Fermi Motion Effects Research Articles

Local field corrections to the pion-nucleus optical potential

We present an evaluation of the local field corrections to the pion optical potential in nuclear matter. These corrections, which amount to including an infinite subset of rescattering terms in the π-nucleus optical potential, are shown to give an improved description of the coupling of the nuclear ground state to the 1p−1h, 2p−2h, …, np− nh excited states. Furthermore it is shown that, if in the evaluation of these local field corrections, the nucleon pole in the p 11 channel is correctly taken into account, one obtains a realistic description of true pion absorption processes with one, two or more nucleons knocked out. Numerical estimations of these corrections, incorporating the nucleon's Fermi motion, recoil corrections and correlations effects are presented. Their density, energy and momentum dependence is studied and the influence of different factors (correlation, π-nucleon range, recoil corrections) is investigated. Finally, these theoretical predictions are compared to the phenomenological values obtained by a fit to the experimental data.

Fermi-motion effects in deep-inelastic lepton scattering from nuclear targets

The ratio of deep-inelastic structure functions of nuclear targets to the sum of free neutron and proton structure functions has been calculated using a modified form of the Atwood-West technique for deuterium. Fermi-gas momentum distributions were used with modifications to include high-momentum tails resulting from nucleon-nucleon correlations. Tables of smearing ratios for ${W}_{1}$, ${W}_{2}$, and ${W}_{3}$ are given as a function of $x$ and ${Q}^{2}$ for deuterium and several heavy nuclei. We find that for $xg0.5$ the scaling violations for heavy nuclei are smaller than those for free nucleons. The shapes of the antiquark distributions are also changed.

Smearing corrections on deuterium targets

In this paper we show and clarify further the consistency (which in the conventional, non-relativistic approach is lacking) of the recent, simple, spinless and relativistic approach (the nucleon-parton approach) to Fermi motion effects on deuterium targets. In analysing the recent high-energy cross-section data of πD, πN and ND, NN processes, we find that the conventional approach leads to a non-negligible smearing correction σ s. On the other hand, the nucleon-parton approach, by using a simple ansatz for the truncated n-p-d relativistic vertex function, leads to a vanishing σ s. We show, also, that the inclusion of a spin degree of freedom in the formalism for leptonic scattering, where smearing effects are very important, does not affect the previous (spinless) results.

Theory of large anglep-nucleus scattering. I.pdelastic scattering and deuteron form factor at largeq2

We derive a new approach to the proton-nucleus large angle scattering which accounts in a systematic way for Pauli, binding, and Fermi motion effects. We concentrate here on $\mathrm{pd}$ scattering. After antisymmetrization of incident and target protons, the $\mathrm{pd}$ amplitude can be separated into (1) the standard multiple scattering series with the last $\mathrm{pp}$ amplitude antisymmetrized and (2) a neutron exchange amplitude which includes rescattering terms. The optimal approximation designed to minimize corrections is derived for the antisymmetrized $\mathrm{pd}$ amplitude. The single scattering amplitude is factorized into an on-shell (antisymmetrized) $\mathrm{pN}$ amplitude and the deuteron form factor, and it is found to play a main role in the large angle $\mathrm{pd}$ scattering at sufficiently high energy. The results are applied to analysis of $\mathrm{pd}$ elastic scattering data (${T}_{p}\ensuremath{\gtrsim}300$ MeV) which are well reproduced by the calculations. This analysis also permits an extraction of the deuteron body form factor for values of ${q}^{2}$ which far exceed those measured in $\mathrm{ed}$ elastic data.NUCLEAR REACTIONS Proton-nucleus scattering. Pauli, binding, and Fermi motion effects. Antisymmetrization. Minimization of corrections. $\mathrm{pd}$ large angle scattering. Comparison with data. Deuteron form factor.

