Translationally Invariant Calculations of Form Factors,Densities and Momentum Distributions for Finite Nuclei with Short–Range Correlations Included: A Fresh Look

The approach proposed in the 70s (Dementiji et al. in Sov J Nucl Phys 22:6–9, 1976), when describing the elastic and inelastic electron scattering off ⁴ He, and elaborated in (Shebeko et al.in Eur Phys J A27:143–155, 2006) for calculations of the one-body, two-body and more complex density matrices of finite bound systems has been applied (Shebeko and Grigorov in Ukr J Phys 52:830–842, 2007; Shebeko et al. in Eur. Phys. J. A48:153–172, 2012) in studying a combined effect of the center-of-mass motion and nucleon–nucleon short-range correlations on the nucleon density and momentum distributions in light nuclei beyond the independent particle model. Unlike a common practice, suitable for infinite bound systems, these distributions are determined as expectation values of appropriate intrinsic operators that depend upon the relative coordinates and momenta (Jacobi variables) and act on the intrinsic ground–state wave functions (WFs). The latter are constructed in the so-called fixed center-of-mass approximation, starting with a mean–field Slater determinant modified by some correlator (e.g., after Jastrow or Villars). Our numerical calculations of the charge form factors (F _CH (q)), densities and momentum distributions have been carried out for nuclei ⁴ He and ¹⁶ O choosing, respectively, the 1s and 1s?1p Slater determinants of the harmonic oscillator model as trial, nontranslationally invariant WFs.     

Spherical Hartree-Fock calculations with linear-momentum projection before the variation

The hole spectral functions and from these the spectroscopic factors have been calculated in a Galilei-invariant way for the ground-state wave functions resulting from spherical Hartree-Fock calculations with projection onto zero total linear momentum before the variation for the nuclei ⁴He, ¹²C, ¹⁶O, ²⁸Si, ³²S and ⁴⁰Ca. The results are compared to those of the conventional approach which uses the ground states resulting from usual spherical Hartree-Fock calculations subtracting the kinetic energy of the center-of-mass motion before the variation and to the results obtained analytically with oscillator occupations.     

Two-body density matrix for closed s-d shell nuclei

The two-body density matrix for ⁴He,¹⁶O and ⁴⁰Ca within the Low-order approximation of the Jastrow correlation method is considered. Closed analytical expressions for the two-body density matrix, the center of mass and relative local densities and momentum distributions are presented. The effects of the short-range correlations on the two-body nuclear characteristics are investigated. Received: 11 September 1999 / Revised version: 20 December 1999     

Nucleon momentum distributions in doubly closed shell nuclei

The momentum distributions in nuclei like⁴He,¹⁶O and⁴⁰Ca are explicitly calculated within a phenomenological model which includes dynamical short range and tensor correlation effects. The common behaviour of such distributions in the high momentum region, already established in light nuclei, is extended to the medium weight region. Comparison with existing calculations is discussed and, for completeness, also form factors are evaluated within the same framework.     

Spherical Hartree-Fock calculations with linear-momentum projection before the variation

Spherical Hartree-Fock calculations with projection onto zero total linear momentum before the variation are performed for the nuclei ⁴He, ¹²C, ¹⁶O, ²⁸Si, ³²S and ⁴⁰Ca using a density-independent effective nucleon-nucleon interaction. The results are compared to those of usual spherical Hartree-Fock calculations subtracting the kinetic energy of the center-of-mass motion either before or after the variation and to the results obtained analytically with oscillator occupations. Total energies, hole energies, elastic charge form factors and charge densities and the mathematical Coulomb sum rules are discussed.Received: 7 April 2003, Published online: 18 December 2003PACS: 21.60.-n Nuclear structure models and methods     

Brueckner theory and Jastrow approach for finite nuclei

The results of Jastrow variational calculations and of Brueckner theory in lowest order (Brueckner-Hartree-Fock) are compared. The comparison is made for the calculations of ground state properties like binding energy and charge-radius of light and medium weight, closed shell nuclei (⁴He, ¹⁶O, ⁴⁰Ca). For the nucleon-nucleon interaction rather simple forces are used (Brink-Boeker potential B1, Afnan-Tang potential S3). For all cases considered it turns out that the results of the two different methods are in fairly good agreement, with the binding energy calculated in the Brueckner-Hartree-Fock approximation being always slightly below the corresponding upper bounds calculated in the Jastrow variational approach. This good agreement between the two methods indicates, that for light and medium weight nuclei the Jastrow variational approach and the Brueck-ner-Hartree-Fock approximation can be considered as reasonable approximations to a complete solution of the many-body problem.     

A study of short-range correlation effects on nuclear spatial and momentum distribution

The relation between the nuclear spatial and momentum densities is studied by means of their integral representations in terms of uniform distributions. Examples of this approach are given for the harmonic oscillator model of⁴He and¹⁶O, with and without Jastrow correlations, and for the single-particle potential model of⁴⁰Ca.     

