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.     

Constrained variational calculations of closed shell nuclei with the reid potential

Lowest-order constrained variational calculations with harmonic oscillator wave functions are carried out for ⁴He, ¹⁶O, and ⁴⁰Ca nuclei with the Reid potential. The results obtained with this simple method are in very close agreement with those obtained by renormalized Brueckner-Hartree-Fock theory with orthogonalized plane wave intermediate states. The A-dependence of the difference between the experimental and calculated binding energies for A = 2, 3, 4, 16, 40 and ∞ can be explained by a three-body cluster term coming either from a wrong off-energy-shell behavior of the Reid potential or a three-body force. The calculated radii of ¹⁶O and ⁴⁰Ca are ≈ 10% too small, indicating that the Reid potential may not be very realistic.     

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.     

A quantal calculation of nucleon transfer contribution in nucleus-nucleus absorptive potential

We present here a model to obtain the quantum-mechanically defined particle-transfer flux in scattering between two heavy nuclei. This flux is calculated from the time-dependent single-particle wave functions in the field of two moving potential pockets. From the calculated flux, we obtain the absorptive potentials for ¹⁶O+⁴⁰Ca and ⁴⁰Ca+⁴⁰Ca, which compare favourably with phenomenological values. In contrast with other similar microscopic calculations, the present results show a weak energy dependence of the absorptive potential as has been observed in phenomenological analyses.     

A treatment of renormalization,potential energy of intermediate states and re-arrangement effects in brueckner calculations of closed shell nuclei

In the Brueckner-Hartree-Fock theory of finite nuclei, the problems of renormalization, potential energy of intermediate states and re-arrangement effects are examined. For practical calculations it is shown that satisfactory solutions to them can be obtained if the parametrization chosen for the single-particle potential in finite nuclei is linked closely to nuclear matter results. We make the link with similar problems in the density dependent Hartree-Fock theory and emphasize the possibility of such a parametrization by recalling the existence of a formal solution for the Hartree-Fock single-particle potential if the effective interaction is of the δ-function type. A method of solution of the Bethe-Goldstone equation is then presented which separates the intermediate states into those of an “open shell” and of a “continuum”. Finally results of model calculations of ¹⁶O and ⁴⁰Ca with harmonic oscillator functions are presented in which the parametrization chosen for the BHF single-particle potential is taken from the Skyrme-Vautherin δ-function force. A self-consistent determination of certain parameters of this form of force leads to values in close agreement with the empirical estimate made by Vautherin and Brink in ¹⁶O, with the exception of the spin-orbit splittings. Limitations and possible improvements of this type of approach are discussed for ⁴⁰Ca.     

Generator coordinate calculations of monopole and quadrupole vibrations with Skyrme's interaction

Skyrme's interaction is shown to lead to significant simplifications in generator coordinate calculations. As an illustration monopole and quadrupole modes are calculated in ⁴He, ¹⁶O and ⁴⁰Ca using pure oscillator wave functions.     

Properties of nuclear matter andN=Z closed-shell nuclei

A simple three-parameter density dependent effective interaction is used to study the properties of nuclear matter, neutron matter and some bulk properties such as ground state energies and rms charge radii of three double-closed shell nuclei⁴He,¹⁶O and⁴⁰Ca. The three parameters of the effective interaction are determined by requiring to fit the binding energy and density of infinite nuclear matter at saturation density as well as ground state energy of¹⁶O in the first order perturbation theory. This interaction gives correct saturation in nuclear matter with a value of 283 MeV for compressibility. The symmetry coefficienta _T atk _F=1·36 fm^–1 is 28·58 MeV. The energy per particle in neutron matter is calculated in the range of nuclear matter densities and it compares well with those ofNemeth andSprung. Groundstate energies and rms charge radii of⁴He,¹⁶O and⁴⁰Ca are calculated using oscillator eigen functions as single particle wave functions. Results for ground state energies are in good agreement with empirical values and rms charge radii are slightly better than those obtained byMoszkowski with the MDI.The authors are thankful to the Computer Centre, Utkal University, Bhubaneswar for providing computational facilities for this work.     

