A correspondence between IBA-1 and IBA-2 models and electromagnetic transitions in the decay of some erbium isotopes

The interacting boson approximation IBA-1 model space, in which neutron and proton degrees of freedom are not distinguished, can be considered as a subspace of the IBA-2 model space. Using the microscopic background of the IBA-2 model, a correspondence can be established between IBA-1 and IBA-2 model space. Since the space of the IBA-1 model can be regarded as a subspace of the IBA-2 model there is a unique way to ‘Project’ the operators of the IBA-2 model onto those of IBA-1. This projection can be carried out using theF-spin formalism. In the IBA-2 model, the lowest states are indeed fully symmetric, and using the calculations with the help of this projection, we explore the energy levels and the electric quadrupole transition probabilitiesB(E2; {ie393-01}) and γ-ray E2/M1 mixing ratios for selected transitions of^{162, 164, 166, 168, 170}Er. Owing to admixtures of non-fully-symmetric states in IBA-2, we renormalized the parameters (ε) and (κ). This is the first time we show that this projection can be applied to some heavier isotopes and the results obtained for^{162, 164, 166, 168, 170}Er isotopes are reasonably in good agreement with the previous experimental values.     

EPM calculations of E2/M1 mixing ratios for even samarium isotopes 146–152Sm

The M1 operator and the E2/M1 mixing ratios, Δ(E2/M1), are discussed in the framework of the extended phonon projection model (EPM). The model is applied to ^{146, 148, 150, 152}Sm to describe their M1 properties. Comparison is made between the EPM, the consistent-Q formalism (CQF) of the basic interacting-boson approximation (IBA-1) and the experimental Δ(E2/M1) data. Both theoretical approaches give reasonably good results and parameter systematics.     

Effective boson number calculations in Mo and Cd isotopes

The effects of the neutron-proton interaction on the low-lying levels of Mo and Cd isotopes have been considered in the frame of the IBA-1 model by taking into account an effective boson number (N _eff). Both an empirical procedure based on previous IBA-2 mixing calculations and theN _p N _n scheme provide comparableN _eff values. Level spectra and electromagnetic transitions are investigated. The results support the idea that IBA-1 calculations with a suitableN _eff can largely simulate IBA-2 mixing calculations, taking advantage of simplicity and a smaller number of parameters.     

Spin and orbital contributions to collective M1 transitions in 46,48Ti

Low- and high-lying K^π = 1⁺ states and M1 transitions in ^46,48Ti are studied. The model hamiltonian is treated in the quasi-particle particle random phase approximation (QRPA) with an exact restoration of its rotational invariance. A considerable spin contribution to the transition matrix elements is found for the low-energy (about 4 MeV) strong M1 transition (the orbital contribution being 30–70% of the spin one), although the microscopic structure of this state in ⁴⁶Ti is typical for an orbital isovector excitation. The calculated energies and B(M1) values are in good agreement with the experimental data. The results are compared to the estimates of the isovector scissor model.     

Description of even-even Krypton isotopes by the Interacting Boson Approximation

Energy spectra andB(E2) values of the even-even Kr isotopes^74–82Kr have been analyzed in terms of the IBA-2 model of Otsuka, Arima and Iachello [6, 7]. We can reproduce the variation of collectivity along the isotope chain by means of a consistent set of 5 parameters, essentially only one of them has been varied with neutron number. The low lying 0 ₂ ⁺ states have been found to be strongly influenced by shell effects. Predictions are given for the not yet observed gamma bands of^74,76Kr.     

Nuclear structure of 144Nd in IBM and DPPQ model

The dynamic deformation theory based on the pairing-plus-quadrupole model is employed for a detailed study of the N = 84 vibrational nucleus ¹⁴⁴Nd. The level energies, shape parameters, B(E2) values, ρ(E0), X(E0/E2) and δ(E2/M1) values are compared with available data. The level structure is also analyzed in IBM-1 and their F-spin character in IBM-2 is discussed. The A-dependence of the IBM-2 parameters is compared with those of the DPPQ model and the E2 moments of ^146–150Nd are presented.     

Multi-nucleus IBA calculations in the NπNν scheme

IBA-1 calculations for ～ 100 nuclei in three mass regions from A = 100–200 are carried out according to a recently proposed N_πN_ν parametrization. The calculations use a total of only seven constants consisting of four parameters which are constant for all 100 nuclei and one parameter which takes on a separate value for each of the three regions. The changes in nuclear structure for each region are reproduced and the detailed values of individual energy levels and B(E2) values are reasonably well accounted for. The implications of the extreme economy in parameters are discussed and the usefulness of the present approach for other collective models is pointed out.     

