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.     

On the left-hand cut in potential scattering

Partial wave $\text{N}\text{D}$ equations in potential scattering are solved for the exponential, Hulthén and Morse potentials. The driving terms are taken to be either the contributions of a finite number of Born terms or the total contributions of only the nearest singularities (first n poles). For repulsive potentials one observes ghosts, anomalous bound states or resonances if the order of approximation is small with respect to the potential strength. The origin and meaning of these unphysical phenomena are explained. For attractive potentials such anomalies occur only at very large potential strengths if at all. Input-equivalent Bargmann potentials are employed to determine the quality and nature of the approximate $\text{N}\text{D}$ solutions. Rough criteria for the validity of approximations within the $\text{N}\text{D}$ approach are formulated.     

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.     

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.     

Theory of kaonic atoms

A nonlocal energy-dependent self-consistent kaon-nucleus optical potential is derived for kaonic atoms. Energy level shifts and widths are calculated for several light nuclei, and the results are compared with experiment. The sensitivity of the results to changes in parameters of the nuclear matter distribution is studied. Nonlocality and off-energy-shell effects are examined.The optical potential is derived by means of a Brueckner-type many-body theory with the independent pair approximation for the kaon and the nucleon. The two-body interaction on which the optical potential depends is represented by separable potentials of the Yamaguchi form. Coupled channels ($\text{K}\text{N}$ and Σπ) are used for the I = 0 states, which are dominated by the $\text{Y}{}_0{}^1$ resonance, and only a single channel ($\text{K}\text{N}$) is used for the I = 1 state.Calculations are carried out in three levels of approximation of the nonlocal energy-dependent optical potential. In no approximation is the potential found to be proportional to the nuclear density. Indeed, the real part of the potential changes sign in the nuclear surface. Sensitivity of the results to variations in the nuclear matter distribution is investigated and found to be on the order of experimental error. Nonlocality and off-energy-shell effects are estimated to be at least as large as this error, so that these effects must be included if one wishes to extract information about the nuclear surface from the existing experimental data. The use of correct nucleon wavefunctions and binding energies is similarly found to be essential in the calculation.     

Inelastic scattering and reaction data analysis for the 3He+44,48Ca systems

Analysing-power and cross-section distributions for the inelastic and single-nucleon transfer reactions involving the scattering of 33 MeV polarised helions from ^44,48Ca targets have been measured, and analysed with the DWBA framework. The inelastic scattering distributions are found to be insensitive to the ambiguities in the spin-orbit component of the distorting optical-model potential — which are shown to exist when fitting the elastic data. The deformation length is found to be a better representative of the nuclear shape than the deformation parameter. For the (³$\text{He}$, d) and (³$\text{He}$, α) reactions, both the zero-range and finite-range (within the local-energy approximation) calculations have been carried out and several effects such as the inclusion of non-locality correction, the j-dependence of the distributions, and the sensitivity of the predictions to the entrance- and exit-channel distorting potentials are discussed. The deduced spectroscopic factors are consistently lower than the values reported in the literature by other investigators, although the relative spectroscopic factors are shown to be in good agreement with those of others. The data measured for the charge-exchange reactions, ⁴⁸Ca(³$\text{He}$, t), populating a number of levels in ⁴⁸Sc are also presented.     

Antiproton-nucleus scattering with self-consistent model for medium corrections

The antiproton-nucleon t-matrix with propagation in the nuclear medium is calculated selfconsistently and applied to the construction of optical potentials for the elastic scattering of antiprotons from nuclei. We find that this treatment gives results that are sensitive to medium corrections even though the strong absorption acts to mask these corrections partially. The agreement with scattering experiments at 46.8 MeV on ¹²C is very good. We compare potentials containing medium corrections to those based on free $\text{p}\text{N}$ amplitudes for ¹²C and ⁴⁰Ca. The local approximation to the optical potential is found attractive at low energies, becoming shallower with increasing bombardment energy in the range considered here (up to about E_L = 120 MeV).     

