Exponential convergence and acceleration of Hartree-Fock calculations

We show that we can expect an exponential behaviour for the convergence of the Hartree-Fock solution during the HF iteration procedure. We use this property to extrapolate some collective degrees of freedom, in this case the shape, in order to speed up the self-consistent calculation. For axially deformed nuclei we apply the method to the quadrupole moment which corresponds to a simple scaling transformation on the single particle wave functions. Results are shown for the deformed nuclei ²⁰Ne and ²⁸Si with a Skyrme interaction.     

An application of the theory of Bi-determinantal intrinsic states to28Si

For some 4n nuclei in the 2s?1d shell, Hartree-Fock (HF) theory with most two-body interactions predicts nearly degenerate prolate and oblate Intrinsic states. The spectrum ofJ ^π states obtained from these Intrinsic states by projection is too compressed in relation to the observed levels. For such systems with a two-fold degeneracy of HF solutions, a Bi-Determinantal Intrinsic state (BDIS) is the more apt variational state. The equations of the theory of BDIS, which were first derived by B. Bremond, are here simplified and cast in a form suitable for numerical solution. The transformation operators introduced by Bremond are given a suitable representation, compatible with the symmetries of these 4n nuclei, and an independent definition is then given of self-consistent (SC) Hamiltonians. These equations are then iteratively solved in a tripyl-SC way, by the method of diagonalizing the SC Hamiltonions, for the problematic nucleus²⁸Si. By angular momentum projection from this BDIS, the low-lying spectrum is obtained. The discrepancy between this projected spectrum and the observed levels suggests that²⁸Si is not describable by a BDIS. The present results are in reasonable agreement with those of other Multi-Determinantal Theories for this nucleus.     

Properties of isovector 1+ states in A=28 nuclei and nuclear muon capture

A method is proposed for taking into account, in a calculation of partial rates of muon capture by nuclei, experimental information about strength functions for Gamow-Teller and isovector M1 transitions. The method, which amounts to choosing an orthogonal transformation that acts in the subspace of wave functions for excited states, requires neither modifying transition operators nor introducing effective charges. The matrix of the above transformation is constructed as a product of the matrices of reflection in a plane. All calculations are performed on the basis of the multiparticle shell model. Numerical results are obtained for isovector states in A=28 nuclei. Strength functions for Gamow-Teller and isovector M1 transitions in ²⁸Si are considered, and the lifetimes of 1⁺ states in ²⁸Al and the branching fractions for gamma decays of this state are calculated. Owing to taking into account experimental information about the properties of isovector states, the branching fractions for the γ decays of the 1⁺ state at 2.201 MeV in ²⁸Al are successfully described for the first time. The above transformation of the wave functions changes substantially the distribution of partial rates of allowed muon capture by a ²⁸Si nucleus among the 1⁺ states of the final nucleus ²⁸Al in relation to the results of the calculations with the eigenfunctions of the Hamiltonian of the multiparticle shell model. The muon-capture rates calculated with the transformed functions agree well with experimental data.     

Core polarization effects on transition densities in medium-heavy nuclei

We propose a microscopic model to study the core-polarization effects of giant resonances on the transition densities of open-shell nuclei. We use the Hartree-Fock-RPA method for the calculation of the single-particle wave functions and the response function of the giant resonances. Particle-vibration coupling is applied to take into account the core polarization effect on the valence many-body wave functions. We apply our model to the quadrupole transitions in the several medium-heavy nuclei. Valence many-body wave functions are calculated with the generalized seniority scheme and with the shell model. Results for the proton and neutron effective charges and the Coulomb form factors for the N = 82 isotones and for ¹¹⁶Sn and ¹¹⁰Pd are presented and discussed. The effective coupling hamiltonian is determined by the Skyrme interaction SGII which is used also in the HF and RPA calculations. The calculated core polarization charges show some state dependences. The average theoretical values are δe_p = 0.4–0.5 and δe_n = 0.6–0.7 compared to typical empirical values of δe_p = 0.6 and δe_n = 1.2.     

Shape-excited states of28Si

The Hartree-Fock (HF) minima for the nucleus²⁸Si were obtained for the prolate, oblate and spherical shapes using the interaction obtained by Preedom and Wildenthal. The interaction gives rise to large energy separation between the prolate and the oblate shapes. The spherical solution is just 2 MeV above the lowest HF (oblate) minimum. The spectrum projected from the oblate HF state is in good agreement with the experimental spectrum. The transition probabilities for the different energy levels also agree reasonably well. The configuration mixing calculations performed on the basis of states projected from the three shapes indicate that there is no significant mixing of different projected states. The second 0 ₂ ⁺ state, thus obtained, corresponds to the third 0 ₃ ⁺ state in the experimental spectrum and stems dominantly from the spherical HF state. It is seen that the structure of the energy levels of²⁸Si, especially the second 0 ₂ ⁺ level is very sensitive to the two body interaction. The results are compared with those obtained using the renormalised interaction of Kuo.     

