Effective three-nucleon forces from the folded diagram expansion

The folded diagram expansion for the effective hamiltonian of a system of three valence nucleons in the 1s0d shell is investigated. Beside the one-body and two-body operators, which already occur in the folded diagram expansion for systems with one and two valence nucleons, also folded diagrams involving three nucleons are obtained. These terms, which can be interpreted as a contribution to an effective three-nucleon force, yield a non-negligible contribution which is repulsive for the low-lying states. Two different techniques are studied for the summation of the folded diagrams. A very good convergence is obtained using the Lee-Suzuki iteration scheme.     

Three-body forces in sd-shell nuclei

The influence of three-body forces on the excitation spectra of nuclei with 3 valence nucleons in the sd-shell is investigated. Three-body forces are considered, which arise from an intermediate excitation of the interacting nucleons to the Δ(3, 3) resonance. Besides these real three-nucleon forces, effective three-body interactions are taken into account which are due to the restriction of the nuclear structure calculation to sd-shell configurations. Significant cancellations are observed between the different contributions to the effective three-nucleon force. The resulting three-body matrix elements yield only a small influence on the spectrum of the A = 19 systems. The typical size of the matrix elements, however, is large enough to expect a serious influence on the results of shell-model calculations with more than three valence nucleons.     

A semimicroscopic model of nuclear vibrations with separable forces and the giant dipole resonance of 12C

In the generalized ph basis generated for the realistic nuclear ground state, the nuclear hamiltonian is approximated by an expression containing the separable effective forces and terms which can be estimated from the spectroscopic data. The model hamiltonian obtained is used to describe normal nuclear vibrations. The application of this approach in the shell-model theory of nuclear reactions is discussed. The S-matrix for nucleon-nucleus scattering is calculated explicitly. The ¹²C photodisintegration calculation is made as a test example.     

Effective operator for the transition density

The effective shell-model operator for electric transition densities is evaluated. The influence of the choice of single-particle wavefunctions, residual interaction (G-matrix) and correlations within the valence shell as well as core-polarisation effects, is investigated. The excitation of the lowest 2⁺ in ⁵⁸Ni is considered as an example. The shape of the transition density can be reproduced only after renormalisation of the effective transition density operator to account for core polarisation. The possible importance of screening terms and many-body terms in the expansion for the effective operator are discussed.     

An exact fermion model with monopole and quadrupole pairing

A hamiltonian for valence nucleons is shown to give exact eigenstates that consist of monopole and quarupole pair creation operators acting on u unpaired nucleons. he quantum numbers and eigenenergies for u = 0 are shown to be the same as those given by the interacting boson approximation in the vibrational limit.     

Proton-neutron interaction near closed shells

Odd-odd nuclei around double shell closures are a direct source of information on the proton-neutron interaction between valence nucleons. We have performed shell-model calculations for doubly odd nuclei close to ²⁰⁸Pb, ¹³²Sn, and ¹⁰⁰Sn using realistic effective interactions derived from the CD-Bonn nucleon-nucleon potential. The calculated results are compared with the available experimental data, attention being focused on particle-hole and particle-particle multiplets. While good agreement is obtained for all the nuclei considered, a detailed analysis of the matrix elements of the effective interaction shows that a stronger core-polarization contribution seems to be needed in the particle-particle case.     

Structure functions of bound nucleons: From the EMC effect to nuclear shadowing

We describe the nuclear structure functions in the whole range of the Bjorken variablex, by combining various effects in a many-step procedure. First, we present a QCD motivated model of nucleons, treated, in the limit of vanishingQ ², as bound states of three relativistic constituent quarks. Gluons and sea quarks are generated radiatively from the input valence quarks. All parton distributions are described in terms of the confinement (or nucleon's) radius. The results for free nucleons are in agreement with the experimental determinations. The structure functions of bound nucleons are calculated by assuming that the main effect of nucleon binding is stretching of nucleons. The larger size of bound nucleons lowers the valence momentum and enhances the radiatively generated glue and sea densities. In the small-x region the competitive mechanism of nuclear shadowing takes place. It also depends on the size of the nucleons. By combining stretching, shadowing and Fermi motion effects (the latter confined to very largex), the structure function ratio is well reproduced. Results are also presented for theA-dependence of the momentum integral of charged partons, the nuclear gluon distribution and the hadron-nuclei cross sections.     

Five-dimensional quasispin and the spectra of odd-parity yrast configurations in the Ba region

The 5-dimensional quasispin formalism is used to evaluate the (n, T)-dependent factors of the matrix elements of abnormal parity operators of relevance for yrast-band spectroscopy. The formalism allows the leading pseudo-SU(3) representation of the normal (N) parity part of a shell-model configuration to be coupled to low-seniority states of the abnormal (A) parity high-j intruder part of the configuration. To illustrate the importance of the n_A, T_A dependence of the (A) to (N) space-coupling matrix elements this model is applied to the negative-parity yrast spectra of ¹²⁷La and ¹²⁷Ba, using hamiltonian parameters which give a reasonable fit for ¹²⁶Ba. The ¹²⁷La spectrum is reproduced with a seniority-one truncation of the h_{$\text{11}\text{2}$} configuration; the ¹²⁷Ba spectrum indicates a need for higher-seniority admixtures.     

