Fixed Scatterer Approximation Research Articles

Few-body multiple scattering calculations forHe6on protons

The elastic scattering of the halo nucleus $^{6}\mathrm{He}$ from a proton target at 717 MeV/nucleon is investigated within three different multiple-scattering formulations of the total transition amplitude. The factorized impulse approximation (FIA) and the fixed scatterer approximation (FSA) of the multiple-scattering expansion are used to evaluate accurately the single-scattering terms and to test the validity of a few-body Glauber approach. The latter also includes terms beyond single scattering and the importance of these terms is investigated. The differential cross section is calculated for proton scattering from $^{6}\mathrm{He}$ at 717 MeV in inverse kinematics and compared with recent data.

Recoil and correlations in pion double scattering

The contribution to the optical potential due to double scattering of a pion by a pair of correlated nucleons has been calculated for nuclear matter taking account of nucleon and delta kinetic energies. The calculation includes a variety of field theoretical effects due to crossed processes and all possible time sequences of the two scattering events including overlap in time. For sequential processes, where the first scattering is complete before the second starts, it is found that inclusion of kinetic energies of the nucleon and delta reduces the contribution drastically. The contribution of these sequential processes is about 2% of the first-order optical potential as contrasted with about 50% at ${T}_{\ensuremath{\pi}}=140$ MeV when the fixed scatterer approximation is employed. The most important correlated double scattering contributions arise from the field theoretical effects which yield about 8% of the first-order optical potential at ${T}_{\ensuremath{\pi}}=0 \mathrm{and} 140$ MeV. Thus, the net contribution of correlated double scattering processes is about 10% of the first-order optical potential. Since the double scattering of the pion by a correlated nucleon pair is a major contributor to the Ericson-Ericson effect, the present work suggests reduced importance of the effect.NUCLEAR REACTIONS Correlations, fixed scatterer approximation, LLEE effect, double scattering, recoil, pion scattering.

S-wave repulsion in the pion-nucleus optical potential and the subthreshold pion-nucleonTmatrix

A covariant theory of the pion-nucleus interaction has been used to calculate off-shell matrix elements of the first-order pion-nucleus optical potential in momentum space. The matrix elements $〈k|{v}_{l}(W)|k〉$ are calculated for $^{16}\mathrm{O}$ and $^{40}\mathrm{Ca}$, as a function of $k$, the momentum in the $\ensuremath{\pi}$-nucleus center-of-mass system, for $W={m}_{\ensuremath{\pi}}+{M}_{N}$. (As in our previous studies we have carried out a full integration over the Fermi motion of the target nucleons. Off-shell effects related to nuclear binding are treated carefully.) As this calculation is sensitively dependent upon the subthreshold $\ensuremath{\pi}NT$ matrix, two different models for the $T$ matrix are studied. One is the well known separable model of Londergan, McVoy, and Moniz. The second model includes terms describing the direct and crossed Born terms and therefore has the appropriate $s$- and $u$-channel nucleon poles. (In the calculations using this model both the pion and the nucleon are allowed to go off the mass shell.) The remaining part of the amplitude in the second model is represented by a separable form. For small $k$, the matrix elements of the optical potential calculated with these two models are significantly larger than those obtained when use is made of the fixed-scatterer approximation. Phenomenological optical potentials which describe low-energy pion-nucleus scattering data or pionic-atom data exhibit $s$-wave repulsion which is about 2-8 times that obtained from calculations made in the fixed-scatterer approximation. (The uncertainty in this enhancement factor reflects the uncertainty in the free-space values of the $\ensuremath{\pi}N$ scattering lengths). We conclude that the explanation of a significant fraction of the $s$-wave repulsion in the low energy pion-nucleus interaction may lie in a careful treatment of off-shell effects using a good model for the subthreshold $\ensuremath{\pi}NT$ matrix.NUCLEAR REACTIONS Pion-nucleus interactions, off-shell effects in the pion-nucleus interaction at low energy; pionic atoms.

Theory of the optical potential for strongly interacting particles

We use the insight gained from the study of analyticity and unitarity to suggest a new method for the construction of the optical potential for strongly interacting particles. The calculational procedures suggested in this work are such as to associate a distinct physical process with each term in our expansion of the optical potential. By starting with various decompositions of the A-body target wave function we can obtain theories of varying complexity. Introducing a decomposition with respect to states of the (A-1) -body system, we obtain a construction of the optical potential based upon the single-scattering approximation. If we introduce a decomposition of the target ground state with respect to states of the (A-2) -body system we obtain a more complex theory. In the extended theory, the first-order optical potential of the simpler model is readily extracted along with a second-order potential that depends on the detailed structure of the two-body density matrix of the target. The role of Pauli and short-range correlations in modifying the calculation of the second-order potential is discussed. The physical picture that emerges from this analysis is different from that obtained from multiple-scattering theory, although the two theories are in general agreement as to themore » nature of the first-order optical potential. When the fixed-scatterer approximation is used to simplify calculations based upon multiple-scattering theory, the second-order optical potential is found to depend on the two-body correlation function. In contrast, for the theory presented here, we find that the inclusion of Pauli and/or short-range correlations is a relatively minor aspect in the construction of the second-order potential. Indeed, we would obtain a second-order potential for an uncorrelated system.« less

