Identical Bands Around the Isobaric Rare-Earth Even–Even Nuclei with the Mass Number of ${A=164}$

Abstract

Eight pairs of rare-earth normally—deformed (ND) nuclei around the isobaric nuclei with $A=164$ and identical values of $F$-spin, $\pm F_{0}$ and $N_{p}N_{n}$ ($N_{p}$ and $N_{n}$ are the number of valence protons and valence neutrons, respectively) have been studied. These pairs of identical bands (IB’s) cover 16 mass units and are classified as (i) 3 pairs of nuclei separated by $(2p,2n)$: (${}^{162}$Yb–${}^{166}$Hf), (${}^{162}$Er–${}^{166}$Yb), (${}^{162}$Dy–${}^{166}$Er), (ii) 2 pairs of nuclei separated by $(4p,4n)$: (${}^{160}$Dy–${}^{168}$Yb), (${}^{160}$Er–${}^{168}$Hf), (iii) 2 pairs of nuclei separated by $(6p,6n)$: (${}^{158}$Er–${}^{170}$W), (${}^{158}$Dy–${}^{170}$Hf) and (iv) one pair of nuclei separated by $(8p,8n)$: (${}^{156}$Dy–${}^{172}$W). We suggested a theoretical collective rotational formula containing three parameters (CRF3) as an extended version of Bohr–Mottelson model to calculate the ground state positive parity excitation energies. Also, the $sd$-version of the interacting boson model (IBM) has been used to describe the nuclear shapes by using the intrinsic coherent state. The optimized model parameters for each nucleus are adjusted by using a simulation search program to minimize the root mean square deviation between the theoretical calculated and experimental excitation energies. The best adopted model parameters of the CRF3 are used to calculate the rotational frequencies $\hbar\omega$, the kinematic $J^{(1)}$ and dynamic $J^{(2)}$ moments of inertia and the evolution of $J^{(1)}$ and $J^{(2)}$ with increasing $\hbar\omega$ are systematically analyzed. A smooth gradual increase in both moments of inertia was seen. The calculated results agree excellently with the experimental ones which give strong support to the suggested CRF3. The adopted IBM parameters are used to calculate the potential energy surfaces (PES’s) which describe the nuclear deformation. The PES’s for our nuclei show two wells corresponding to prolate and oblate sides which indicate that these nuclei are deformed and have rotational behaviors. The correlation quantities which identify the IB’s are extracted. It is found that the nuclei having $N_{p}N_{n}/\triangle$, where $\triangle$ is the average pairing gap, exhibit identical excitation energies and energy ratios in their ground state

rotational bands.