Oscillator Shell Model Research Articles

Alpha spectroscopic factors for light nuclei

In recent papers (see Fliessbach 1975, 1976) it was shown that a consistent reaction theory requires a modification of the usual definition for alpha spectroscopic factors. A simple formula for the new spectroscopic factors is derived in the framework of the harmonic oscillator shell model. These factors are calculated for a number of light nuclei.

Cluster spectroscopic factors for thep-shell nuclei

A convenient method of calculation has been developed to determine the cluster spectroscopic factors for the ejection of excited and complex fragments ($^{4}\mathrm{He}$, $^{6}\mathrm{Li}$, $^{7}\mathrm{Li}$, $^{8}\mathrm{Be}$, etc.) from the low-lying levels of the $p$-shell nuclei followed by transition both to the states of the lowest shell configuration of the daughter nucleus and to its hole states. The method can be used to analyze the cluster knock-out, transfer reactions induced by heavy ions, etc.NUCLEAR STRUCTURE $p$-shell nuclei; calculated cluster spectroscopic factors. Harmonic oscillator shell model.

Pion exchange currents and nuclear charge form factors

The effect of the pion exchange current due to the Born term in the photoproduction amplitude on the charge form factors and distributions of 4He, 16O and 40Ca is calculated in the simple harmonic oscillator shell model. The exchange current contribution is most notable in the case of the α-particle while it is rather unimportant in 16O and 40Ca. The contribution to the nuclear mean square radius is small (≈ 0.03 fm 2) and roughly independent of the mass number.

Optical potentials for elastic heavy ion scattering

The real part of the optical potential is calculated in a microscopic model for the reactions4He-4He,16O-16O and40Ca-40Ca at several scattering energies. The two scattering partners are represented by ground state wave functions of an oscillator shell model localized at a given separation distance. The relative motion of the nuclei is approximated by a plane wave. For the nucleon-nucleon interaction we use the Skyrme force. The dependence of the optical potential on the scattering energy is investigated, both in the sudden and in the adiabatic approximation.

Scale transformations for renormalized field operators: Discussion of the soluble model: J. Lukierski. International Centre for Theoretical Physics, Trieste, Italy. And A. Ogielski. Institute of Theoretical Physics, University of Wrocław, Poland

Within the framework of the closure approximation dispersion corrections to elastic electron scattering from 12C and 16O have been calculated using the Harmonic Oscillator Shell Model and keeping only the Coulomb contributions to the dispersive part of the second-Born amplitude. Center-of-mass effects and recoil have been taken into account. By means of an expansion scheme, the first-order corrections to the closure approximation have also been calculated and under certain conditions, it is believed that the corrected expansion is a valid one. It is found that the effective excitation energy, defined by the expansion, is approximately one oscillator spacing. The relation between this calculation and a previous one by deForest [26] is discussed.

Spin-orbit splitting in nuclei near closed shells: (I). Contribution of the two-body spin-orbit interaction

We have studied the spin-orbit (s.o.) splitting arising from the effective two-body s.o. interaction in the Brueckner-Hartree-Fock (BHF) approximation, considering all of the most commonly used realistic nucleon-nucleon (NN) interactions and three different nuclear models. We conclude that the s.o. interaction alone cannot account for the observed s.o. splitting in nuclei. We derive the one-body s.o. potential arising from the effective NN s.o. interaction in the BHF approximation. The derivation applies to the interaction of a nucleon with a spin-saturated (s.s.) core of nucleons. The expression we obtain is more than a factor of two stronger than some similar expressions obtained by previous investigators. The enhancement results from the exchange term, which had previously been ignored. We discuss, and obtain expressions for, the effective s.o. interaction arising, in the local-density approximation, from the most commonly used realistic NN potentials, namely the Reid soft core (RSC), Reid hard core (RHC), Hamada-Johnston (HJ), Yale, and Gammel-Thaler (GT) potentials. We find that they yield two distinct sets of values for the strength, − 1 2 πS 30 , of the one-body potential: the RSC, RHC, and HJ potentials give − 1 2 πS 30 ≈ 59 MeV · fm 5 while the Yale and GT potentials give − 1 2 πS 30 ≈ 73 MeV · fm 5 . We have calculated the s.o. splitting for three different nuclear models: (a) the spherical harmonic oscillator (h.o.) shell model, (b) a hybrid model which uses a Fermi distribution to describe the nucleons in the s.s. core and h.o. orbitals to describe the single-particle levels, and (c) the self-consistent Brueckner model of Negele. For each of these, quantitative results for the splitting are obtained for the five modern NN potentials listed above. Significant differences between the three models are found, models (b) and (c) giving better agreement with experiment than (a) for the heavier nuclei. We have also used the h.o. model to verify the numerical accuracy of our approach. We find that the effective s.o. interaction gives splittings which are roughly compatible with experiment for the normally unoccupied levels of the entirely s.s. nuclei 16O and 40Ca. For all the other levels of the nuclei studied, it appears that the s.o. interaction alone lacks sufficient strength to account for the entire s.o. splitting.

