Monopole Pairing Force Research Articles

Shell-model-like approach based on cranking covariant density functional theory with a separable pairing force

The shell-model-like approach (SLAP) based on cranking covariant density functional theory (CDFT) with a separable pairing force is developed. The developed cranking CDFT-SLAP with separable pairing force is applied to investigate the rotational spectra in $^{60}$Fe, including the positive-parity yrast band and two negative-parity signature partner bands, in comparison with the cranking CDFT-SLAP with monopole pairing force calculations. Excellent agreement with the available data is achieved.

Fission fragment mass and total kinetic energy distributions of spontaneously fissioning plutonium isotopes

The fission-fragment mass and total kinetic energy (TKE) distributions are evaluated in a quantum mechanical framework using elongation, mass asymmetry, neck degree of freedom as the relevant collective parameters in the Fourier shape parametrization recently developed by us. The potential energy surfaces (PES) are calculated within the macroscopic-microscopic model based on the Lublin-Strasbourg Drop (LSD), the Yukawa-folded (YF) single-particle potential and a monopole pairing force. The PES are presented and analysed in detail for even-even Plutonium isotopes with A = 236–246. They reveal deep asymmetric valleys. The fission-fragment mass and TKE distributions are obtained from the ground state of a collective Hamiltonian computed within the Born-Oppenheimer approximation, in the WKB approach by introducing a neck-dependent fission probability. The calculated mass and total kinetic energy distributions are found in good agreement with the data.

Improved treatment of blocking effect at finite temperature

The blocking effect caused by the odd particle on the pairing properties of systems with odd number of fermions at finite temperature interacting via the monopole pairing force is studied within several approximations. The results are compared with the predictions obtained by using the exact solutions of the pairing Hamiltonian. The comparison favors the approximation with the odd particle occupying the top level, which is the closest to the Fermi surface and whose occupation number decreases with increasing temperature.

DYNAMICAL COUPLING OF ROTATION WITH THE PAIRING FIELD IN HEAVY NUCLEI

The macroscopic–microscopic model with the Lublin–Strasbourg Drop, the Yukawa–folded single-particle potential and a monopole pairing force is used together with the cranking model to describe rotational bands in even–even actinide ( Ra – Cn ) isotopes. The pairing strength is adjusted for every nucleus to reproduce the experimentally known energy of the rotational 2+ state. The average pairing strength obtained in this way is used to predict the rotational states in superheavy No – Cn nuclei. A simple mechanism which takes into account a dynamical coupling of rotation with the pairing field is explained in detail.

Rotational states and masses of heavy and superheavy nuclei

The macroscopic-microscopic model with the Lublin-Strasbourg drop, the Strutinsky shell-correction method, and the BCS approach for pairing correlations is used with the cranking model to describe nuclear masses and rotational bands in even-even Ra to Cn isotopes of actinide and transactinide nuclei. The single-particle levels and potential-energy surfaces are calculated with the Yukawa-folded single-particle potential using the “modified funny hills” (c, h) shape parametrization. A monopole pairing force is used in our calculations. At equilibrium deformation the pairing strength is adjusted for every nucleus so as to reproduce the experimentally known rotational E2+ state within the cranking model. The pairing strength obtained in this way is used to predict the masses and rotational states in superheavy No to Cn nuclei. It is also shown that the rotational states with L 10 in Ra to No nuclei evaluated using a simple rotational formula agree quite well with the data. We propose a simple mechanism which takes into account a dynamical coupling of rotation with the pairing field that then allows one to obtain an excellent agreement with the data up to the states with the largest angular momenta (L 30) measured in this mass region.

Nuclear scissors mode with pairing

The coupled dynamics of the scissors mode and the isovector giant quadrupole resonance are studied using a generalized Wigner function moments method taking into account pair correlations. Equations of motion for angular momentum, quadrupole moment and other relevant collective variables are derived on the basis of the time dependent Hartree-Fock-Bogoliubov equations. Analytical expressions for energy centroids and transitions probabilities are found for the harmonic oscillator model with the quadrupole-quadrupole residual interaction and monopole pairing force. Deformation dependences of energies and $B(M1)$ values are correctly reproduced. The inclusion of pair correlations leads to a drastic improvement in the description of qualitative and quantitative characteristics of the scissors mode.

Effect of pairing correlations on the nuclear scissors mode

The dynamics of the scissors mode and isovector giant quadrupole resonance has been studied using the generalized method of Wigner function moments, taking into account pairing correlations. Analytical expressions for the transition energies and probabilities have been obtained within the harmonic oscillator model with the quadrupole-quadrupole residual interaction and monopole pairing forces. The dependences of the energies and B(M1) values on deformation are correctly reproduced. Consideration of pairing correlations leads to radical improvement in the description of both qualitative and quantitative characteristics of the scissors mode.