The mass parameters for the average mean-field potential

Starting from a mean-field hamiltonian with pairing interaction, we use the generator coordinate method (GCM) and a generalized gaussian overlap approximation to derive a multidimensional collective hamiltonian for large-amplitude motion. Numerical calculations are performed for Nilsson and Woods-Saxon potentials with BCS pairing. The BCS wave function is taken as the generator function and the deformation parameters of the single-particle mean field are used as the generator coordinates. We find that the GCM mass parameters on the average are smaller than those of the cranking (+ BCS) model by a factor of $\text{～}\text{2}\text{3}$. In the present approach, the zero-point energy correction to the collective potential is shown to vanish identically.     

Two body reactions and the quark model

A simple quark model for two-body hadronic amplitudes is presented and applied to pp ($\text{p}\text{p}$) and K^±p elastic scattering. Hadronic processes are described in terms of quark-quark scattering, using Glauber theory to take into account the effects of multiple scattering. Exchange degenerate Regge poles are introduced at the quark level. A predictive model for Regge cuts is obtained. It has some features in common with the dual absorption model in its correlation of dip structure and shrinkage in elastic processes with the exoticity of the s-channel but also significant differences, for example in the interpretation of the $\text{p}\text{p}$ dip. The qualitatively different structure in the pp differential cross section is also reproduced. It breaks exchange degeneracy in a well-defined way and in doing so offers an explanation of the differences between pp and K⁺p total cross sections. The difference in their differential cross sections is a natural consequence of the model and structure similar to the pp structure is predicted to appear in the K⁺p differential cross section in the region t ≈ ?2(GeV/c)².     

Generalized fresnel model for very heavy ion scattering: (III). Dynamic polarization effects

It is shown that the large deviations from the typical Fresnel shape of heavy ion angular distributions, observed recently in the elastic scattering from deformed nuclei, can be described quite adequately by a very simple modification of the closed formulae for the differential cross section ratio $\text{gs(θ)}\text{σ}{}_{\mathrm{R}}\text{(θ)}$ of the generalized Fresnel model presented in parts I and II of this work. Our treatment rests basically on the relatively long range of the dynamic polarization potential that describes the Coulomb excitation to which the large deviations from the normal Fresnel pattern are attributed. We also calculate the effects of nuclear coupling in an adiabatic approximation and show that these lead to another modification, this time of the damping function F(z), which is however much smaller than that for Coulomb excitation and hardly discernible in angular distributions for very heavy ion scattering.     

Role of the 12C-core exchange in the 16O-12C elastic scattering

The exchange of the ¹²C core is taken into account explicitly in ¹⁶O-¹²C elastic scattering to explain the backward rise of the angular distributions. The equation for the relative motion of the colliding particles contains the non-local kernel due to the exchange effect. The equation is solved by the iterative method. The angular distributions are calculated at $\text{E}{}_{\mathrm{<mtext>L</mtext>}}\text{(}{}^{16}\text{O}\text{) = 24, 42, 65}\text{and}\text{80}\text{MeV}$. The results are compared with those of finite-range DWBA calculations. The effects of the multi-step transfer process, the α¹ transfer process and of the non-orthogonality term are studied. The equivalent local potential to the non-local kernel is derived under the adiabatic approximation. The resulting local potential shows the parity dependence with the exponentially decreasing behaviour at the asymptotic regions and is found to describe the low-energy scattering phenomena quite well.     

Spin-12 particles in the field of monopoles

The equations of motion of a $\text{spin}\text{-}\text{1}\text{2}$ particle in the field of a point-like (Abelian and non-Abelian) monopole are solved; helicity states and scattering solutions are constructed. The motion in an extended monopole and dyon source is discussed: charge-exchange scattering amplitudes are computed in the distorted-wave Born approximation. By the introduction of suitable spinors, the separation of variables is accomplished for any value of isospin. The non-existence of fermion zero modes in the field of an SU(2) monopole when $\text{T > 1, J >}\text{1}\text{2}$ is shown.     