Elastic and inelastic deuteron scattering from 152Sm and 162Dy

Angular distributions are measured for 12 MeV deuterons scattered to the 2⁺ and 4⁺ states in ¹⁵²Sm and ¹⁶²Dy. By including the couplings of the ground state to the low-lying excited states, it is shown that the experimental results can be explained by using optical model parameters form a spherical nucleus. Deformation amplitudes are deduced and compared to those found from inelastic α-particle scattering and from Coulomb excitation.     

(α, 2nγ) Studies of γ-vibrational and other side-bands in 162, 164, 166Er

Levels in ^{162, 164, 166}Er have been studied using the (α, 2nγ) reaction at an energy of 24 MeV. Singles spectra, γ-γ coincidence spectra, and angular distributions were obtained using Ge(Li) detectors. Transitions from levels in the γ-vibrational bands up to the 8^+' in ^{162, 164}Er and 10^+' in ¹⁶⁶Er were observed and M1/E2 mixtures were determined for many of these transitions. There is a relative shifting upward of the even-spin levels in the γ-band of ¹⁶⁶Er while the analogous levels of ^{162, 164}Er are shifted downward with the effect being most pronounced for¹⁶²Er. The standard phenomenological band-mixing parameters z₂ and z₀₂ were obtained from γ-ray branching ratio data and the values are probably correlated with the staggering of levels in the γ-bands. The ratios of the intraband and interband E2 transition strengths which are related to the intrinsic quadrupole moments of the ground-state and γ-bands are discussed. A number of other levels are observed in ¹⁶², 164Er and some of these correspond to negative parity states reported in decay studies.     

Studies of the electric dipole transition of deformed rare-earth nuclei

Spectrum and electric dipole transition rates and relative intensities in ^152–154Sm, ^156–160Gd, ^160–162Dy are studied in the framework of the interacting boson model with s,p,d,f, bosons. It is found that E1 transition data among the low-lying levels are in good agreement with the SU(3) dynamical symmetry of the spdf interacting boson model proposed by Engel and Iachello to describe collective rotation with octupole vibration. These results show that these nuclei have SU(3) dynamic symmetry to a good approximation. Also in this work many algebraic expressions for electric dipole transitions in the SU(3) limit of the spdf-IBM have been obtained. These formulae together with the formulae given previously exhaust nearly all the E1 transitions for low-lying negative parity states. They are useful in analyzing experimental data.     

α particle preformation and shell effect for heavy and superheavy nuclei

The α particle preformation factor is extracted within a generalized liquid drop model for Z = 84-92 isotopes and N = 126,128,152,162,176,184 isotones.The calculated results show clearly that the shell effects play a key role in α particle preformation.The closer the proton and neutron numbers are to the magic numbers,the more difficult the formation of the α cluster inside the mother nucleus is.The preformation factors of the isotopes reflect that N = 126 is a magic number for Po,Rn,Ra,and Th isotopes,but for U isotopes the weakening of the influence of the N =126 shell closure is evident.The trend of the factors for N = 126 and N = 128 isotones also support this conclusion.We extend the calculations for N = 152,162,176,184 isotones to explore the magic numbers for heavy and superheavy nuclei,which are probably present near Z = 108 to N = 152,162 isotones and Z = 116 to N = 176,184 isotones.The results also show that another subshell closure may exist after Z = 124 in the superheavy nuclei.This is useful for future experiments.     

Long <Emphasis FontCategory="NonProportional">K</Emphasis><Superscript>π</Superscript> = 0<Superscript>+</Superscript> and 4<Superscript>−</Superscript> bands in deformed nuclei

Experimental data on some long bands in N=88–98 deformed nuclei are analyzed on the basis of a method systematizing the energies of levels and inertial parameters and a method employing the analogy between neighboring. Beta bands built on low-lying 0 ₂ ⁺ levels are identified in N=90 isotones, including the ¹⁵⁸Er, ¹⁶⁰Yb, and ¹⁶²Hf nuclei. Some of intermediate members of these bands have not been found yet. The systematic properties of 4^? bands are presented, and the dynamics of the inertial parameters of these bands in N=88–98 isotones is revealed. The bands are identified in ¹⁵⁸Yb, ¹⁵⁶Dy, ¹⁵⁶Er, ¹⁶²Yb, and ¹⁶⁶Hf.     