THE EFFECTS OF DEFORMATION AND PAIRING CORRELATIONS ON NUCLEAR CHARGE FORM FACTOR

A set of moderately deformed s–d shell nuclei is employed for testing the reliability of the nuclear ground state wave functions which are obtained in the context of a BCS approach and offer a simultaneous consideration of deformation and pairing correlations effects. In this method, the mean field is assumed to be an axially symmetric Woods-Saxon potential and the effective two-body interaction is a monopole pairing force. As quantities of main interest we have chosen the nuclear form factors, the occupancies of the active (surface) orbits and the Fermi sea depletion, which provide quite good tests for microscopic descriptions of nuclei within many body theories. For our comparisons with results emerging from other similar methods, an axially deformed harmonic oscillator field is also utilized.

The spin-quadrupole interaction and the 0+ excitations in 158Gd

In this work, the effect of spin-quadrupole forces on the 0+ sates in 158Gd has been investigated. For this purpose, the model Hamiltonian including monopole pairing, quadrupole-quadrupole and spin-quadrupole forces has been diagonalized in one phonon basis. In conclusion, for the distribution of energies of the states and their collective properties, fairly good results have been obtained.

Collective excitations and a backbending phenomenon in 156Dy

We propose a self-consistent practical method to study collective excitations in rotating nuclei within the cranking + random phase approximation approach. It consists in solving the cranking Hartree-Bogolyubov equations for the modified Nilsson potential + monopole pairing forces. Further, the mean field results are used to construct collective vibrations treated in the random phase approximation (RPA). Special attention is paid to fulfill all conservation laws in the RPA to separate spurious and physical solutions. We demonstrate that the backbending in 156Dy can be explained as a result of the disappearance of collective γ vibrations of the positive signature in the rotating frame.

Differences between pairing and zero-range effective interactions for nuclear binding energies

In this paper, we show that, although the spectroscopic properties of the monopole pairing force and a zero-range delta-function interaction are very similar, their saturation properties are quite different. In particular, the predictions for binding energies when filling up a major shell are radically different past mid-shell. This has significant consequences for understanding the masses and binding energies of long isotopic chains of nuclei that will be accessible with advanced exotic beam facilities.

Nuclear Chaotic Behavior in a Two-j Shell Coupled with a Rotor Model**The project supported by National Natural Science Foundation of China, the Major State Basic Research Development Program under Grant No. G2000-0774-07 and the CAS Knowledge Innovation Project under contract No. KJCX2-N11

The chaotic properties for six particles interacting by a monopole pairing force in a two-j shell model coupled with a deformed core are studied in the frame of particle-rotor model. The nearest-neighbor distribution of energy levels and spectral rigidity in the two-j shell are compared with those in the single-j case. The results show that the system is more regular in the two-j model than that in the single-j case.

Solution of the Dyson equation for nucleons in the superfluid phase

We investigate the role the interweaving of surface vibrations and nucleon motion has on Cooper pair formation in spherical superfluid nuclei. A quantitative calculation of the state-dependent pairing gap requires to go beyond the quasiparticle approximation, treating in detail the breaking of the single-particle strength and of the associated poles. This is done solving self-consistently the Dyson equation, including both a bare nucleon–nucleon interaction (which for simplicity we choose as a monopole-pairing force of constant matrix elements g) and an induced interaction arising from the exchange of vibrations (calculated microscopically in QRPA) between pairs of nucleons moving in time reversal states. Both the normal and anomalous density Green functions are included, thus treating self-energy and pairing processes on equal footing. We apply the formalism to the superfluid nucleus 120Sn. Adjusting the value of g so as to reproduce, for levels close to the Fermi level, the empirical odd–even mass difference ( Δ≈1.4 MeV), it is found that the pairing gap receives about equal contributions from the monopole-pairing force and from the induced interaction. This result is also reflected in the fact that if one were to reproduce the observed Δ allowing the nucleons to interact only through the bare monopole-pairing force, a value of g≈0.233 MeV (≈28/ A MeV) is needed, 50% larger than the value g≈0.166 MeV (≈20/ A MeV) needed in the full calculation. To keep in mind that the bare and the induced pairing contributions to Δ enter the corresponding equations in a very nonlinear fashion. It is furthermore found that self-energy processes reduce the contribution of the phonon induced interaction to the pairing gap by ≈20% as compared to the value obtained by only phonon exchange without taking into account the breaking of the single-particle strength.

Details of nuclear masses away from stability: pairing and deformation effects

Exact many-body calculations have been performed in a large Nilsson-model basis using a two-body monopole-pairing force. The diagonalisations are reduced to a manageable size by exploiting the symmetry of the general pairing Hamiltonian and time-reversal invariance. The results have been successfully applied to a study of multi-quasiparticle states in the Hf-W region. An extension of this work, calculating nuclear masses, allows a comparison with the experimental neutron-pairing energiesP N. These have been studied in detail for the Kr isotopes. Two important conclusions emerge (i) the monopole-pairing term must be generalized to a state-dependent interaction; good results being obtained for a delta-function force (ii) the small nuclear deformations relevant to this mass region appear to vary smoothly, as given by laser spectroscopy experiments, rather than showing the fluctuations of microscopic-macroscopic calculations.