Antisymmetrization effects on the effective internuclear potential

The effects of internuclear antisymmetrization in nucleus-nucleus scattering are studied by examining the structure of the exchange-Hamiltonian kernel function in the resonating group formulation. By investigating the features of the effective local potentials which are constructed to yield the same Born scattering amplitudes as the various nucleon-exchange terms in this kernel function, it is found that, among all exchange terms, the one-exchange and core-exchange terms have the largest influence. In addition, this investigation shows that the one-exchange terms, giving rise to a Wigner-type effective potential, are generally important in all scattering systems and over a wide energy range, while the core-exchange terms, giving rise to a Majorana-type effective potential, are generally important only when the nucleon-number difference of the interacting nuclei is rather small. Based on these results, it can therefore be concluded that, if a local-potential model is adopted to analyze experimental scattering results, then the real central part of the effective potential in this model must, in general, contain a Majorana exchange component or an odd-even l-dependence. Explicit resonating-group calculations in ${}^3\text{He}\text{+ α}\text{and}\text{α +}{}^{16}\text{O}$ systems, where contributions from individual exchange terms are studied, have also been performed. From this study, one finds that the conclusions mentioned above, reached in the Born approximation, may in fact be valid even in the lower-energy region.     

Solution of the Schrödinger equation in terms of classical paths

An expression in terms of classical paths is derived for the Laplace transform $\text{Ω(s)}$ of the Green function G of the Schrödinger equation with respect to $\text{1}\text{h}\text{?}$. For an analytic potential V(r), the function Ω is also analytic in the plane of the complex action variable s; its singularities lie at the values $\text{S}$ of the action along each possible (complex) classical path, including the paths which reflect from singularities of the potential. Accordingly, G may be written as a sum of terms, each of which is associated with such a classical path, and contains the factor $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This expansion formally solves the problem of constructing waves out of the corresponding (complex) classical paths. A similar expression, in terms of closed paths, is derived for the density ? of eigenvalues of the Schrödinger equation. In situations when the eigenvalues are dense, ? is well approximated by the contributions of the shortest closed paths: while the path of vanishing length corresponds to the Thomas-Fermi approximation and its smooth corrections, the other paths yield contributions which oscillate and are damped as $\text{exp}\text{(}\text{iS}\text{h}\text{?}\text{)}$. This result also holds for nonanalytic potentials V(r). If the spectrum is continuous, closed classical paths yield oscillations in the scattering phase-shift. The analysis is also extended to multicomponent wave functions (describing, e.g., motion of particles with spin, or coupled channel scattering); along a classical path, the internal degree of freedom varies adiabatically, except through points at which it is not coupled to the potential, where it may undergo discrete changes.     

Inclusive hadron-nucleus scattering at high energy

The multiple scattering series of Glauber and Matthiae for hadron-nucleus scattering is summed to a simple analytical expression. It reproduces quantitatively the main features of the experimental cross section for the reaction ${}^4\text{He}\text{+}{}^{\mathrm{A}}\text{Z →}{}^4\text{He + X}$ at 1 GeV/nucleon.     

The manifold of solutions of Roy's S- and P-wave equations for pion-pion scattering: The neighbourhood of the physical amplitudes

We study the solutions of the coupled S- and P-wave Roy equations for low-energy pion-pion scattering which are in the neighbourhood of a given solution. We give a general method for constructing these solutions for fixed inelasticities and driving terms and variable S-wave scattering lengths. In the case of the neighbourhood of the physical amplitudes, our procedure leads to a seven-dimensional manifold of solutions. If we omit variations affecting only the I = 0 S-wave around the $\text{K}\text{K}$ threshold, the number of dimensions is reduced to five, in accordance with the results of the phenomenological analysis of low-energy pion-pion scattering. Our solutions allow a study of the correlations between S- and P-waves implied by the Roy equations. This work completes a previous one dealing with the single-channel problem of the I = 1 P-wave.     

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.