Microscopic analysis of octupole shape transitions in neutron-rich actinides with relativistic energy density functional

Quadrupole and octupole deformation energy surfaces, low-energy excitation spectra, and electric transition rates in eight neutron-rich isotopic chains – Ra, Th, U, Pu, Cm, Cf, Fm, and No – are systematically analyzed using a quadrupole-octupole collective Hamiltonian model, with parameters determined by constrained reflectionasymmetric and axially-symmetric relativistic mean-field calculations based on the PC-PK1 energy density functional.The theoretical results of low-lying negative-parity bands, odd-even staggering, average octupole deformations β_3,and B(E3;3_1~- →0_1~+) show evidence of a shape transition from nearly spherical to stable octupole-deformed, and finally octupole-soft equilibrium shapes in the neutron-rich actinides. A microscopic mechanism for the onset of stable octupole deformation is also discussed in terms of the evolution of single-nucleon orbitals with deformation.     

A measurement of B(E3;0+ → 31−) and some E2 transition probabilities in 132, 134, 136, 138Ba using Coulomb excitation

Reduced transition probabilities, B(E2) and B(E3), have been measured for low-lying 2⁺ and 3^? states in ^{132, 134, 136, 138}Ba using Coulomb excitation by 40 MeV ¹²C ions. The B(E2) values are in general consistent with previous measurements and the B(E3;0⁺ → 3₁^?) values are 0.176 ± 0.022, 0.148 ± 0.018, 0.155 ± 0.018 and 0.133 ± 0.013 e² · b³ for ^{132, 134, 136, 138}Ba respectively. These B(E3) values correspond to about 24 to 17 W.u. and such enhancements suggest that these 3^? states have an essentially collective character which may be attributed to octupole vibrations.     

Number-conserving treatment of the BCS-Tamm-Dancoff approximation for deformed rare-earth nuclei

The formulation for the exact number-conserving treatment of the BCS-Tamm-Dancoff approximation (NTDA) in deformed even nuclei is given. It is applied to theK ^π=0^? octupole vibrations of rare-earth nuclei and is also compared with the ordinary TDA or RPA method based on the Bogoliubov transformation which has the defect of mixing of the particle number in the wave functions. The excitation energies andB(E3) calculated by NTDA method show rather stronger dependence on the Nilsson orbits than those calculated by the usual TDA or RPA methods, especially atN=94 nuclei.     

Dynamic deformation theory of the B(E2) values and the quadrupole shape characteristics of even gadolinium nuclei

Dynamic deformation theory, combined with the pairing-plus-quadrupole model, is employed to calculate the level energies and the B(E2) values of even ^152–160Cd. Calculated energies of the 2_β, 2_γ states are too high by ≈ 0.5 MeV, but the absolute B(E2) values for the excitation of these as well as several other states (2_g, 4_g, 2_2β, 0_β) are reproduced remarkably well. Various static and dynamic shape characteristics of the Gd nuclei are discussed.     

Semimicroscopic description of giant octupole resonances in deformed nuclei

The strength functions b(E3, ω) are calculated, and the positions and widths of giant octupole resonances in deformed nuclei are found. It is shown that the giant octupole isoscalar resonances have energies (19–20) MeV for the rare-earth nuclei and (17–18) MeV for the actinides and the widths (5–7) MeV. The energies of the giant octupole isovector resonances are defined by the value of the isovector constant κ⁽³⁾₁.     

Nuclear pears: Recent developments and future prospects

We report studies of examples of reflection-asymmetric nuclei which are difficult to access using compound nucleus reactions. The octupole radium isotopes withN>132 and radon isotopes are not accessible by reactions employing stable targets and beams; we have shown that multinucleon transfer reactions can populate these nuclei with sufficient yield for their structure to be determined. We report high-spin studies in^{218, 220, 222}Rn and^{222, 224, 226, 228, 230}Ra: these show that the Ra isotopes withA<228 have the characteristics of octupole deformed nuclei whereas the Rn isotopes behave like octupole vibrators. Measurements of theB(E1)/B(E2) ratios indicate that the electric dipole moment in these nuclei is constant with spin. The most octupole deformed nuclei are predicted to be uranium isotopes withN≈132; measurements of the very fissile nucleus²²⁶U suggest that it is octupole deformed and has a large intrinsic electric dipole moment. Finally, we speculate that the best examples of pear shapes are the hyperdeformed minima predicted to lie low in uranium isotopes withN≈140; their signature of high-multiplicity low-energyE1 photon cascades should be detectable using present-day high-efficiency germanium arrays.     

Magnetic moments of K isomers as indicators of octupole collectivity

The relation between the quadrupole-octupole deformation and the structure of high-K isomers in heavy even-even nuclei is studied through a reflection asymmetric deformed shell model including a BCS procedure with constant pairing interaction. Two-quasiparticle states with K ^?? = 4^?, 5^?, 6^?, 6⁺ and 7^? are considered in the region of actinide nuclei (U, Pu and Cm) and rare-earth nuclei (Nd, Sm and Gd). The behaviour of two-quasiparticle energies and magnetic dipole moments of these configurations is examined over a wide range in the plane of quadrupole and octupole deformations (?? ₂ and ?? ₃. In all considered actinide nuclei, the calculations show that there is pronounced sensitivity of the magnetic moments to the octupole deformation. In the rare-earth nuclei, the calculations for ^{154, 156}Gd show stronger sensitivity of the magnetic moment to the octupole deformation than in the other considered cases.     