High energy scattering from a bound nucleon: Unitarity and closure approximation

The scattering of high energy hadrons from nuclei is treated in a generalized closure approximation for inclusive cross sections which takes into account some Fermi motion and binding potential effects. The consistency with unitarity is demonstrated.NUCLEAR REACTIONS High energy scattering, single scattering amplitude, inclusive cross section, closure approximation, unitarity.

Fermi motion and off-shell effects in the reaction 3He( γ, π+) 3H

The reaction 3He( γ, π +) 3H is studied in the impulse approximation using realistic wave functions obtained as solutions of the Faddeev equations with the Reid soft core potential. The influence of Fermi motion on the resonance structure is treated exactly. The previously used assumption 〈 k 1〉 = − Q 3 , defining an average nucleon momentum proportional to the overall momentum transfer Q , is shown to be a good approximation over a large range of photon energies and momentum transfers. The dependence of the results on different off-shell prescriptions is investigated for two models of the elementary pion photoproduction amplitude.

Pion production in nuclear collisions with Fermi motion effects

The independent multiple collision model for pion production in nuclear collisions is extended to include the effects of Fermi motion. We have developed an analytic expression for the distribution of invariant mass for colliding nucleons whose momentum distributions are in relative motion. This expression is tractable if the colliding distributions are described in terms of Gaussian functions. We make use of this formalism to calculate Fermi averaged cross sections for pion production, using as input multi-Gaussian fits to momentum distributions obtained from nuclear structure calculations. We have calculated charged pion multiplicity distributions and inclusive cross sections for proton-nucleus and nucleus-nucleus collisions. We use a phenomenological method for incorporating the effects of energy loss by participant nucleons and pion reabsorption. Sample calculations are presented and compared with available experimental data. We discuss the striking effect of Fermi motion on pion production at bombarding energies below the nucleon-nucleon pion production threshold.NUCLEAR REACTIONS Particle production, pion multiplicities, pion inclusive cross sections, Fermi motion effects, absorption and energy loss effects, intermediate energy particle-nucleus and nucleus-nucleus collisions.

Particle production in nucleon-nucleus and nucleus-nucleus collisions: Fermi motion versus collective tube mechanisms

Within the general framework of multiple scattering theory for particle production in nucleon-nucleus and nucleus-nucleus collisions we investigate the results of two limiting situations: independent multiple collisions with Fermi motion effects versus the collective tube model with the assumption of universality for the elementary cross sections. Dramatically different predictions may be obtained. We summarize the results for pion multiplicity distributions in nucleus-nucleus collisions to assess the possible evidence for a coherent-collective mechanism and find that no strong conclusion is possible with the currently available pion multiplicity data. Attention is then focused on potentially more conclusive tests---subthreshold (in the nucleon-nucleon kinematics) production of massive particles, $\overline{p}$, $\frac{\ensuremath{\psi}}{J}$, and $W$ as examples. Evidence for a collective mechanism is found by examining subthreshold $\overline{p}$ production data in particle-nucleus collisions. However, owing to the limited scope of these data a firm conclusion is probably premature. Further $\overline{p}$ production experiments are proposed. With the assumption that the collective mechanism can be proved, it is worth studying the possibility of producing other massive particles, especially in nucleus-nucleus collisions where the collective effects have large subthreshold consequences. Within the (limited) framework of universality we then present predictions for the subthreshold production of $\frac{\ensuremath{\psi}}{J}$ and the $W$ boson. Owing to the qualitative nature of our assumptions and since these approaches bracket the $\overline{p}$ data, we expect that the collective tube model provides an optimistic limit and independent multiple collisions a pessimistic limit for $W$ production cross sections.NUCLEAR REACTIONS Nucleon-nucleus and nucleus-nucleus at high energies; particle production subthreshold; pion, antiproton, $\frac{\ensuremath{\psi}}{J}$, $W$ production, Fermi motion and collective effects.

