Analysis of correlations between dipole transitions $1^-_1 ightarrow 0^+_1$ and $3^-_1 ightarrow 2^+_1$ based on the collective model

The purpose of the work is to evaluate effect of the isovector dipole and quadrupole-octupole modes coupling on the $B(E1;1^-_1\rightarrow 0^+_1)/B(E1;3^-_1\rightarrow 2^+_1)$ ratio. The Hamiltonian of the phenomenological collective model is used to calculate mixing of the isovector dipole and quadrupole and octupole modes. The effect of the admixture of the giant dipole resonance to the low-lying collective quadrupole and octupole modes is estimated. It is shown that the coupling of the quadrupole and octupole collective modes to giant dipole resonance leads to decrease of the ratio $B(E1;1^-_1\rightarrow 0^+_1)/B(E1;3^-_1\rightarrow 2^+_1)$ relative to the value 7/3 predicted by the pure collective quadrupole-octupole model.

💡 Research Summary

The paper addresses the longstanding discrepancy between the experimentally observed ratio of electric‑dipole transition strengths, R = B(E1; 1⁻₁ → 0⁺₁)/B(E1; 3⁻₁ → 2⁺₁), and the value predicted by the pure quadrupole‑octupole collective model (7/3). The authors employ a phenomenological collective Hamiltonian that explicitly includes three components: a quadrupole term (H_quad), an octupole term (H_oct), and an isovector dipole term (H_dip). The dipole part is further divided into an independent‑particle contribution and a residual interaction that couples the giant dipole resonance (GDR) to the quadrupole and octupole phonons.

Using the Tamm‑Dankoff approximation, the GDR is treated as a collective 1⁻ phonon p⁺₁µ with energy ω_GDR. The dipole transition operator contains two pieces: D(1)₁µ, which creates particle‑hole excitations (the GDR component), and D(2)₁µ, which incorporates the quadrupole (α₂) and octupole (α₃) collective coordinates. The coupling strength between the GDR and the low‑lying modes is parametrized by a constant C, derived from microscopic considerations (C ≈ √(35/3)(1 + χ)C_LD A Z, with χ = ‑0.7). Because the actual residual interaction may not be perfectly in phase with the factorized dipole term, the authors introduce a reduced constant C′ (< C) to simulate a weaker effective mixing.

Perturbation theory yields mixing amplitudes x₁, x₂, x₃, and x₀ that describe how much GDR admixture enters the wave functions of the 1⁻₁, 2⁺₁, 3⁻₁, and 0⁺₁ states. These amplitudes are proportional to Γ C′/(ω_GDR ‑ ω_i), where Γ is fixed by the normalization of the GDR phonon and ω_i are the energies of the low‑lying states (ω₀ ≈ 41 A⁻¹⁄³ MeV, ω₂, ω₃). The resulting wave functions are linear combinations of pure quadrupole‑octupole components and GDR‑mixed components.

From these mixed wave functions the authors derive an analytical expression for the transition‑strength ratio:

R = (7/3) ×