Quantum theory of inelastic multiple scattering in phase space

We propose a general framework in which inclusive inelastic scattering of medium- or high-energy hadrons on nuclei can be discussed, without resorting either to the fixed scatterer approximation, or to a frankly classical description. A combination of Watson's multiple scattering theory and the Wigner representation—which had to be extended for this purpose—enables us to visualize the reaction in phase space as a series of inelastic two-body collisions, separated by propagations in an optical potential. The inclusive multiple scattering series can be summed up in principle by solving a one-body integral equation in phase space. This “cascade equation” is derived in a simple nuclear model, and shown to be equivalent to a transport equation for the Wigner density matrix of the projectile. Quantum-mechanical interference and wave effects can be treated systematically within this scheme. In the static, eikonal approximation, we recover the results of Glauber theory, and in the classical limit a particular form of the Boltzmann equation.

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.

Scattering lengths of pionic 3He and 4He

Scattering lengths of pionic 3He and 4He and the charge exchange contribution to the 1s width of pionic 3He are calculated within the fixed scatterer approximation of the multiple scattering formalism. Particular attention is focussed on the nuclear physics part and on πN p-wave contributions. For the first time triple scattering and double-spin-flip contributions have been included. We find significant deviations from previous estimates and calculations. Good agreement is achieved with the experimental π −3He scattering length, whereas in the case of 4He a repulsive dispersion contribution is clearly needed. We propose to use the measured 1s level shift of pionic 3He as a constraint to deduce a precise value of the isoscalar πN scattering length. Furthermore, we find that multiple scattering reduces the impulse approximation value for Γ 1s( π −3He → π 0 3H) by more thsn 20 %. This result casts some doubt on impulse approximation calculations of radiative pion capture as well.

Comments on the fixed-scatterer approximation toe−-H scattering

It is pointed out that the fixed-scatterer approximation (FSA) does not underestimate the e/sup -/-H scattering cross section near the forward direction. The FSA result rather suffers from a logarithmic divergence in the forward direction which comes, as is well known, from the double-scattering term.

On the Δ-resonance in the two-nucleon system

The excitation spectrum of the two-nucleon system in the region of the Δ-resonance is studied by means of a simple field-theoretical model. It is shown how the properties of the free Δ-resonance can be strongly modified by the presence of another nucleon. The total crosssection for the elastic scattering of a pion on the two-nucleon system is calculated in the fixed scatterer approximation in various channels and in the case of the deuteron. Finally the effective Δ-N potential is deduced and the results are discussed also in connection with other similar previous works on the subject.

How useful is the fixed-scatterer approximation in pion physics?

How useful is the fixed-scatterer approximation in pion physics?

Correlation expansion of the optical potential

The multiple scattering theory for the optical potential is examined. This series is arranged according to the number of target particles struck in forming the optical potential. The term which involves two target particles is summed as a three-body problem. Explicit formulas for calculating the optical potential in the fixed scatterer approximation are presented. Corrections to the fixed scatterer approximation, one a correction to closure, another a correction due to nonlocality in the two-body interaction, are presented. The relation between this work and other formal,rearrangements of the multiple scattering series is presented.

Can multiple-scattering theory explain the low-energy pion-nucleus interaction?

We discuss the 50 MeV data for the elastic scattering of pions by $^{4}\mathrm{He}$ and $^{12}\mathrm{C}$ and show that a careful treatment of off-shell effects and the effects of pion absorption, in the context of a multiple-scattering theory, is able to explain the data. We present a discussion of the importance of nuclear binding effects and describe how the Fermi motion influences the calculation of the optical potential. (In this work we have performed a complete integration over the variables which describe the Fermi motion of the target nucleons.) It may be seen from this discussion how the neglect of these features in the fixed scatterer approximation leads to large discrepancies between the results of theoretical calculations and the experimental data.[NUCLEAR REACTIONS Low-energy $\ensuremath{\pi}$-nucleus elastic scattering; 50 MeV $\ensuremath{\pi}\ensuremath{-}^{4}\mathrm{He}$ and $\ensuremath{\pi}\ensuremath{-}^{12}\mathrm{C}$ differential cross sections. Influence of Fermi motion, nuclear binding, and pion absorption.]

Simple Form of the Fixed-Scatterer Approximation: An Application toe−-H Scattering

A tractable form of the fixed-scatterer approximation retaining up to double-scattering terms has been proposed. This formalism can be employed to investigate the elastic as well as inelastic processes at intermediate and high energies. The results for the differential cross section for the elastic scattering and for excitation of the $n=2$ states of H atom in ${e}^{\ensuremath{-}}$-H scattering are found to be very encouraging, computational labor involved for $s\ensuremath{-}s$ transitions being comparable to that of the first Born calculations.