Momentum distributions in A = 3 and 4 nuclei

We develop a general Monte Carlo method to study momentum distributions of nucleons and nucleon clusters in nuclei. The method is used to calculate the momentum distributions of protons and neutrons in A = 3 and 4, d + p amplitudes in ³He, and t + p and d + d amplitudes in ⁴He nuclei, with improved variational wave functions. The nucleon and d + p momentum distributions in ³He are also calculated from a five-channel Faddeev wave function. The calculations are based on realistic hamiltonians that include the three-nucleon interaction, and give reasonable binding energies and densities for light nuclei and nuclear matter. The calculated proton and d + p momentum distributions in ³He at low k are in good agreement with the results obtained by the Saclay analysis of the electron scattering data in the plane-wave impulse approximation; however, at higher values of k, the calculated momentum distributions are larger. The calculated values of the asymptotic D- to S-wave ratio of the d+n and d+d amplitudes are also in agreement with the values obtained from (d, t) and (d, α) reactions. The number of deuterons is found to be 1.38 and ∼2.4 in A = 3 and 4 nuclei, while the number of tritons in ⁴He is ∼1.6. This large value of the triton number reflects the large contribution (more than 90% at small k) of the t+p state to the total proton momentum distribution in ⁴He.     

Models and approximations for the momentum distribution in nuclear matter

As a step toward development of a framework for phenomenological analysis of nuclear momentum distributions in terms of dynamical interparticle correlations, the momentum distribution n(k) of symmetrical nuclear matter described by a Jastrow wave function is studied within a variety of approximation schemes. In particular, two simple low-cluster-order approximations are proposed, one of which may be readily adapted to finite nuclei. For two choices of pair correlations tailored respectively to soft and moderately stiff repulsive cores, the results based on these approximations compare favorably with the standards set by Fermi-hypernetted-chain evaluation and (for the kinetic energy) by Monte Carlo integration — even at densities somewhat beyond the empirical equilibrium density of nuclear matter. The crucial role played by the Jastrow wound parameter, as a determinant of the overall behavior of n(k), emerges clearly from these calculations.     

Quasielastic knockout of light fragments from 12C and 16O nuclei by intermediate-energy pions

Quasielastic deuteron and triton knockout from ¹²C and ¹⁶O nuclei has been studied infull kinematics using a 0.72-GeV/c pion beam. The momentum distributions of the intranuclear quasideuteron motion, excitation-energy spectra of the residual nuclei, and the effective numbers N _d ^eff of quasideuterons are determined. The parameters of the quasideuteron intranuclear motion are in reasonable agreement with the results obtained in other beams. The N _d ^eff in the nuclei from ⁶Li to ¹⁶O measured in full kinematics are virtually independent of the atomic number in contrast to the analogous values in the inclusive deuteron-knockout reaction induced by protons. The phenomenon of triton knockout from these nuclei is observed, which makes possible estimation of the cross section of backward pion-triton elastic scattering in yet unexplored regions of energy and momentum transfer.     

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.     

Energy and angular momentum dependence of heavy ion optical potentials

The influence of bombarding energy and angular momentum on the depth and shape of the real part of the optical ion-ion potential is studied in a model which uses oscillator wave functions for the ground states of the interacting nuclei and takes into account the relative motion of the nuclei by a multiplication with a plane wave factor. The calculations were done for α+α,¹⁶O+¹⁶ O,⁴⁰Ca+⁴⁰Ca,α +¹⁶O,α +⁴⁰Ca and¹⁶ O +⁴⁰Ca with the Skyrme force as interaction.     

Momentum distributions of <Superscript>4</Superscript>He nuclei from the <Superscript>6</Superscript>He and <Superscript>6</Superscript>Li breakup

Experimental data on the momentum distributions of ⁴He nuclei originating from ⁶He and ⁶Li breakup on various targets are presented over a wide beam energy range. The experiment with ⁶He was performed at the DRIBs accelerator complex for radioactive beams at the Joint Institute for Nuclear Research (JINR, Dubna). The intensity of the ⁶He beam used was 5 × 10⁶ particles per second and its energy was 10 MeV per nucleon. The momentum distributions of breakup products were measured by means of the MSP-144 magnetic spectrometer. The distribution width was shown to be virtually independent of the target mass. A small value of this width, σ ～ 28 MeV/c, confirms the presence of a halo in ⁶He. The measurements performed with ⁶Li beams of energy 18 and 46 MeV per nucleon at the U-400M accelerator yielded a width value of σ ～ 50 MeV/c for the momentum distributions of ⁴He nuclei, which is intermediate between that for ⁶He and those for stable nuclei. A compilation of the widths of the momentum distributions of fragments originating from the breakup of various nuclei is presented versus the binding energy of one or two neutrons in these nuclei, the target mass and the projectile energy.     

Self-consistent wave functions,center-of-mass motion and high-energy particle-nucleus scattering

Elastic electron and hadron scattering by ⁴He and ¹⁶O is calculated using for the latter nucleus Brueckner-Hartree-Fock (BHF) wave functions obtained in the harmonic oscillator representation. The effects of the center-of-mass motion spuriosity are estimated.     

