barrier is predicted from a simple extension of the Glauber theory for high energy particle-nucleus scattering. The only parameters needed are the “measured” nuclear density distribution function and the measured forward amplitude for free nucleon-nucleon scattering. An extensive comparison between theory and experiment is presented. It is demonstrated that the nuclear absorption needed to reproduce the measured differential cross section for elastic scattering can be well reproduced by means of the measured free nucleon-nucleon total cross section and the nuclear density distribution as “measured” for instance by electron scattering. Remarkable agreement between theory and experiment is demonstrated.     

Interference effects in proton scattering on <Superscript>15</Superscript>N nuclei at intermediate energies

The differential cross section for proton scattering on ¹⁵N nuclei is calculated within Glauber diffraction theory at energies of 0.2, 0.6, and 1.0 GeV. Use is made of the shell-model wave function for the ¹⁵N nucleus. The contribution of single and double collisions to the Glauber operator Ω is taken into account. The sensitivity of the differential cross sections to the contributions of scattering on nucleons from different shells, to the parameters of the elementary pN amplitude, and to the energy of projectile protons is investigated. It is shown that the interference between amplitudes corresponding to different collision multiplicities, as well as between the amplitudes for scattering on nucleons from different shells, determines special features of the cross section.     

On the description of center of mass motion in the Glauber model

The problem of center of mass motion of nucleus is discussed in the framework of the Glauber model, starting from a translation invariance condition. Theπ-⁴He coherent scattering amplitude is calculated to illustrate how predictions of the model are sensitive to ways of handling recoil corrections.     

Study of Different Forms of Density Distributions in Proton–Nucleus Total Reaction Cross Section and the Effect of Phase in NN Amplitude

Proton total-reaction cross-section (σR), measurements for about five nuclei in the range ¹²C to ²⁰⁸Pb at beam energies spanning 40–800 MeV have been analyzed in a systematic way by using the optical limit approximation of the Coulomb-modified Glauber multiple scattering theory. Two different phenomenological nuclear density distributions of the target nucleus in addition to the realistic one have been used in the present analysis. By applying the energy dependence in the slope parameter of nucleon–nucleon (NN) scattering amplitude, it is found that in general, the predictions of σ _R with the phenomenological Gambhir and Patil density distribution agree fairly well with the experimental data. The inclusion of phase in the NN amplitude improves the theoretical results. Our analysis shows that the calculated total reaction cross sections closely reproduce the measured data over the whole range of energy considered in this work. To validate our analyses, we have also obtained a fairly good representation of elastic p-nucleus differential scattering cross section data. The effect of a Coulomb energy shift in the proton beam has also been studied.