Potential Energy Surfaces for the Fission of the Actinide Nuclei

Collective potential energy surfaces have been systematically calculated for the symmet­ ric fission of the five isotopes of the elements Th, Pu, Cm, Cf, Fm and No and the eight isotopes of the element U. The calculation is performed on the basis of Strutinsky's pre­ scription in which the liquid drop model of v. Groote and Hilf and the modified two-center harmonic oscillator shell model are used for macroscopic and microscopic parts, respectively. Effects of mass asymmetry at the second saddle point are investigated. The properties of the ground state, the second minim urn and the first and second saddle points along the static fission path are discussed in comparison with the results of Moller and Nix. The structure of the potential energy surface is found to come mainly from the shell structure which is strongly related to the distance between the mass centers of the nascent fragments. Special attentions are given to the fragment mass distributions and the constancy of the heavy fragment masses is partially explained on the basis of the properties of the second barrier. § I. Introduction Since the late 1960's, a renewed interest has been aroused 111 the structure of the potential energy surface for the fission process. It was motivated mainly by the discoveries of the fission isomer!) and the related resonant structure of the fission cross section. 2J The calculation based on the liquid drop model was proved not to be able to reproduce these featuers and in 1967 a new method for the calculation of the potential energy-- the macroscopic-microscopic model'J __ was proposed. Since then, calculations based on this model have been widely made and the fission isomer has been theoretically explained as a shape isomer. It is caused by the shell structure of a largely deformed nucleus and the fact that the shell structure appears at such a large deformation has made a drastic change m our knowledge of the nuclear structure. 4J The structure of the potential energy surfaces of the heavy nuclei is at present fairly well known. 5 J~sJ For the actinide nuclei, a typical feature is that there exists a second minimum and as a result the fission barrier is split into two, the inner barrier and the outer barrier. The calculated heights of these barriers and the second minimum were compared with the experimental data and overall agreement

Isospin-forbidden electroexcitation of the 1 −, T = 0 state at 7.12 MeV in 16O at low momentum transfers

The isospin-forbidden electroexcitation of the 1 −, T = 0 state at 7.12 MeV in 16O has been studied for momentum transfers q = 0.22 − 0.49 fm −1. The form factor at these momentum transfers is determined by the sum of the isospin T = 0 part and a T = 1 contribution. A single particle-hole harmonic oscillator shell model calculation reproduces the q-dependence of the measured form factor which shows that the isovector contribution interferes destructively with the isoscalar part and has a strength of about 2% of the T = 0 amplitude.

Dispersion corrections to elastic electron scattering and the closure approximation

It is shown that, in contrast to the pure closure approximation, the corrected closure approximation of Friar and Rosen is not very sensitive to the value of the excitation energy $omega$ and is in good agreement with the exact results as long as $omega$ is reasonably small. Calculations of the effective excitation energy which should be used in the pure closure approximation show that this quantity tends to be quite small and, except for very small momentum transfers, has little resemblance with the intuitive concept of an average excitation energy. Arguments have been presented which indicate that the results of the present calculation are probably representative of all dispersion calculations and not simply a feature of the harmonic oscillator shell model which we have used. (NL)

Quasimolecular states in the elastic α-scattering on 40Ca

Resonance states seen in α- 40Ca elastic scattering are treated in a microscopic model which describes the fragments by displaced oscillator shell model functions. The minima of the energy expectation value for various angular momenta are in good agreement with the experimental resonance energies, thus confirming the concept of an underlying quasimolecular structure.

Dispersion corrections to elastic electron scattering by 12C and 16O II. On the use of the closure approximation

Within the framework of the closure approximation dispersion corrections to elastic electron scattering from 12C and 16O have been calculated using the Harmonic Oscillator Shell Model and keeping only the Coulomb contributions to the dispersive part of the second-Born amplitude. Center-of-mass effects and recoil have been taken into account. By means of an expansion scheme, the first-order corrections to the closure approximation have also been calculated and under certain conditions, it is believed that the corrected expansion is a valid one. It is found that the effective excitation energy, defined by the expansion, is approximately one oscillator spacing. The relation between this calculation and a previous one by deForest [26] is discussed.

Alpha particle spectroscopic amplitudes in the j- j coupled shell model

Formulae for calculating spectroscopic amplitudes for α-transfer reactions are developed in the framework of the j- j coupled harmonic oscillator shell model. It is shown that the amplitudes can be written as a sum of coupled two-neutron and two-proton spectroscopic amplitudes, illustrating that many properties of α-transfer reactions can be understood in terms of our experience with two-nucleon transfer reactions.

