Anomalous collective modes in atomic nuclei within the proton-neutron interacting boson model

Novel collective modes characterized by a $B_{4/2}$ ratio ($\equiv B(E2;4_1^+\rightarrow 2_1^+)/B(E2;2_1^+\rightarrow 0_1^+)$) less than 1.0 that were observed recently have been identified within the proton-neutron interacting boson model (IBM-2) using the consistent-$Q$ Hamiltonian. These modes are shown to give rise to triaxial spectral features, including significant band mixing. The results provide a compelling explanation for the deeply suppressed $B_{4/2}$ ratio observed in $^{166}$W, $^{168,170}$Os, and $^{172}$Pt, offering new insights into the $B(E2)$ anomaly phenomenon in neutron-deficient nuclei.

💡 Research Summary

This paper presents a theoretical investigation into anomalous collective modes observed in atomic nuclei, characterized by a B4/2 ratio—defined as B(E2; 4₁⁺ → 2₁⁺) / B(E2; 2₁⁺ → 0₁⁺)—that is less than 1.0. This phenomenon contradicts the established rule for conventional collective modes (spherical vibrator, axial rotor, γ-unstable rotor), where B4/2 is always greater than 1.0. The anomaly has been experimentally detected in neutron-deficient nuclei near N=90 and the proton dripline, such as ¹⁶⁶W, ¹⁶⁸,¹⁷⁰Os, and ¹⁷²Pt, presenting a challenge to existing nuclear models.

The authors employ the proton-neutron interacting boson model (IBM-2) to explain this anomaly. Unlike its simpler counterpart IBM-1, which treats proton and neutron degrees of freedom identically, IBM-2 distinguishes between proton (π) and neutron (ν) bosons. This explicit two-fluid system description allows IBM-2 to naturally accommodate triaxial shapes arising from the different quadrupole deformations of proton and neutron systems. The study utilizes the widely adopted consistent-Q Hamiltonian form within IBM-2, focusing on the dominant proton-neutron quadrupole-quadrupole interaction.

Numerical examinations are conducted for a system corresponding to ¹⁶⁸Os, with boson numbers N_π=3 and N_ν=5. The key finding is that an unconventional mode with B4/2 < 1.0 emerges within IBM-2 when the quadrupole shape parameters for protons and neutrons are set with opposite signs, specifically when χ_ν = -χ_π and χ_c (where χ_c = -χ_π) is less than approximately -1.1. This condition microscopically corresponds to situations where proton bosons are particle-like (below mid-shell) and neutron bosons are hole-like (above mid-shell), or vice versa. The evolution of the B4/2 ratio as a function of the parameter η (controlling the weight of the spherical U(5) limit in the Hamiltonian) shows a rapid transition from values below 1.0 to values above 1.0, resembling a phase-transition-like behavior between the anomalous and normal collective modes.

A detailed analysis of the level structure and E2 transition strengths for a parameter set yielding B4/2=0.73 reveals the presence of low-lying γ and β bands, indicative of triaxiality. Crucially, the analysis of inter-band E2 transitions demonstrates significant mixing between the yrast (ground-state) band and the γ band. The suppressed B4/2 value is primarily attributed to this band mixing effect, which redistributes the E2 strength from the 4₁⁺ → 2₁⁺ transition to other pathways, such as 4₁⁺ → 2₂⁺. In contrast, for a parameter set with a larger η value that yields a “normal” B4/2 > 1.0, the band mixing is substantially reduced.

The paper also discusses the mean-field structure of IBM-2 using the coherent state method, highlighting the complexity introduced by the four deformation variables (β_π, γ_π, β_ν, γ_ν). In summary, the work successfully identifies the conditions within the microscopic IBM-2 framework that give rise to the anomalous B(E2) mode. It provides a compelling explanation for the experimentally observed suppressed B4/2 ratios in specific nuclei, linking the anomaly to the emergence of triaxial shapes and significant band mixing effects in coupled proton-neutron systems.