Effective nucleus-nucleus potentials derived from the generator coordinate method

The equivalence of the generator coordinate method (GCM) and the resonating group method (RGM) and the formal equivalence of the RGM and the orthogonality condition model (OCM) lead to a relation connecting the effective nucleus-nucleus potentials of the OCM with matrix elements of the GCM. This relation may be used to derive effective nucleus-nucleus potentials directly from GCM matrix elements without explicit reference to the potentials of the RGM. In a first application local and l-independent effective potentials are derived from diagonal GCM matrix elements which represent the energy surfaces of a two-centre shell model. Using these potentials the OCM can reproduce the results of a full RGM calculation very well for the elastic scattering of two α-particles and fairly well for elastic ¹⁶O-¹⁶O scattering.     

Off-shell effects in O

We calculate the binding energy of ¹⁶O for a set of phase-shift-equivalent potentials previously studied in nuclear matter. Off-shell variations of up to 2.8 MeV per particle occur compared with about a 10 MeV per particle variation in nuclear matter. As in nuclear matter calculations a nearly linear relation exists between the variations in the binding-energy results and the wound integral k. We compare the ¹⁶O results with a nuclear matter calculation at the “equivalent” nuclear matter density of k_F = 1.13 fm⁻¹. This “equivalent” density reflects the fact that ¹⁶O has a surface and hence a lower average density than nuclear matter. The ¹⁶O and nuclear matter off-shell variations are comparable once one takes into account the lower average density of ¹⁶O and the suppression of the relative D-wave interaction — also a surface effect. We present a method of computing the correlated wave functions of finite nuclear systems and display such wave functions for ¹⁶O. The correlated wave functions of ¹⁶O and of nuclear matter are strikingly similar for all of the potentials studied.     

Density dependent interactions and the consistency of folding estimates of nucleon-nucleus and nucleus-nucleus potentials

It is shown that density dependence in the interaction does effect the consistency of the results obtained in folding calculations of nucleon-nucleus and nucleus-nucleus potentials. A modified density dependent version of the new G matrix interaction of Bertsch et al. gives reasonable results for both.     

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.     

Scattering solutions to non-local Hamiltonians in terms of generalized Bessel-Neumann series

We develop and test a method for the numerical solution of partial-wave scattering equations with partially non-local potentials, as arising in microscopic theories of the nucleus-nucleus interaction. The method is complementary to the usual Robertson discretization in that the amount of computation required decreases with increasing energy. It is based on a Bessel-Neumann expansion of the x-space scattering wave function and on its analogue for Coulomb functions. As a by-product, we obtain a new, general representation of the half-off-shell t-matrix and discuss the class of special functions arising in this context.     

Compact and loosely bound structures in light nuclei

The role of various components in the wave function of loosely bound light nuclei is considered in terms of the cluster model by taking into account orbital polarization. We show that several structures corresponding to particular modes of nucleon motion can be concurrently important for such structures. Specific examples of simple and fairly flexible trial wave functions are given for the ⁸Be and ⁶He nuclei. Explicit expressions are derived for the microscopic wave functions of these nuclei and employed to calculate basic nuclear parameters for commonly used central exchange NN potentials.     

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.     

Accuracy of simple folding model in the calculation of the direct part of real α−α interaction potential

The direct part of real α?α interaction potential is calculated in the simple folding model using density-dependent Brink–Boeker effective interaction. The simple folding potentials calculated from the short- and finite-range components of this effective interaction are compared with their corresponding double folding results obtained from the oscillator model wave function to establish the relative accuracy of the model. It is found that the direct part of real α?α interaction potential calculated in the simple folding model is reliable.     

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.     

Value of the absolute cross section for the reaction 40Ca(16O, 12C)44Ti

The reaction ⁴⁰Ca(¹⁶O, ¹²C)⁴⁴Ti(g.s.) is calculated making use of a microscopic finite-range form factor and full recoil. The states connected by the reaction are described in terms of shellmodel wave functions. They were obtained making use of Kuo-Brown matrix elements in the (p, sd) and (f, p) subspaces for the case of ¹⁶O and ⁴⁴Ti, respectively. The predicted value of the cross section is a factor of the order of 100 smaller than the experimental value.     

A simple and efficient numerical scheme to integrate non-local potentials

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrödinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method was proposed several years ago, with the trivially equivalent potential method, where the non-local potential is replaced by an equivalent local potential, which is state dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of ¹⁶O using the finite-range N³LO realistic interaction.     

Alpha scattering from nonclosed shell nucleus 12C and alpha-nucleus interaction

In order to understand the α-nucleus interactions for the α+¹²C～α+¹⁶O systems systematically, α scattering from ¹²C is studied in a microscopic coupled-channel model using a folding potential and realistic microscopic wave functions of ¹²C. The experimental angular distributions of elastic and inelastic scattering to the 2⁺(4.43 MeV), 3^? (9.64 MeV), and 0 ₂ ⁺ (7.66 MeV) states in the range of E _α = 41–172.5 MeV are analyzed.     

Elastic <Superscript>16</Superscript>O<Superscript>16</Superscript>O scattering at <Emphasis Type="Italic">E</Emphasis>(<Superscript>16</Superscript>O) = 350 MeV and new method for a model-independent determination of the scattering matrix

A new approach that can be used to extract the scattering matrix S(l) as a complex-valued function of the orbital angular momentum l directly from experimental data on elastic nucleus-nucleus scattering at intermediate energies without resort to additional model assumptions is developed on the basis of the evolutionary algorithm. Owing to the fact that the behavior of the derivatives of S(l) is automatically monitored within this approach, the scattering matrix obtained for ¹⁶O¹⁶O interaction at an energy of 350 MeV is determined by its absolute value and the nuclear phase shift, which are smooth monotonic functions of the orbital angular momentum. Moreover, the result involves no distortions, the form of the quantum deflection function being typical of the nuclear-rainbow pattern. It is shown that the ultimate form of S(l) is independent of input representations of the scattering matrix, which were usually taken on the basis of various phenomenological models.     

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.     

A new method for evaluating the density matrix and its application to the ground state form factors of 4He and 16O

The form factors of ⁴He and ¹⁶O are calculated using wave functions given by solutions of a generalized Hartree-Fock equation. No free parameters exist in the theory except the parameters of various phenomenological nucleon-nucleon potentials. The c.m. motion is treated exactly in the framework of a many body theory. The results strongly depend on the nucleon-nucleon potentials.     

Neutron-proton transfer reactions leading toT=1 particle-hole states of56Co

Zero range DWBA analysis for two-particle (neutron and proton) transfer reactions is carried out, using simple shell model structure wave functions for⁵⁴Fe,⁵⁶Co and⁵⁸Ni, with⁵⁶Ni inert core. In this structure calculation, a microscopic set of two-body interaction matrix elements derived from the non-local separable potential of Tabakin are employed. These matrix elements include in the perturbation theory two corrections (i) the second-order Börn term and (ii) the appropriate core excitations. Unlike the situation in many two-particle transfer reactions, the fragmentation of the reaction strengths to the excited states with respect to the lowest states of same spin and parity in the above transfer processes is satisfactorily borne out from this analysis.     

One-nucleon exchange and the effective local potential in nucleus-nucleus scattering

The influence of one-nucleon exchange on the local nucleus-nucleus potential is investigated using the resonating group method. Neutron scattering on ⁴He and ¹⁶O nuclei is examined in detail. The results are compared to microscopic calculations.