Renaissance of the Nilsson-model approach to light nuclei: The case of the A = 26 system

The farthest reaching interpretation of a light nuclear system in terms of the Nilsson-model is obtained for A = 26. The T = 1 and T = 0 levels of either parity in ²⁶Mg and/or ²⁶Al are shown to be grouped in more than 35 rotational bands together containing some 55 levels in ²⁶Mg and 170 in ²⁶Al. The Nilsson-model configurations and deformations of bands are deduced from measured spectroscopic factors and electromagnetic transition rates. The experimental data are amended by shell-model calculations in the complete s-d basis space based on the universal s-d Hamiltonian. The rotational structure of positive-parity states is completely contained in the shell-model results. The Nilsson diagram gives a natural explanation of all bands. An important facet is the observation of particle-hole configurations with large K which arise from the complete alignment of particle and hole angular momenta along the major axis of deformation.     

Monte Carlo simulation of finite mass nucleons interacting via a neutral,scalar boson field

A recently proposed Monte Carlo method to solve the Schrödinger equation when expressed in Fock space is applied to the hamiltonian which describes the interaction of nucleons via a neutral, scalar boson field. The fact that a nucleon has finite mass is taken into account and a gaussian cut-off for the nucleon form factor is adopted. The problem is solved for systems with A = 1 and 2 sources (nucleons) in the three-dimensional continuous space. From the results for A = 1 a bare nucleon mass, m_Bc² = 962.58 ± 0.06 MeV, is obtained. This value is used to determine the binding energy for an A = 2 system by means of this new algorithm. The result, B(2) = 2.14 ± 0.50 MeV, is consistent with the value corresponding to the static potential approximation.     

On the valence model for radiative capture

We give several parametrizations for the elastic scattering and radiative capture cross sections for low neutron bombarding energy and discuss the relationship between the corresponding resonance parameters. We then perform an extensive investigation of the valence radiative capture model of Lane and Lynn. This model is formulated here in the frame of the shell-model approach. We exhibit the similarities and differences between our results and those derived from the R-matrix approach by Lane and Lynn on the one hand and from the optical-model approach by Lane and Mughabghab on the other hand. Particular attention is paid to the choice of the average potential well in the shell-model approach, in relation to the proper way to identify theoretical quantities and phenomenological parameters. We show that practically equivalent results can be obtained from a complex average potential well and from a suitably chosen real potential well, respectively. The following topics are investigated formally and numerically: dependence of the various theoretical expressions on the choice of the (real or complex) average potential well; relative importance of external and internal capture; dependence of photon widths and background cross section on mass number (for thermal energy and for E = 100 keV); dependence of the resonance parameters and background cross sections on energy, for A = 60; comparison between experimental data and theoretical values for radiative capture on ⁵⁶Fe and ⁶⁰Ni. We discuss the conditions of validity of the valence capture model. In particular, we investigate the role of the giant dipole resonance and of the closed channels. We argue that the success of the valence capture model is intimately related to the importance of external capture. The contribution of the low-lying excited target states is investigated formally and numerically; it increases with mass number and tends to diminish the correlation between neutron and photon widths, which is implied by the valence capture model.     

Microscopic theory of multiple scattering for open-shell nuclei

We consider the scattering of a distinguishable projectile from a nucleus assuming that the underlying interaction Hamiltonian is a sum of two-body potentials. We show that the effective interaction of the projectile with the nucleus in a truncated nuclear model space can be calculated as a linked-cluster expansion. The rules for evaluating this expansion are given in terms of the nucleon-nucleon and projectile-nucleon potentials and the exact eigenstates of the (effective) shell-model interaction. The shell-model interaction is required to be an energy-independent, Hermitian potential; its expression in terms of the underlying two-body potential is given by folded diagrams. The terms in the expansion of the effective projectile-nucleus interaction must also contain folded diagrams but, unlike the shell-model potential, these are energy-dependent in order to describe the singularities associated with the crossing of the scattering thresholds as the projectile energy is varied. Once the effective interaction is known, elastic and inelastic scatterings may be evaluated numerically by solving a finite-dimensional coupled-channel equation.     

An isospin invariant form of the interacting boson model for odd nuclei (IBFM-3)

The interacting boson-fermion model IBFM for odd nuclei is extended to incorporate isospin symmetry in nuclei where valence neutrons and protons occupy the same orbits. A mapping is established with shell-model states. A boson-fermion hamiltonian is derived for the special case of charge-independent pairing forces and is found to reproduce the shell-model energies exactly. A comparison of the extension (IBFM-3) with the earlier version IBFM-2 shows that the latter omits some low-lying shell-model states.     

Nuclear reaction matrix calculations with a shell-modelQ

The Barrett-Hewitt-McCarthy (BHM) method for calculating the nuclear reaction matrixG is used to compute shell-model matrix elements forA=18 nuclei. The energy denominators in intermediate states containing one unoccupied single-particle (s.p.) state and one valence s.p. state are treated correctly, in contrast to previous calculations. These corrections are not important for valence-shell matrix elements but are found to lead to relatively large changes in cross-shell matrix elements involved in core-polarization diagrams.     