Scattering from bound states and the validity of fixed scatterer models

For hadron-nucleus scattering one frequently employs the fixed scatterer approximation (FSA). We show that for non-local interactions this approximation violates Galilean invariance, leading to both formal and numerical inconsistencies. However we have verified with a simple but realistic model the accuracy of the closure approximation on the target Hamiltonian both with separable and local interactions. Only for local interactions is the FSA equivalent to closure.

Multiple-scattering analysis of theπ−Hesystem at high energies

Elastic scattering of negative pions by $^{4}\mathrm{He}$ nuclei at 1.12 GeV is analyzed using the multiple scattering theory. The contribution of the higher order optical potentials is found to be large in the region of the second diffraction maximum; it comes mainly from the center of mass pair correlations enhanced by a large relativistic kinematic correction. The effects of the dynamical correlation, the fixed scatterer approximation as a result of the closure, and the off-shell property of the $\ensuremath{\pi}N$ amplitude are found to be negligible. The calculated cross section is in fair agreement with the experiment for ${q}^{2}\ensuremath{\lesssim}0.4$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$.NUCLEAR REACTIONS ${\ensuremath{\pi}}^{\ensuremath{-}}^{4}\mathrm{He}$ elastic scattering at 1.12 GeV; analyzed by multiple-scattering theory, effect of higher-order optical potentials.

Covariant calculation of pion-He4elastic scattering in the (3, 3) resonance region

Differential cross sections for ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{-}}^{4}\mathrm{He}$ elastic scattering were calculated at energies between 110 and 260 MeV. The analysis is completely novel since we have used a covariant multiple scattering theory, and a complete integration over the motion of the target particles has been carried out (instead of invoking the fixed-scatterer approximation). The full dynamical effects of nuclear binding and Fermi motion are considered as well as some modification of the free $\ensuremath{\pi}N$ interaction. Theoretical results are in good agreement with the data.NUCLEAR REACTIONS Covariant optical potential, ${\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{-}}^{4}\mathrm{He}$ elastic scattering from 110 to 260 MeV.

Model for scattering from the bound state of a two-body system

An exactly solvable model for scattering of a particle from the bound state of a two-body sysem is studied. In this model the motion of all particles is confined to one dimension. It is also assumed that the interaction between the projectile and the target is nonlocal and separable, but the potential between the target particles is local. Depending on the form of the latter potential two cases are considered: (1) the target is the ground state of a system whose eigenvalues are discrete and eigenfunctions localized, and (2) the target has only one bound state. In both cases the problem can be solved with overlapping as well as nonoverlapping potentials. This model is used to test the accuracy of the fixed scatterer approximation. It turns out that in this approximation the two cases yield similar results.NUCLEAR REACTIONS One-dimensional model. Separable forces. Fixed scatterer approximation.

Optical potentials derived from microscopic separable interactions including binding and recoil effects

We first consider a projectile scattering from a nucleon bound in a fixed potential. A separable Galilean invariant projectile-nucleon interaction is adopted. Without using the fixed scatterer approximation or using closure on the intermediate target nucleon states we obtain various forms for the projectile-bound nucleon $t$ matrix. Effects due to intermediate target excitation and nucleon recoil are discussed. By making the further approximations of closure and fixed scatterers we make connection with the work of previous authors. By generalizing to projectile interaction with several bound nucleons and examining the appropriate multiple scattering series we identify the optical potential for projectile elastic scattering from the many-body system. Different optical potentials are obtained assuming different projectile-bound nucleon $t$ matrices and we study the differences predicted by these dissimilar optical potentials for elastic scattering. In a model problem, we study pion-nucleus elastic scattering and compare the predictions obtained by adopting procedures used by (1) Landau, Phatak, and Tabakin and (2) Piepho-Walker to the predictions obtained in a less restrictive, but computationally difficult treatment.NUCLEAR REACTIONS Effects of different approximations on optical potentials and calculated angular distributions. Model problem studies. Nucleon recoil. Momentum distribution. Intermediate nuclear excitation effects.

Nonstatic theory of pion-nucleus scattering

A multiple scattering description of π-nucleus scattering that does not make use of the standard fixed-scatterer approximation (FSA) is presented. A generalized version of the impulse approximation is derived. This is then used to obtain an improved first-order optical potential, which is studied for pion energies around the 3-3 resonance. One of the most important features of the nonstatic description is to take into account the intermediate propagation of the interacting π- N system (Δ). This propagation induces a nonlocality in the optical potential. Its effect is studied for pion propagation in nuclear matter, which is found to be dominated by two collective modes of the π-nucleus system. The static description represents a well-defined limit of our formalism that allows the determination of the range of validity of the FSA. Binding corrections to the elementary amplitude are investigated and it is established that they represent, within the nonstatic formalism, the Δ-residual nucleus interaction. Their influence on π-nucleus scattering is discussed. Pion absorption is shown to be inherently contained in the nonstatic description of π-nucleus scattering.

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.