Schematic Forces Research Articles

BruecknerG-matrix approach for neutron-proton pairing correlations in the deformed BCS approach

Ground states of even-even Ge isotopes with mass number $A=64\text{--}76$ have been studied in the deformed Bardeen-Cooper-Schrieffer (BCS) theory by taking neutron-proton $(np)$ pairing correlations as well as neutron-neutron $(nn)$ and proton-proton $(pp)$ pairing correlations. The $np$ pairing has two different modes $J=0,\phantom{\rule{4pt}{0ex}}T=1$ (isotriplet) and $J=1,\phantom{\rule{4pt}{0ex}}T=0$ (isosinglet). In this work, the Brueckner $G$ matrix, based on the CD-Bonn potential, has been exploited to reduce the ambiguity regarding nucleon-nucleon interactions inside nuclei compared to the results by a simple schematic phenomenological force. We found that the $G$ matrix plays important roles to obtain reasonable descriptions of even-even nuclei compared to the schematic force. The $np$ pairing strength has been shown to have a clear correlation with quadrupole deformation parameter ${\ensuremath{\beta}}_{2}$ for the isotopes, and affects the smearing of the Fermi surfaces of not only $N=Z$ nuclei but also $N\ensuremath{\ne}Z$ nuclei. In particular, the coexistence of the like particle ($nn$ and $pp$) and the $np$ pairing modes was found to become more salient by the $G$-matrix approach than by the schematic force approach.

Evolution of the nuclear structure approachingNi78:βdecay ofCu74−78

A \ensuremath{\beta}-decay study of the even mass $^{74,76,78}\mathrm{Cu}$ isotopes toward levels in $^{74,76,78}\mathrm{Zn}$ was performed at the ISOLDE mass separator. The copper isotopes were produced in proton- or neutron-induced fission reactions on $^{238}\mathrm{U}$, laser ionized, mass separated, and sent to a $\ensuremath{\beta}\ensuremath{-}\ensuremath{\gamma}$ detection system. Half-lives, decay schemes, and possible spin configurations were obtained for the copper isotopes. The results are compared with calculations using schematic forces as well as large-scale shell-model calculations with realistic forces.

Mean field at finite temperature and symmetry breaking

For an infinite system of nucleons interacting through a central spin–isospin schematic force we discuss how the Hartree–Fock theory at finite temperature T yields back, in the T=0 limit, the standard zero-temperature Feynman theory when there is no symmetry breaking. The attention is focused on the mechanism of cancellation of the higher order Hartree–Fock diagrams and on the dependence of this cancellation upon the range of the interaction. When a symmetry breaking takes place it turns out that more iterations are required to reach the self-consistent Hartree–Fock solution, because the cancellation of the Hartree–Fock diagrams of order higher than one no longer occurs. We explore in particular the case of an explicit symmetry breaking induced by a constant, uniform magnetic field B acting on a system of neutrons. Here we compare calculations performed using either the single-particle Matsubara propagator or the zero-temperature polarization propagator, discussing under which perturbative scheme they lead to identical results (if B is not too large). We finally address the issue of the spontaneous symmetry breaking for a system of neutrons using the technique of the anomalous propagator: in this framework we recover the Stoner equation and the critical values of the interaction corresponding to a transition to a ferromagnetic phase.

Fermion-boson interactions and quantum algebras

Quantum Algebras (q-algebras) are used to describe interactions between fermions and bosons. Particularly, the concept of a su_q(2) dynamical symmetry is invoked in order to reproduce the ground state properties of systems of fermions and bosons interacting via schematic forces. The structure of the proposed su_q(2) Hamiltonians, and the meaning of the corresponding deformation parameters, are discussed.

Systematic study of nuclear beta decay.

\ensuremath{\beta}-decay properties of nuclei are studied in the framework of proton-neutron quasiparticle random-phase approximation with a schematic Gamow-Teller residual interaction. Particle-hole and particle-particle terms of the separable Gamow-Teller force are consistently included for both ${\mathrm{\ensuremath{\beta}}}^{+}$ and ${\mathrm{\ensuremath{\beta}}}^{\mathrm{\ensuremath{-}}}$ directions, and their strengths are fixed as smooth functions of mass number A of nuclei in such a way that the calculation reproduces observed \ensuremath{\beta}-decay properties of nuclei. Using the fixed interaction strengths, \ensuremath{\beta}-decay half-lives of nuclei up to A=150 are calculated, and generally good agreement with experiment is obtained. A schematic force which is relevant to the unique first-forbidden decay is also included in a similar way to the Gamow-Teller force, and its effects are studied. \textcopyright{} 1996 The American Physical Society.