Orbital rearrangement in light nuclei

Orbital rearrangement effects due to the removal of a proton are studied. Calculations in the nuclei¹⁶O,²⁰Ne, and⁴⁰Ca are performed using the Yale potential, and the Brink Boeker forceB1. The odd mass nuclei are not restricted to time reversal symmetry which prevents a large overestimation of the rearrangement energies. Although the orbital rearrangement energies are small (<1.5 MeV) utilizing density independent forces one can find larger changes in the wave functions due to the rearrangement as exhibit the discussion of the quadrupole moments and the mass distributions.     

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.     

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.     

Translationally invariant treatment of the charge density in nuclei

An expression is derived for the translationally invariant (transition-) charge density in the center of mass rest frame of the nucleus in between arbitrary wave functions. An explicit formulation is given for the charge density of spherically symmetric Hartree-Fock type ground states. The method is applied to the nuclei⁴He,¹⁶O,⁴⁰Ca and⁹⁰Zr using Woods-Saxon single nucleon orbits for the construction of the ground state wave functions. For the two lighter nuclei in addition realistic Hartree-Fock wave functions are investigated. The results are compared to those obtained with various approximate treatments of the center of mass motion. It turns out that at least in the lighter nuclei the usual center of mass correction, which is based on the assumption of pure non-spurious oscillator configurations and commonly used in the analysis of, e.g., electron scattering data, does not produce reliable (or even predictable) results. Thus here indeed the full translationally invariant treatment seems to be required. For the heavy nucleus⁹⁰Zr on the other hand, as expected, the center of mass effects become negligibly small.     

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.     

Study of many-nucleon correlations in32S as a model for fragment shell effects in fission

The formation and break-up of substructures is studied in³²S as a calculable microscopic model for analogous long-range many-nucleon correlations in the fission of actinides. The calculations are performed for alpha-cluster wave functions with Volkov and Brink-Boeker interaction allowing forα,¹²C,¹⁶O and²⁰Ne clustering. It turns out that the correlated motion of large magic numbers of nucleons in two groups (¹⁶O+¹⁶O) is energetically favorable already at relatively small deformations. In the second minimum the¹⁶O+¹⁶O substructure occurs with high probability (about 80%). In analogy to these results the “pre-formation” of fragments and “fragment shell” effects occuring in the fission of actinides are explained in terms of probability statements for the formation of the corresponding heavy clusters in the many-body wave function.     

Translationally invariant calculations of form factors, nucleon densities and momentum distributions for finite nuclei with short-range correlations included

Relying upon our previous treatment of the density matrices for nuclei (in general, nonrelativistic self-bound finite systems) we are studying a combined effect of center-of-mass motion and short-range nucleon-nucleon correlations on the nucleon density and momentum distributions in light nuclei (⁴He and ¹⁶O). Their intrinsic ground-state wave functions are constructed in the so-called fixed center-of-mass approximation, starting with mean-field Slater determinants modified by some correlator (e.g., after Jastrow or Villars). We develop the formalism based upon the Cartesian or boson representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators for oscillatory quanta in the three different space directions, and get the own “Tassie-Barker” factors for each distribution and point out other model-independent results. After this separation of the center-of-mass motion effects we propose additional analytic means in order to simplify the subsequent calculations (e.g., within the Jastrow approach or the unitary correlation operator method). The charge form factors, densities and momentum distributions of ⁴He and ¹⁶O evaluated by using the well-known cluster expansions are compared with data, our exact (numerical) results and microscopic calculations.     

Effect of the Pauli exclusion principle and the polarization of nuclei on the potential of their interaction for the example of the 16O+16O system

On the basis of the energy-density method, the effect of simultaneously taking into account the Pauli exclusion principle and the monopole and quadrupole polarizations of interacting nuclei on their interaction potential is considered for the example of the ¹⁶O + ¹⁶O system by using the wave function for the two-center shell model. The calculations performed in the adiabatic approximation reveal that the inclusion of the Pauli exclusion principle and the polarization of interacting nuclei, especially their quadrupole polarization, has a substantial effect on the potential of the nucleus-nucleus interaction.     

Coupling the Lorentz Integral Transform (LIT) and the Coupled Cluster (CC) Methods: A Way Towards Continuum Spectra of “Not-So-Few-Body” Systems

Here we summarize how the LIT and CC methods can be coupled, in order to allow for ab initio calculations of reactions in medium mass nuclei. Results on ¹⁶O are reviewed and preliminary calculations on ⁴⁰Ca are presented.     

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.