Subshell closure effects on the collectivity of 36 88 Kr52 and the beta decay of88Br to levels in88Kr

The level structure of⁸⁸Kr has been investigated by measurements on chemically separated bromine samples. A total of 146γ-rays have been assigned to the decay of⁸⁸Br, and a level scheme with 58 levels is proposed. The collectivity of⁸⁸Kr has been investigated within the framework of the Interacting Boson Approximation (IBA-2 model). Comparison of our detailedγ-ray data and calculations have led to the establishment of a protonhole boson desciption rather than protonparticle nature. This supports recent works establishingZ=40 as a major subshell closure.     

Improved description of M1 transitions by special hamiltonians of the proton-neutron interacting boson model

The proton-neutron interacting boson model (IBM-2) describes energies, B(E2) and B(M1) values of nuclei. In order to reduce the great number of free IBM-2 parameters two special IBM-2 hamiltonians are proposed which allow a decoupling of the energy and B(E2) fit from the determination of the B(M1) values and the energy of the lowest mixed symmetry 1⁺ state. This property allows a simple fit procedure of the IBM-2 parameters in both cases.     

The change of sign of E2 and M1 matrix elements,and of δ-values of166Er at high spins

Calculations of the Dynamic Deformation Model for the E2/M1 mixing ratios for ψ-band to ground-band transitions in¹⁶⁶Er have been extended to higher spins (up to I=20). Previous comparisons with the experimental values and with the IBA-1 predictions, available at present up to spins 8 only, are also included for the sake of completeness. Additional comparison with the Frankfurt Model is also given. While all three models give reasonable magnitudes of the mixing ratios, only the DDM gives the sign changes at the correct spin values. Predictions are presented for additional sign changes of the mixing ratios, as well as those of the E2 and the M1 transition moments.     

Excited states in124Xe

The isomer shift in the optical transition Eu II 4f⁷(⁸S ₇ ^2/o )⁶p_3/2 ^(7/2,3/2)4?4f⁷ (⁸S ₇ ^2/o )5d⁹D ₄ ^o , λ=6O4.95 nm, has been measured between the isotope^152gEu(3^?) and its isomer^152mEu(0^?). From the valuev(^152gEu)–v(^152mEu)=736(10) MHz the deformation parameter of^152mEu has been estimated to be^152mβ?+0.25. This value is smaller than^152gβ as obtained from isotope shift measurements and the spectroscopic quadrupole moment. The influence of the shape difference on the decay of^152mEu is discussed.     

Ein Teilchen-Rotor-Modell mit VMI-Rumpf für Backbending-Kerne

A deformed VMI-core plus two particles is considered as a simple model for even-even nuclei. In a basis of three antisymmetrized wave functions, which representK=0-, 1-, 2- bands, the Hamiltonian of this system is diagonalized. After an additional minimalisation, which must take place according to the VMI-model, one gets the energies of the yrast-band depending on the two VMI-parameters and two further parameters, which rule the back-bending-behavior. Fitting these parameters with the experimental data nearly total agreement between theory and experiment is obtained for the backbending nuclei¹⁵⁸Er,¹⁶⁰Er,¹⁶²Er and¹⁶⁶Yb.     

The nature of three-pion enhancements in the minimal K-matrix model

The square of the magnitude of the inverse Fredholm determinant associated with the minimal K-matrix integral equations describing three-pion to three-pion scattering was calculated as a function of three-pion mass ¢M from threshold to 1900 MeV for the 0 ≦ I ≦ 2, 0 ≦ J ≦ 2 channels of the three-pion system. The input for these equations consisted of the on-shell π-π t-matrices t₀⁰, t₀², t₁¹, and t₀² obtained from phenomenological phase-shift analyses. For the minimal K-matrix model, the only structure in the natural parity states is an enhancement of kinematical origin at 1284 MeV in the ω-channel. Spurious enhancements appear in all the unnatural parity states considered at roughly 1150 MeV corresponding to the ρπ → ρπ Peierls singularity. In the I = 1 unnatural parity channels, broad enhancements at $\text{1450}\text{MeV}\text{≦ ¢M ≦ 1675}\text{MeV}$ occur. These correspond to the ρπ → fπ fit generalized Peierls singularity.