The charge exchange IVQ strength and the role of nonlocality forces

We evaluate commutators involving a general hamiltonian H and one body hermitian operators in the TDHF-RPA scheme and give explicit forms for such commutators in the case of schematic forces of separable type. The suggestion that nonlocal forces could be responsible for shifting the IVQ strength in the charge exchange channels and thus explain the nonobservation of this strength is investigated and found to be unjustified. In particular, we show that the relations used in that approach do not take into consideration the proper symmetry character of the interaction and lead to a breakdown of the sum rules. We then present the corrected relations for hamiltonians involving separable forces of mixed type and refer to an alternative explanation of the problem regarding the IVQ strength.

On some aspects of the semiclassical approach to giant resonances of rotating nuclei

Quadrupole and isovector dipole resonances of rotating nuclei are investigated in the frame-work of Vlasov equations transformed to a rotating system of reference, which are based on the time-dependent Hartree-method for schematic forces. The parameter free model of the self-consistent vibrating harmonic oscillator potential for the quadrupole mode is extended to a coupling to rotation, which also includes large amplitude behaviour. A generalization to an exactly solvable two-liquid model describing the isovector mode is established; for rotating nuclei Hilton's explicit result for the eigen-frequencies is obtained. — The advantage of using the concept of the classical kinetic momentum in a rotating system also in quantum-mechanical descriptions is demonstrated. It completes the standard transformation of density matrices by a time-odd part realized in a phase-factor and permits a more direct interpretation of rotation effects in terms of the classical forces of inertia. — In its generalization from constant angular velocity to constant angular momentum, our model is used to demonstrate that cranking calculations of rotating giant resonances should be corrected for an oscillation of the cranking parameter to assure angular-momentum conservation.

Shifts of the spherical single-particle levels

Spherical HFB calculations are reported for a wide range of nuclei using a variety of “realistic” and schematic forces. It is shown that the realistic interactions lead to substantial shifts of the spherical single-particle levels as a function of N and Z, large enough in some cases to change the “magic” numbers away from the line of nuclear stability. A simple force is presented which can simulate these effects.

A schematic model for collective states including non-locality effects: (I). Isoscalar quadrupole and rotational modes

Schematic forces including non-locality terms are derived in a self-consistent way in the framework of the time-dependent Hartree-Fock formalism. The usual dispersion relation of the vibrating potential model is generalized to include current effects. The model is explicitly solved for the isoscalar quadrupole and rotational modes and the role of the effective mass is pointed out.

Nuclear level density with schematic forces

For a system of fermions interacting via a pure pairing force, exact level densities are obtained in a semirealistic model space. The results are transmitted through graphs and tables so that they are readily available for comparison with various approximate methods. A fairly detailed comparison with a particular approximate method using the grand partition function is also made. The consequences of adding the quadrupole force to the Hamiltonian are discussed. Variational calculations done in the specific case of Sn isotopes suggest that although these nuclei are spherical in the ground state (zero temperature), it is possible that they prefer deformation at finite but low temperature. This can lead to very large changes in the calculated values of level densities.[NUCLEAR STRUCTURE $^{116}\mathrm{Sn}$, $^{117}\mathrm{Sn}$; calculated level density using quasispin formalism, grand partition method, and pairing-plus-quadrupole model at finite temperature.]

Can phenomenological particle-particle and particle-hole forces be distinguished?

It is attempted to answer the question whether or not the simple, schematic forces widely used in nuclear structure calculations should be given particle-particle (hole-hole) character, particle-hole character or possibly a definite mixture. If the phenomenology of the forces is fixed by requiring a reproduction of experimental energy systematics then even a sensitive measure of the correlations as the two-nucleon transfer does not conclusively distinguish the mentioned types of interaction. The reaction 208Pb(t,p) 210Pb is used as an example.

The 2+ and 3− collective states in the Sn and Zr isotopes using schematic forces

The energies and transition probabilities of the low-lying 2+ and 3− states in the Sn and Zr isotopes are calculated by the Random Phase Approximation method. The results forδ-force and surface delta interaction are compared. The properties of 3− states are practically independent of the residual interaction used while for the 2+ states the adequate agreement with experimental data was obtained usingδ-force only. The influence of the interaction in the particle-particle channel proves to be important in the case of 2+ excitations only.

Schematic forces and number-conserving approach to spherical nuclei

In the frame of a method originally proposed by Jean, an exact expression is found for the eigenvalues of some residual interaction Hamiltonians in the degenerate model of spherical nuclei. The limits of validity of the isopairing force and other similar schematic forces are examined and extended. For the nondegenerate model, in a particular case (small energy gap between the various shells) a simple approximate formula for the eigenvalues of the pairing Hamiltonian is derived.