Folding model analysis of the nucleus–nucleus scattering based on Jacobi coordinates

This paper presents the results of scattering of ¹⁶O+²⁰⁹Bi interaction near the Coulomb barrier. The interaction potential between two nuclei is calculated using the double folding model with the effective nucleon–nucleon (NN) interaction. The calculations of the exchange part of the interaction were assumed to be of finite-range and the density dependence of the NN interaction is accounted for. Also the results are compared with the zero-range approximation. All these calculations are done using the wave functions of the two colliding nuclei in place of their density distributions. The wave functions are obtained by the D-dimensional wave equation using the hyperspherical calculations on the basis of Jacobi coordinates. The numerical results for the interaction potential and the differential scattering are in good agreement with the previous works.     

Neutral transverse momentum spectra in 60 and 200A·GeV16O+nucleus and proton+nucleus reactions

Transverse momentum (p _T) distributions fo inclusive photons and neutral pions at midrapidity are measured with a lead glass calorimeter in 60 and are measured with a lead glass calorimeter in 60 and 200A·Gev¹⁶O+nucleus and and proton+nucleus reactions. Inclusive photon distributions are compared for central and peripheral reactions. The degree of centrality is determined either from the charged particle multiplicity or from the remaining projectile energy in the forward direction. Deviations from a nucleus+nucleus interaction model based upon linear extrapolation from p+p reactions are observed in central¹⁶O+Au data. The variation of theaverage transverse momentum is investigated as function of centrality. The target-mass and energy dependence of π⁰ p _T distributions are presented. For¹⁶O+Au a change of slope in these distributions is observed atp _t ≈0.8 GeV/c compatible with hydrodynamic expansion models.     

Occupation probabilities and momentum distributions in closedS-D shell nuclei

Single-particle occupation probabilities and the depletion of the nuclear Ferrmi sea (caused by the short-range nucleon-nucleon correlations) as well as momentum distributions in closeds-d shell nuclei are calculated within the Jastrow correlation method in its low-order approximation. The theoretical results for the occupation probabilities are discussed in relation to experimental data from (e, e′ p) reactions and to other theoretical estimates.     

Intense-field double ionization of helium: identifying the mechanism

We present quantum mechanical calculations of the electron and ion momentum distributions following double ionization of a one-dimensional helium atom by ultrashort laser pulses (780 nm) at various intensities. The two-electron momentum distributions exhibit a clear transition from nonsequential to sequential double ionization. We provide strong evidence that rescattering is responsible for nonsequential ionization by calculating the momentum spectrum of the He2+ recoil ions-which we find in excellent agreement with recent experiments-and by analyzing the electronic center-of-mass motion via Wigner transforms.     

Implication of proton-nucleus scattering for density distributions of unstable nuclei

The use of elastic proton scattering at intermediate and high energies to obtain information about the density distributions of unstable nuclei is investigated. A comparison between the relativistic impulse approximation (RIA) and Glauber model for proton scattering from ¹⁶O, ⁴⁰Ca, ⁴⁴Ca and ⁴⁸Ca at medium energies is performed. We used density distributions derived from the relativistic mean-field theory, employing the recent relativistic force NL-RA1, as well as experimental and phenomenological densities. It is found that the eikonal approximation can describe the cross-section only at small scattering angles and is weakly sensitive to the density distributions, while the RIA nicely produced the experimental cross-sections, even at medium and larger angles, and was very sensitive to the nuclear densities. Furthermore, the RIA better describes the isospin dependence of the cross-section. We used the RIA to investigate the density distribution of ⁵⁸Ca for proton scattering at different energies. It is found that the cross-section strongly depends on the parameters of the density distribution even at a small scattering angle. These results are important in extracting information about the structure of unstable nuclei. We also investigated the RIA and its sensitivity in describing halo nuclei such as ⁶He. We used for ⁶He a no-halo Gaussian density and a realistic-halo density that derived in the cluster orbital shell model approximation and contains the extended distribution of the valence nucleons. Comparison with the recent experimental data at GSI at 717 MeV/nucleon shows that the RIA successfully described the data at all considered range of the momentum transfer and on the other hand favor the halo structure of ⁶He. Received: 1 December 2001 / Accepted: 31 July 2002 / Published online: 11 February 2003 RID="a" ID="a"e-mail: mrashdan@hotmail.com Communicated by P. Schuck     

Single-nucleon transfer reactions induced by16O on14C

Angular distributions of the proton and neutron transfer reactions¹⁴C(¹⁶O,¹⁵N)¹⁵N and¹⁴C(¹⁶O,¹⁷O)¹³C leading to the ground states of the final nuclei were measured atE _lab=20, 25 and 30 MeV. A DWBA analysis was performed using the no-recoil approximation of Buttle and Goldfarb. All angular distributions, including the pronounced structures of the proton transfer arising from the fact that the final nuclei are identical, are well reproduced. The spectroscopic factor for the neutron transfer is in agreement with shell model calculations whereas the proton transfer into aj _<-state yields a value which is too high. Exact finite-range calculations do not show this discrepancy, indicating that recoil effects are important even for light targets and lower energies. Contributions of the nonnormall-transfer, however, are small.