Particle plus octupoleM2/E3 isomers and high-spin particle-hole states in147Tb and148Tb

The level structures of theN=82 andN=83 nuclei¹⁴⁷Tb and¹⁴⁸Tb have been studied by means of (α, 8n) and (α,7n) reactions induced by 68 to 110 MeVα particle bombardments of¹⁵¹Eu targets. In-beam conversion electron measurements have established that isomers withT _1/2=4.8(6)ns in¹⁴⁷Tb andT _1/2=22(1)ns in¹⁴⁸Tb decay byM2+E3 transitions to the ground states. The measuredB(E3) values show that the isomeric states arise from the coupling of the valence nucleon(s) to the¹⁴⁶Gd core octupole. Particlephonon coupling in these nuclei and in the one-neutron nucleus¹⁴⁷Gd is discussed and compared with well known cases involving the²⁰⁸Pb core. The higher lying yrast states in the two Tb nuclei are described as shell-model particle-hole excitations using empirical single particle energies and nucleon-nucleon interactions.     

Search for octupole deformation in neutron-rich Xe isotopes

A search for octupole deformation in neutron-rich Xe isotopes has been conducted through prompt gammaray spectroscopy of secondary fragments produced in the spontaneous fission of²⁴⁸Cm. The spectrometer consisted of the Eurogam 1 array and a set of 5 LEPS detectors. Level schemes were constructed for Xe isotopes with mass number ranging from 140 to 144 and excited states for^143,144Xe nuclei were observed for the first time. None of the level schemes exhibit an alternating parity quasimolecular band, a feature usually expected in nuclei in which octupole correlation effects are strong enough to produce stable octupole deformation. For several isotopes, structures observed in the level schemes are consistent with an octupole softness of the nuclei.     

Signature-dependent ml and e2 transition probabilities in 155ho and 157ho

¹⁵⁵Ho and ¹⁵⁷Ho have been populated in the reactions ¹⁴¹Pr(¹⁸O,4n) and ¹⁴⁶Nd(¹⁵N, 4n) at 85 and 74 MeV, respectively. In both nuclei bands built on the $\text{7}\text{2}$?[523] configuration were established to spin values considerably above the first backbend. A signature dependence in the excitation energies as well as in the ratio of M1 to E2 transition rates is observed below, but not above, the backbend in both nuclei. In ¹⁵⁷Ho lifetimes were measured with the recoil-distance method. The ΔI = 2; E2 transition probabilities obtained show very little variation with either signature or spin and no irregularity at the backbend. The signature dependence and strong rise in the ratio B(M1)/B(E2) observed at the backbend in ¹⁵⁷Ho therefore must be caused by the B(M1) values. A signature dependence in the B(E2, I → I ?1)/B(E2, I → I ?2) ratios also found in ¹⁵⁷Ho below the backbend is mainly the result of signature dependence in the ΔI = 1 ; E2 transition rates. Qualitatively, most of the features observed can be explained by nonaxial deformations, which change from large negative to slightly positive values of γ at the backbend.     

Fission barriers and other characteristics of nuclei from the uranium region

Fission barriers in nuclei belonging to the uranium region and their other characteristics are calculated on the basis of the FaNDF⁰ energy density functional. In particular, the neutron-separation energies S _n and S _2n, the proton-separation energies S _p, and the beta-transition energiesQ _β are calculated for uranium, neptunium, and plutonium isotopes. In addition, the deformation energies and parameters of these nuclei are presented along with their radii. A comparison with the predictions of the Skyrme–Hartree–Fock method implemented with several versions of the Skyrme energy density functionals is performed. The role of the octupole deformation β ₃ is studied for the ²³⁸U nucleus. It is shown that this deformation does not have any significant effect on the first-barrier height B ⁽¹⁾ _f or ground-state properties. At the same time, the second-barrier height B(2) f decreases by a factor of about two upon taking into account β ₃. A phase transition at A ~ 260 is found for the three isotopic chains being considered: this point is a bifurcation point at which B ⁽¹⁾ _f (A) forks into two curves. Of these, the curve B ⁽²⁾ _f (A) splits from it, prolonging the former curve for B ⁽¹⁾ _f (A) almost continuously, whereas the curve for B ⁽¹⁾ _f (A) itself goes down sharply.     

Octupole correlations in the actinides and the spdf-IBA model

The octupole bands in the Z～88, N～134 region, with their abundance of experimental information, offer a real challenge for different nuclear structure models. In this region, quasimolecular bands, which are thought to indicate the presence of stable octupole deformation, have been observed. The quantities most often reported are the B(E1)/B(E2) branching ratios from these levels. The calculations in the framework of the spdf model show that the 1-pf boson limit (dipole-octupole vibrations) can describe the essential experimental features related to the octupole bands in the Rn-Th region.     