Comparative study of the VMI model expressions for the fitting of the rotational spectrum of a single intrinsic wave function

Skyrme's method is used to introduce a correlation coefficient which tells us which of the VMI model expressions should be considered better in the calculation of the projected energies E_J when one uses a single intrinsic wave function for a set of good J-states. As an application of the correlation coefficient, we have shown that out of several VMI model expressions for the energy which have the same overlap with the Hamiltonian, the one having minimum dispersion should be considered to be the best choice. A remark is passed that the Holmberg-Lipas model can also be derived using Bohr and Mottelson's conditions. Numerical calculations are carried out for the nuclei ²⁸Si and ³⁶Ar using Ripka's Hartree-Fock wave functions to compare Holmberg-Lipas model with the one in which one directly expands the energy E_J in powers of J(J + 1), where J is the total angular momentum.     

Generator coordinate calculations on 28Si

The spectrum of ²⁸Si is calculated by the generator coordinate method for various choices of generating functions obtained by solving the HF equations within the s-d shell with a constraint on the quadrupole or the hexadecapole moment, or with a constraint of the Holzwarth-Yukawa type. The influence of the symmetry of the generating functions is studied. The spectrum if quite sensitive to the choice of generator coordinate. A comparison with the shell-model calculations of Soyeur and Zuker, and of Whitehead and Watt, is attempted.     

Search for cluster emission and extremely deformed shapes using charged-particle spectroscopy

The possible occurrence of highly deformed configurations is investigated in the N=Z nuclei ⁴⁰Ca, ⁴⁴Ti, and ⁵⁶Ni as formed in the ²⁸Si+¹²C, ¹⁶O+²⁸Si, and ²⁸Si+²⁸Si reactions by using the properties of the emitted light charged particles (LCP). The energy spectra, in-plane and out-of-plane angular correlations of LCP’s are analysed for each of the 3 studied reactions within the framework of the statistical model. Strong deformation effects are deduced from Hauser-Feshbach calculations performed with the Monte Carlo code CACARIZO by using a consistent set of parameters with spin-dependent level densities. The analysis of α particles in coincidence with ³²S fragments emitted in ²⁸Si+¹²C shows a strong contribution from a ⁸Be cluster emission. The binary nature of this cluster emission has been verified in three other reactions involving a ¹²C target: ²⁷A1+¹²C, ³¹P+¹²C, and ³²S+¹²C.     

Rotational bands from shell-model Hamiltonians

Shell-model Hamiltonians with eigenstates forming rotational bands are considered. Such states have eigenvalues proportional to J(J+ 1) and can be projected from a Slater determinant of deformed orbitals. The latter are linear combinations of single-nucleon wave functions of j-orbits in a major shell. Conditions on matrix elements and single-nucleon energies are obtained in terms of the deformation parameters. An actual effective interaction is constructed yielding exact ground-state rotational bands for ²⁰Ne and ²⁴Mg which gives reasonable agreement with energies of other sd shell nuclei. Unlike the case of SU(3) symmetry, spin-orbit interaction and different single-nucleon energies can be accommodated and the procedure is not confined to oscillator major shells. Other welcome departures of our effective interaction from the SU(3) picture are the absence of rotational spectra in oxygen isotopes and that the ²⁴Mg ground-state band is projected from a Nilsson-type deformed state with axial symmetry.     

Application of a folding-model optical potential to analyzing inelastic pion–nucleus scattering and the in-medium effect on a pion–nucleon amplitude

The folding-model optical potential is generalized in such a way as to apply it to calculating the cross sections for inelastic scattering of π ^±-mesons on ²⁸Si, ⁴⁰Ca, ⁵⁸Ni, and ²⁰⁸Pb nuclei at the energies of 162, 180, 226, and 291 MeV leading to the excitation of the 2⁺ and 3^? collective states. In doing this, use is made of known nucleon-density distributions in nuclei and the pion–nucleon scattering amplitude whose parameters were obtained previously by fitting the elastic scattering cross sections for the same nuclei. Thus, the values of quadrupole (β ₂) and octupole (β ₃) deformations of nuclei appear here as the only adjustable parameters. The scattering cross section is calculated by solving the relativistic wave equation, whereby effects of relativization and distortion in the entrance and exit scattering channels are taken exactly into account. The cross sections calculated in this way for inelastic scattering are in good agreement with respective experimental data. The importance of the inclusion of in-medium effects in choosing parameters of the pion–nucleon amplitude is emphasized.     

Self-consistent models of collective and single-particle structures and their test by inelastic scattering: The case of 54Fe

Generator coordinate wave functions built from angular momentum and particle number projected Hartree-Fock-Bogoliubov states that were constrained to different quadrupole deformations and allowing two-quasiparticle excitations are tested by their use in analyses of inelastic proton scattering. Transitions to low-lying states in ⁵⁴Fe are used as the specific examples and comparisons are made with appropriate, albeit smaller basis, shell-model predictions.     