Selective excitation in kaon-nucleus inelastic scattering

Abstract The K + meson (kaon) inelastic excitation of low-lying ( E x = 0–15 MeV) T = 0 collective states in 16 O is theoretically studied as a function of energy and momentum transfer. The distorted wave impulse approximation is used to calculate angular distributions and total inelastic cross sections for exciting the first J π = 2 + , 3 − , 4 + and 5 − states at lab energies from threshold to 400 MeV. The distortions are represented in a Kisslinger-type optical potential constructed from elementary K + -nucleon amplitudes. Total nuclear elastic and reaction K + -nucleus cross sections are computed to demonstrate sensitivity to choice in K + -nucleon amplitudes. Fermi motion effects are also assessed using a simple averaging procedure. The weak absorption character of the kaon is reflected in the inelastic calculations which predict selective excitation of low spin states at low momentum transfer and high spin states at high momentum transfer.

Electromagnetic properties of the πNN system : (II). The γD → D π° reaction

By computing the non-relativistic limits of the direct and the pion-nucleon rescattering amplitudes, we have obtained a good accounting for the available experimental data, both at threshold and in the Δ(1236) region. The effects of the Fermi motion were fully taken into account. At threshold the pion-nucleon rescattering diagram is very important, and we have studied the sensitivity of the results to the different ingredients of the calculation: the deuteron wave function, the pion propagator, the elementary operators, etc. In the Δ(1236) region the rescattering effects are small and the cross section is mainly sensitive to the direct term. Here the Fermi motion effects are important and the kinematical ambiguities in the elementary amplitudes are avoided in our treatment. The good overall agreement with the data gives us some confidence in the methods we are using in this series of papers which study the electromagnetic properties of the πNN system.

Kaon-nucleus inelastic scattering

A theoretical investigation of kaon inelastic scattering is conducted for the specific reaction /sup 12/C(K/sup +/,K/sup +/')/sup 12/C* (J/sup ..pi../ = 2/sup +/,E/sub x/ = 4.43 MeV). The analysis utilizes the Kerman-McManus-Thaler multiple scattering formalism and the distorted wave impulse approximation. The distorting parameters are uniquely determined through the empirical K/sup +/-nucleon scattering amplitudes. Numerical results demonstrate a clear sensitivity to choice of amplitudes. The cross sections, calculated over a wide range of energies, display a slow variation. Fermi motion effects are found to be quite important at kaon lab kinetic energies below 300 MeV. The magnitude of the K/sup +/-nucleus elastic S matrix shows that the kaon is not strongly absorbed; however, cut-off calculations, which eliminate all internal form factor contributions, firmly establish the inelastic process is surface localized.

High-energy expansion for nuclear multiple scattering

The Watson multiple scattering series is expanded to develop the Glauber approximation plus systematic corrections arising from three effects: (1) deviations from eikonal propagation between scatterings, (2) Fermi motion of struck nucleons, and (3) the kinematic transformation which relates the many-body scattering operators of the Watson series to the physical two-body scattering amplitude. Operators which express effects ignored at the outset to obtain the Glauber approximation are subsequently reintroduced via perturbation expansions. Hence a particular set of approximations is developed which renders the sum of the Watson series to the Glauber form in the center of mass system, and an expansion is carried out to find leading order corrections to that summation. Although their physical origins are quite distinct, the eikonal, Fermi motion, and kinematic corrections produce strikingly similar contributions to the scattering amplitude. It is shown that there is substantial cancellation between their effects and hence the Glauber approximation is more accurate than the individual approximations used in its derivation. It is shown that the leading corrections produce effects of order ${(2k{R}_{c})}^{\ensuremath{-}1}$ relative to the double scattering term in the uncorrected Glauber amplitude, $\ensuremath{\hbar}k$ being momentum and ${R}_{c}$ the nuclear charge radius. The leading order corrections are found to be small enough to validate quantitative analyses of experimental data for many intermediate to high energy cases and for scattering angles not limited to the very forward region. In a Gaussian model, the leading corrections to the Glauber amplitude are given as convenient analytic expressions.NUCLEAR REACTIONS Multiple scattering theory at intermediate to high energy; leading corrections to Glauber approximation due to eikonal, Fermi motion, and kinematic effects.

