Multi-quasiparticle high-K isomeric states in deformed nuclei

In the past years, we have made many theoretical investigations on multi-quasiparticle high-K isomeric states. A deformation-pairing-configuration self-consistent calculation has been developed by calculating a configuration-constrained multi-quasiparticle potential energy surface (PES). The specific single-particle orbits that define the high-K configuration are identified and tracked (adiabatically blocked) by calculating the average Nilsson numbers. The deformed Woods-Saxon potential was taken to give single-particle orbits. The configuration-constrained PES takes into account the shape polarization effect. Such calculations give good results on excitation energies, deformations and other structure information about multi-quasiparticle high-K isomeric states. Many different mass regions have been investigated.


Introduction
Atomic nuclei can be excited by breaking paired nucleons [1][2][3][4].If the unpaired nucleons couple to a high angular momentum (usually resulting in a large angular momentum projection onto the symmetry axis of the deformed nucleus, named K), the excited state can be an isomer due to the forbiddenness of electromagnetic decays from high-K to low-K states [1,2].For example, the K π = 8 − isomers with the configuration π 2 {7/2 + [404] ⊗ 9/2 − [514]} systematically occur in 170−184 Hf, with half-lives that range from nanoseconds to hours.The well-known 31-yr K π = 16 + isomer at 2.4 MeV in 178 Hf is still attracting interest for possible applications.An abundance of high-K isomers have been observed across the entire chart of nuclides [1,2] including superheavy (see Ref. [5] and references therein) and drip-line mass regions [6,7].
Theoretically, previous calculations were usually performed assuming that isomers have the same deformation as the ground state of the nucleus (see, e.g., [8]).However, high-Ω orbits (Ω is defined as the nucleon angular momentum projection onto the symmetry axis of the deformed nucleus) usually have a strong deformation-driving force, that would make the isomer shape deviate from that of the ground state.Shape polarization is more obvious in soft nuclei [9][10][11][12].Furthermore, the pairing is another crucial factor that drastically affects the excitation energy of multi-quasiparticle states.The reduction of pairing in broken-pair multi-quasiparticle states leads to different pairing energies from the ground state.Configurationconstrained PES calculation takes into account these as-pects in a self-consistent manner with respect to deformation and pairing [3,13,14].

Model
We start from the non-axial deformed Woods-Saxon (WS) potential and use the universal potential parameters [15], with the monopole pairing strength G determined by the average gap method [16].The quadrupole pairing can also be included [17] but its effect on the excitation energies of multi-quasiparticle states is small [18].Due to nucleon pair(s) broken in multi-quasiparticle states, the pairing correlation is reduced remarkably in high-K states.To avoid the spurious phase transition encountered in the BCS approach, we approximate particle number projection by means of the Lipkin-Nogami (LN) pairing method.In the macroscopic-microscopic model, the total energy of a state contains a macroscopic part that is calculated by the liquid-drop model and a microscopic part that can be obtained by the Strutinsky shell correction.This leads to the energy of a configuration that is given by [3] where e k is the energy of the k-th single-particle orbit, and ) is the occupation (or unoccupation) probability of the orbit in the BCS wave function.G is the pairing strength, and Δ is the pairing gap.S is the seniority of the given type of nucleon, i.e., the number of unpaired protons or neutrons (indicated by k j ), and N is the number of protons or neutrons.λ 2 is a Lagrange multiplier that brings the second-order (ΔN 2 ) correction to the fluctuation of the particle number in the BCS wave function.Adiabatic blockings are achieved by tracking the specific single-particle orbits by calculating their average Nilsson numbers that evolve smoothly with changing deformation [3].

Calculations
We have investigated multi-quasiparticle high-K states in different mass regions.Furthermore, PES calculations give excitation energies, deformations, g-factors, and deformation softness.The configurations can be analysed by comparing with experimental observations.Figure 1 shows configuration-constrained PES's for the lowestenergy two-quasiparticle high-K states in neutron-rich zirconium isotopes [34].It can be seen that the calculated PES's show energy minima (giving the deformations of the states) and situations of deformation softnesses.The information is useful for analyses of the structure of a nucleus.Both triaxial deformation and softness can lead to K mixing and hence affect the decay properties of the multiquasiparticle state.It is interesting that high-K isomers can have longer lifetimes than corresponding ground states [35].This is due to the K selection rule in electromagnetic transitions.We have made a systematic investigation of possible high-K isomers in superheavy nuclei [5]. Figure 2 predicts possible two-quasineutron states in superheavy nuclei.The calculated excitation energies are low, around 1.0 − 1.5 MeV.The low-lying high-K states should be isomers that can be observed experimentally.Both calculations [5,38] and experiments [35][36][37] show that high-K isomerism can enhance the stability of superheavy nuclei, that is important for the experimental study of the heaviest nuclei.Figure 3     256 Fm.We see that the isomer has a higher and wider barrier than the ground state.It implies that the isomer should be more stable against fission which is the main decay mode of the heaviest nuclei.Figure 4 predicts some low-lying fourquasiparticle high-K states in Fm, No and Rf isotopes.Experiments have observed two-quasiparticle and fourquasiparticle isomers in 254 No [36].Our configurationconstrained PES calculations give good descriptions of the experimental excitation energies and configuration assignments [5,38].It has also been found that high-spin isomerism can enhance the stability of drip-line nuclei [6], which may extend the borders of the nuclide chart.
High-K states at extremely elongated shapes present another interesting topic related to nuclear isomerism.A large axially-symmetric prolate deformation would provide a favoured condition to preserve the intrinsic angular momentum K, and hence enhance the stability of the metastable state.Experiments have observed many su- perdeformed (SD) rotational bands.The most well-known mass region for the SD rotation is around A ∼ 190.It is an interesting open question whether multi-quasiparticle high-K states exist at superdeformation and can be both populated and detected in experiment.Figure 5 shows that there are many high-Ω single-particle orbits in the SD range of β 2 ≈ 0.4 − 0.5.If SD high-K states are located at low energies (relative to SD rotational bands), they would be populated.
With the configuration-constrained PES method, we have performed systematic calculations to search for possible SD multi-quasiparticle high-K states in the mass A ∼ 190 region [39].Figure 6 gives predicted SD twoquasiparticle high-K states.Their excitation energies are about 1.0 − 2.0 MeV higher than the calculated yrast SD rotational band.Figure 7 shows a scheme for electromagnetic transitions from possible SD high-K bands, which may provide guidance for future experiments.

Summary
We have performed deformation-pairing self-consistent configuration-constrained potential-energy-surface calculations for multi-quasiparticle high-K states in different mass regions.The calculations show that high-K states exist systematically in the superheavy mass region.The calculated excitation energies of high-K states can compete with those of collective rotational states.Experimentally several high-K isomers have been identified in the superheavy region.The present calculations can well reproduce the experimental data.We have also predicted superdeformed high-K states in the A ∼ 190 region where superdeformed rotational bands have been observed.The calculated excitation energies are about  1.0 − 2.0 MeV above the observed yrast superdeformed rotational band.The predictions can provide useful guidance for experiments.The collective rotations of multi-quasiparticle high-K states have also been calculated by the configuration-constrained total Routhian surface method [40,41].

Figure 7 .
Figure 7. Notional decay scheme of calculated high-K SD bands in 196 Pb, showing the main decay paths expected between bands.Energies are in keV, relative to the SD minimum.