Possible solution of the Coriolis attenuation problem

The most consistently useful simple model for the study of odd deformed nuclei, the particle-rotor model (strong-coupling limit of the core-particle coupling model) has nevertheless been beset by a long-standing problem: It is necessary in many cases to introduce an ad hoc parameter that reduces the size of the Coriolis interaction coupling the collective and single-particle motions. Of the numerous suggestions put forward for the origin of this supplementary interaction, none of those actually tested by calculations has been accepted as the solution of the problem. In this paper we seek a solution for the difficulty within the framework of a general formalism that starts from the spherical shell model and is capable of treating an arbitrary linear combination of multipole and pairing forces. With the restriction of the interaction to the familiar sum of a quadrupole multipole force and a monopole pairing force, we have previously studied a semimicroscopic version of the formalism whose framework is nevertheless more comprehensive than any previously applied to the problem. We obtained solutions for low-lying bands of several strongly deformed odd rare-earth nuclei and found good agreement with experiment, except for an exaggerated staggering of levels for $K=$$\frac{1}{2}$ bands, which can be understood as a manifestation of the Coriolis attenuation problem. We argue that within the formalism utilized, the only way to improve the physics is to add interactions to the model Hamiltonian. We verify that by adding a magnetic dipole interaction of essentially fixed strength, we can fit the $K=$$\frac{1}{2}$ bands without destroying the agreement with other bands. In addition we show that our solution also fits ${}^{163}$Er, a classic test case of Coriolis attenuation that we had not previously studied.

Semi-microscopic theory of quantized rotational and intrinsic motions in deformed nuclei

The method of mode-mode coupling for the rotational and intrinsic excitations in deformed nuclei, previously worked out by Kuriyama for a single- j-shell model with pairing plus quadrupole force, is first refined for a many- j-shell model and then extended to a model with the monopole and quadrupole pairing plus quadrupole force. The collective rotation is regarded as an independent branch of intrinsic excitations. Then, the coupling between rotational and intrinsic-excitation modes is derived microscopically. The method consists of succesfully solving the equation of motion for fermion pair operators. The method makes it possible to determine, without involving the concept of forced rotation, all the matrix elements of the operators within and between rotational bands, as well as the moment of inertia, which coincides with the sum of the cranking model term and the Migdal term. The present method is compared with other approaches.

Structure of superdeformed states in AuRa nuclei

Energies of superdeformed states in nuclei around 192Hg are calculated using the Strulinsky shell correction method with an average Woods-Saxon potential and a monopole pairing force. The influence of various terms in the model hamiltonian on the excitation energy of the superdeformed minimum is analysed. The systematics of calculated excitation energies of shape-isomeric minima and barrier heights in even-even HgRa nuclei are given together with predictions for one-quasiparticle band-head energies in odd-A Au, Hg, Tl, Pb and Bi nuclei. Possible occurrence of hyperdeformed states in very neutron-deficient PoRa isotopes is investigated. It is shown that the presence of these exotic configurations is strongly related to the magnitude of pairing correlations at strongly elongated shapes. Finally, the role played by reflection-asymmetric deformations at superdeformed shapes is discussed.

Pairing and repulsive effective interactions

A two-level model with particles interacting via monopole pairing forces is used to show that the ground state can exhibit superfluidity even in cases where the interaction is seemingly repulsive. This mechanism is expected to play a role in the understanding of pairing correlations obtained for meson exchange forces, where the interaction has repulsive as well as attractive components.

Two-octupole-phonon states in 208Pb

Abstract The so-called “two-octupole-phonon states” in 208 Pb are theoretically analyzed by using the Dyson boson mapping method for the collective multi-phonon space consisting of collective octupole phonons, collective quadrupole phonons and collective monopole pairing phonons. The effective hamiltonian used is constituted by a Woods-Saxon-type single-particle potential, monopole pairing forces for neutrons and protons, a quadrupole-quadrupole force and an octupole-octupole force. The numerical calculation reasonably reproduces the experimental low-energy spectrum and E2 and E3 transitions in 208 Pb, and we can get a good understanding on the low-energy collective properties in 208 Pb.

Structure of superdeformed bands in the A ≈ 150 mass region

The structure of superdeformed rotational bands recently discovered around 152Dy is discussed within the deformed shell model based on an average Woods-Saxon potential with a monopole pairing force. A comparison with available experimental data is provided and detailed predictions for yet unobserved cases are given. Pronounced variations in the observed rotational pattern are attributed to the angular momentum alignment of the high- N intruder (quasi)particles.