Giant resonances in exotic spherical nuclei within the RPA approach with the Gogny force

Theoretical results for giant resonances in the three doubly magic exotic nuclei ⁷⁸Ni, ¹⁰⁰Sn and ¹³²Sn are obtained from Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations using the D1S parameterization of the Gogny two-body effective interaction. Special attention is paid to full consistency between the HF field and the RPA particle-hole residual interaction. The results for the exotic nuclei, on average, appear similar to those of stable ones, especially for quadrupole and octupole states. More exotic systems have to be studied in order to confirm such a trend. The low energy of the monopole resonance in ⁷⁸Ni suggests that the compression modulus in this neutron-rich nucleus is lower than the one of stable ones.     

Symmetry-restoring interactions for Kπ = 1+ isovector vibrations

The low-lying isovector 1⁺ states are studied by a symmetry-restoring RPA approach in some rare-earth nuclei. A new velocity-dependent residual interaction is proposed in order to restore the rotational invariance of the hamiltonian in the quasiparticle random-phase approximation with an axially-symmetric Woods-Saxon potential. A new quadrupole interaction is introduced with a self-consistently determined coupling strength. Calculations for six rare-earth nuclei (¹⁵⁴Sm, ^156,158Gd,¹⁶⁴Dy,¹⁶⁸Er,¹⁷⁴Yb) show a good agreement with the experimental energies and B(M1) values. The M1 transitions, corresponding to the experimental strong magnetic dipole states, have in all apart from one case a predominant orbital contribution. The largest orbital contribution (90%) is found in ¹⁵⁴Sm. About half of the low-energy (E < 5MeV, B(M1)↑>0.1μ_N²) states in each nucleus have an orbital character with a (10–40)% spin admixture. The M1 strength is concentrated in the region 2–9 MeV with a maximum around 5 MeV and corresponds to ΔN_osc = 0 transitions.     

Boson description of nuclear collective motion

Nuclear system interacting via quadrupole and octupole particle-hole forces is studied by the boson expansion technique. Energy spectra of the negative parity yrast band and the ground state band are calculated and compared with experiment for¹⁰⁰Ru,¹¹²Cd,¹⁵⁰Sm and¹⁵⁰Gd. ExperimentalB(E1)/B(E2) ratios show strong hindrance for E1 transitions, and are used to deduce the static polarizability of E1 transitions.     

Collective states in even Sn nuclei

0⁺, 2⁺, 4⁴ and 3^? states in ^112–124Sn have been studied with the (p, p′γ) reaction and in Coulomb excitation. Absolute E2 transition rates between these levels have been extracted with the aid of the Winther-de Boer code. For ^116,118Sn, B(E2; 4₁⁺ → 2₁⁺) ≈ 20 W.u., suggesting a two-phonon character of the 4₁⁺ states. For the lighter and heavier isotopes, this value is significantly smaller. All observed values of B(E2; 2₂⁺ → 2₁⁺) and B(E2; 2₃⁺→ 2₁⁺) are about 5 W.u. Also. values of B(E3; 0₁⁺ → 3^?₁) have been measured for all stable even Sn nuclei. In ¹¹⁶Sn the branching ratio (3₁^? → 0₁⁺)/(3₁^? → 2₁⁺) has been measured. From this we obtain a half-life of 0.34±0.07 ps for the first 3^? level in ¹¹⁶Sn and B(E1; 3₁^? → 2₁⁺) = (1.4±0.3) × 10^?5'e² · b, corresponding to a hindrance factor of 10³.     

The breaking of intrinsic reflection symmetry in nuclear ground states

Negative-parity excited states of doubly even nuclei have earlier been attributed to vibrational excitations. This paper shows that an interpretation starting from a reflection asymmetric intrinsic state is more appropriate for certain nuclei in the radium region. Theoretical evidence for stable octupole deformation comes from a deformed shell-model calculation in which we use a single-particle potential with a realistic radial shape and a finite-range interaction for the surface energy. The octupole effect systematically improves the agreement between theoretical and experimental masses. The low-lying O⁺ excitations observed in experiment are compatible with the calculated collective octupole potentials. The possibility of obtaining further evidence from the spectroscopy of odd-mass nuclei is considered in an exactly solvable model, which shows that the smaller energy splitting observed in odd-A parity doublets mainly reflects single-particle fragmentation of the collective mode. The systematics of theoretical shell structure and experimental spectroscopy suggests the presence of other regions of octupole collectivity near the limits of stability.     

Particle–particle Tamm–Dancoff approximation and particle–particle random phase approximation calculations for 18O and 18F nuclei

The nuclear structures of ¹⁸O and ¹⁸F nuclei are studied using particle–particle Tamm–Dancoff approximation (pp TDA) and particle–particle random phase approximation (pp RPA). All possible single-particle states of the allowed angular momenta are considered in the 0p and 1s–0d shells. The Hamiltonian is diagonalized in the presence of Warburton and Brown interactions. The results containing energy-level schemes and transition strength B(E2) are compared with the available experimental data.