Untersuchung von 0+−0+-Übergängen in12C,24Mg,28Si,32S und40Ca durch unelastische Elektronenstreuung

Transitions from the 0⁺ ground states to 0⁺ excited states at 7.65 (¹²C), 6.44 (²⁴Mg), 4.98 and 6.69 (²⁸Si), 3.78 (³²S) and 3.35 (⁴⁰Ca) MeV have been studied with 28 to 60 MeV electrons at scattering angles from 105 to 165°. Matrix elements and transition radii have been deduced, using DWBA-calculations. The monopole excitations can be uniquely distinguished from electric quadrupole excitations by the angular dependence of the cross sections. Some results forE2- andE3-transitions in¹²C,²⁸Si,³²S and⁴⁰Ca are given, too.     

Fast neutron radiative capture in silicon

Using the²⁸Si(n, γ)²⁹Si reaction, transitions to the ground state and first excited state in²⁹Si have been studied in the neutron energy range 3–14 MeV with improved neutron energy resolution (of about 100 keV). The 90° cross sections show considerable structure in the entire neutron energy range. Comparison with theoretical calculations shows that compound-nucleus and direct-semidirect processes account for the non-resonant part (smoothly varying part) of the cross section. A microscopic model is, however, required to describe the resonance structure. Continuum shell-model calculations have proven to be a very promising means towards a better understanding of the capture process in, and below, the giant resonance region in light nuclei. The angular distributions of gamma rays in the neutron energy range 8–14 MeV indicate that the capture reaction is mainly of direct character and that the effect of interference between the electric dipole and isoscalar quadrupole resonance is weak.     

Calculations of the giant dipole resonance for sd-shell nuclei in the open-shell random phase approximation

The systematics of the giant dipole resonance have been calculated in the open-shell RPA for all the self-conjugate sd-shell nuclei, using (i) a phenomenological Rosenfeld interaction, (ii) Barret, Hewitt and McCarthy G-matrix elements and (iii) Kuo G-matrix elements. The excitations are based on shell model ground states for all nuclei except ²⁸Si for which a projected Hartree-Fock ground state was used.     

Angular momentum projection from non-axial intrinsic states in Si

States of definite angular momentum projected from seven Hartree-Fock solution of ²⁸Si are used as a basis in which to diagonalize the Hamiltonian. It is found that each of these HF solutions contributes appreciably to the low energy spectrum.     

Fragmentation of giant multipole resonances in deformed nuclei

The fragmentation of the giant monopole resonance in deformed nuclei is first studied by coupling the monopole oscillation with the quadrupole oscillation by means of the variational procedure for resonance frequencies. It is shown that, for non-axial symmetry, the monopole oscillation couples with both m = 0 and 2 modes of the quadrupole oscillation and the giant monopole resonance is split into three components, whereas for axial symmetry, the fragmentation is given by E₀(1 + 0.86δ² ± 1.25δ³) and E₀(0.74 ± 0.22δ ? 0.21δ² ± 0.57δ³), where E₀ is the g monopole resonance energy for spherical nuclei, δ is the deformation parameter, and the upper and lower signs stand for prolate and oblate deformations, respectively. The initial fragmentation of the giant quadrupole resonance is seen to be little modified by the coupling, except for the m = 0 mode which is split into two components. The variational method is extended to general multipoles for an ellipsoid and the fragmentation of giant multipole resonances in deformed nuclei is investigated for both axial and non-axial symmetries. A brief discussion is also made about the meaning of the energy eigenvalue involved in the model wave equation in terms of multipole sum rules. The giant dipole resonance for the static octupole deformation is shortly considered. The giant E0 and E3 resonances for largely deformed nuclei are finally examined by solving the spheroidal eigenvalue equation and they are compared with the results of the giant dipole and quadrupole resonances.     

Proton spectroscopic factors of Sm isotopes in the pairing-plus-quadrupole model

Proton spectroscopic factors have been calculated for the single-proton states in Sm isotopes with N = 84–92 by using the deformed quasiparticle wave functions obtained in the pairing-plus- quadrupole model. Comparison is given with the experimental data from the pick-up reactions (d, ³He) and (t, ⁴He) to the product nuclei ^145–153Pm. Effects of deformation on the spectroscopic factors are studied.     

An analysis of α-particle inelastic scattering using Hartree-Fock densities

Differential cross sections for the elastic and inelastic scattering of α-particles from ²⁰Ne and ²⁴Mg were calculated, using HF wave functions that had previously been used for (p, p') calculations. We do not use projected wave functions, but we do require consistency between the elastic and inelastic form factors. The result is that although the HF solutions are unable to fit the data, the discrepancy is significantly less than that found in (p, p') calculations.