Structure of odd Ge isotopes with 40 < N < 50

We have interpreted recentlymeasured experimental data of ⁷⁷Ge, and also for ^73,75,79,81Ge isotopes in terms of state-of-the-art shell-model calculations. Excitation energies, B(2) values, quadrupole moments and magnetic moments are compared with experimental data when available. The calculations have been performed with the recently derived interactions, namely with JUN45 and jj44b for f _5/2pg9/2 space. We have also performed calculation for fpg _9/2 valence space using an fpg effective interaction with ⁴⁸Ca core and imposing a truncation to study the importance of the proton excitations across the Z = 28 shell in this region. The predicted results of jj44b interaction are in good agreement with experimental data.     

Folded diagrams and 1s-Od effective interactions derived from Reid and Paris nucleon-nucleon potentials

The sd-shell effective-interaction matrix elements are derived from the Paris and Reid potentials using a microscopic folded-diagram effective-interaction theory. A comparison of these matrix elements is carried out by calculating spectra and energy centroids for nuclei of mass 18 to 24. The folded diagrams were included by both solving for the energy-dependent effective interaction self-consistently and by including the folded diagrams explicitly. In the latter case the folded diagrams were grouped either according to the number of folds or as prescribed by the Lee and Suzuki iteration technique; the Lee-Suzuki method was found to converge better and yield the more reliable results. Special attention was given to the proper treatment of one-body connected diagrams in the calculation of the two-body effective interaction.We first calculate the (energy-dependent) G-matrix appropriate for the sd-shell for both potentials using a momentum-space matrix-inversion method which treats the Pauli exclusion operator essentially exactly. This G-matrix interaction is then used to calculate the irreducible and non- folded diagrams contained in the $\text{Q}\text{?}\text{-}\text{box}$. The effective-interaction matrix elements are obtained by evaluating a $\text{Q}\text{?}\text{-}\text{box}$ folded diagram series. We considered four approximations for the basic $\text{Q}\text{?}\text{-}\text{box}$. These were (C1) the inclusion of diagrams up to 2nd order in G, (C2) 2nd order plus hole-hole phonons, (C3) 2nd order plus (bare TDA) particle-hole phonons, and (C4) 2nd order plus both hole-hole and particle-hole phonons.The contribution of the folded diagrams was found to be quite large, typically about 30%, and to weaken the interaction. Also, due to the greater energy dependence of higher-order diagrams, the effect of folded diagrams was much greater in higher orders. That is, the contribution from higher-order diagrams for most cases was greatly reduced by the folded diagrams. The convergence of the folded-diagram series deteriorates with the inclusion of higher-order $\text{Q}\text{?}\text{-}\text{box}$ processes in the method which groups diagrams by the number of folds, but remains excellent in the Lee-Suzuki method.Whereas the inclusion of the particle-hole phonon was essential to obtain agreement with experiment in earlier work, when the folded diagrams are included the effect of the particle-hole phonon is to reduce the amount of binding. All four approximations to both potentials produce interactions which badly underbind nuclei. The excitation spectra given by these interactions are, however, all rather similar to each other. The Paris interaction produces more binding than does the Reid, but differences between results obtained with the two interactions were often less than differences obtained in the four approximations. Essentially no difference was found between the effective non-central interactions from the Reid and Paris potentials after including the folded diagrams, although these two potentials themselves are quite different, especially in the strength of the tensor force.Comparisons between.calculated spectra and experiment were done for ¹⁸O, ¹⁸F, ¹⁹F, ²⁰O, ²⁰Ne, ²²Ne, ²²Na and ²⁴Mg.     

A hermitian open-shell many-body perturbation theory for treating intruder states

In this paper we have formulated an open-shell many-body perturbation theory (mbpt) that applies to an incomplete model space. The effective HamiltonianH ^eff generated in our theory is hermitian. We follow the resolvent-operator based time-dependent formulation ofmbpt of Banerjeeet al, and show quite generally that, by classifying the various determinants spanning the Hilbert space as model valence spaceP, virtual valence spaceR and virtual spaceQ, a diagrammaticmbpt satisfying size consistency can be developed. The chief new features of the theory are (i) manifest hermiticity ofH ^eff; (ii) presence of disconnected diagrams no part of which is a legitimate diagram (irreducible disconnected diagrams); (iii) presence of folded diagrams with components that may be irreducible disconnected diagrams; (iv) a consistent treatment of the variousn-valence sectors of the Hilbert space that may be interacting throughH ^eft if valence-holes are present—in particular a modified treatment of the core-valence separation. The generalisation afforded by the theory offers useful conceptual as well as computational advantages because the convergence difficulty encountered in a complete valence space formulation may be bypassed here by keeping the offending valence space determinants out of the model space. A brief critique of the developmentvis-a-vis the only other general model space development leading to a nonhermitianH ^eff is also given.