Collective rotations and moments of inertia in the classical mechanics of a deformed oscillator shell model

This paper presents an alternative to the concepts of the rigid rotor or of the liquid drop as points of departure for the description of collective nuclear motions. The Lagrangian and Hamiltonian of a classical system of A bodies of equal mass are expressed in terms of non-redundant dynamical coordinates defined microscopically so as to correspond to the rotational and vibrational degrees of freedom used customarily in models of collective nuclear motions. Three dynamical coordinates describe the orientation of the principal axes of the tensor of inertia, and three coordinates describe the principal moments of inertia. The potential energy is assumed to depend only on the principal moments of inertia, so that the Lagrangian and Hamiltonian have simple separable forms. The orientation of the principal axes and the intrinsic structure are specified as functionals, rather than point functions, of angular coordinates. The relationships between the equilibrium values of the moments of inertia, the total angular momentum, and other constants of the motion are unambiguous and complete, and these form the basis for a classical theory of the mass parameter of collective rotations for axially symmetric systems. A type of collective motion which is neither a rotation of principal axes nor a shape oscillation appears naturally in this approach. The motion may be intuitively pictured as a coherent flow, and a constant of the motion roughly describable as the vorticity of the flow plays a key role in the description of moments of inertia. The calculation of the energy of axially symmetric systems, if the condition of shape equilibrium is postulated, is found to be formally analogous to the procedures of the variable moment of inertia model.

Application of infinite oscillator shell model calculations to the description of nuclear reactions

A simple R-matrix procedure is proposed to provide a means of using nuclear structure information, from calculations in an infinite oscillator well, to describe reaction cross sections. The procedure uses values for the channel radius parameters of the R-matrix theory that are extracted from knowledge of the structure and reaction properties of the neighbouring closed shell nucleus answering one of the main criticisms expressed concerning the dependence of the R-matrix on unphysical parameters. Application of the model to the scattering of neutrons from 15N and comparison to previous theoretical predictions are considered.

A simplified calculation of parity non conserving E1 transitions in the oscillator shell model

Contributions of excitations of the neutron core to the circular polarization of γ emitted in the transition 5 2 +→ 7 2 + in 181 Ta are calculated. We show in a very simple model that these contributions are coherent and multiply by a factor 2 the polarizations calculated previously with proton-neutron parity-violating potentials.

Relativistic corrections to Coulomb energy calculations

Coulomb energies of nuclei have been calculated using a recently introduced relativistic nuclear shell model 1). The results are very close to those of the usual non-relativistic, isotropic harmonic oscillator shell model, showing the most deviation for heavy elements such as lead.

Alpha-decay widths of 20Ne

Based on harmonic oscillator shell model wave functions, α-widths are calculated for the 20Ne ground state band and other levels. The SU 3 shell model wave function is rewritten as an α-cluster form and the tail of the relative 16O and α cluster wave function is improved by introducing a Woods-Saxon type effective potential. The calculated results are in good agreement with experimental data, and detailed features are discussed.

A phenomenological local soft-core nucleon-nucleon potential

The parameters of a phenomenological local charge-independent soft-core nucleon-nucleon potential with central, spin-orbit and tensor terms were fitted to the experimental data for the four two-nucleon states. The radial dependences are represented by sums of Gaussian functions. A potential for this kind is particularly suited for calculations of nuclear states and reactions, because it can be handled easily. Especially, it simplifies calculations in the harmonic oscillator shell model and in the refined cluster model.

The use of SU(3) in the elimination of spurious center of mass states

SU(3) recoupling techniques are used to give explicit constructions of the spurious states in the SU(3) scheme of the harmonic oscillator shell model. A compact formula is derived for all spurious states of excitation 1&Z.xl;hω in 1p shell nuclei. Spurious states of excitation 1&Z.xl;hω in heavier nuclei can be written down from a general formula whenever the needed fractional parentage coefficients are known. Calculation of spurious states of excitation 2&Z.xl;hω are illustrated with the simple example, 16O. It is shown how the purely non-spurious parts can be projected out of specific core excited states such as those of the weak coupling model for hole and particle states of high SU(3) symmetry. Most of the SU(3) recoupling coefficients needed for the construction of spurious states can be simply related to ordinary SU(2) Racah coefficients.

Description of nuclear reactions in the harmonic oscillator shell-model function scheme

The general reaction theory formalism given by Heiss and this author — based on the Kohn variational principle — is applied to the case that the two reaction partners in every channel are described in the harmonic oscillator shell model. One of them should be large so that translational invariance may be neglected; the other one consists of a maximum number of four nucleons so that its wave function is given by the translationally invariant shell model without difficulty. The Hamiltonian used in this theory is composed from the elementary nucleonnucleon interactions. It is shown how some matrix elements of this Hamiltonian can be computed; these matrix elements are the basic quantities in the reaction equation.