Connected kernel methods in nuclear reactions: (III). Effective interactions

The recently derived connected kernel equation (CKE) for N-body scattering operators is applied to direct nuclear reactions. A spectral representation is derived for the kernel of the CKE in order to obtain manageable approximations. This allows the kernel to be split into orders corresponding to the propagation of different numbers of bound clusters. By formally solving one part of the kernel at a time, the CKE is written as a hierarchy of nested equations in increasingly many variables. The first equation of this hierarchy is a set of coupled channel Lippmann-Schwinger equations coupling together all two-cluster channels. These equations reduce to the usual coupled channel equations for inelastic scattering and to the coupled channel Born approximation for rearrangement reactions when weak coupling assumptions are made. The second equation of the hierarchy is a two-variable integral equation for the effective interactions appearing in the coupled channel equations. The driving terms and kernel of this integral equation are obtained from the third equation of the hierarchy which is a three-variable integral equation and so forth. The use of the spectral expansion results in a renormalized theory in the sense that the bound state and reaction problems are separated. This permits the inclusion of nuclear models in the theory in a straightforward manner. The hierarchy is applied to a particular example, that of nucleon-nucleus scattering. For this case the hierarchy is truncated at the level allowing no more than three clusters in the continuum. By suppressing exchange and keeping only one-particle transfer and single-nucléon knockout channels, a set of equations for the optical potentials and transfer operators is obtained. These equations provide a three-body treatment of the single scattering approximation to the optical potential. Iteration of the equations yields the usual single scattering approximation in first order including three-body off-shell effects. After suppression of Fermi motion and off-shell effects, the standard impulse approximation is recovered. Modifications of the method for other cases are discussed and other possible applications suggested.

Measurement of Charged-Pion Production by 7.28-GeV Nitrogen Ions

Cross sections for charged-pi-meson production by 7.28-GeV nitrogen ions on elemental targets were measured at a laboratory angle of 15\ifmmode^\circ\else\textdegree\fi{} at the Princeton Particle Accelerator. At pion energies of \ensuremath{\sim} 260 and \ensuremath{\sim} 415 MeV, the data show the ${A}^{\frac{1}{3}}$ dependence predicted by an "effective number of nucleons" model. For ${\ensuremath{\pi}}^{\ensuremath{-}}$ energies above \ensuremath{\sim} 415 MeV, the measured cross sections exceed those calculated from nucleon-nucleus data, even after the inclusion of Fermi-motion effects.

Fermi-motion effects in neutron stripping cross-section from relativistic deuteron-nucleus collisions

The angular distribution of the proton in neutron stripping reactions, from relativistic collisions of deuterons with complex nuclei, is computed by means of Glauber’s theory. The effect of Lorentz contraction on Fermi motion is explicitly taken into account and the result is expressed as an incoherent multiple-scattering series.

Measurement of the reaction γd → π−pp, and determination of cross sections for the reaction γn → π−p, at photon energies between 0.2 and 2.0 GeV

The reaction γd→ π −pp has been measured in a bubble chamber experiment at DESY with a bremsstrahlung beam of E γ max = 5.5 GeV. Total and differential cross sections are presented for the energy region from 0.2 to 2.0 GeV. The extraction of cross sections on the neutron is discussed in detail. The impulse approximation and the closure relation, with corrections for the exclusion principle and the Fermi motion effects, as well as the Chew-Low extrapolation technique are applied. Total cross sections and angular distributions for the reaction γn→ π −p, deduced by these methods, are presented in graohs and tables. The results are compared with other experiments and with various predictions. No positive evidence for an isotensor electromagnetic current is found. The results disagree with some of the measurements of radiative pion capture, if detailed balance is assumed to connect the two reactions.