Axially Deformed Proton-Neutron Relativistic Quasiparticle Finite Amplitude Method for Charge-Exchange Transitions

TL;DR

This work develops a relativistic proton-neutron QFAM (pnQFAM) within axially deformed RHB theory to describe charge-exchange transitions, validated against spherical pnQRPA and applied to Zn isotopes to examine deformation effects on IAR and GT transitions. The method uses the DD-PC1 density functional with isovector TV/TPV couplings and a finite-range separable pairing force, enabling efficient treatment of deformed nuclei without matrix diagonalization. Key findings show significant GT strength fragmentation and a systematic K-splitting: prolate shapes favor $K=1$ strength and exhibit lower $K=1$ centroids, while oblate shapes enhance $K=0$ strength with lower $K=0$ centroids; isoscalar pairing shifts GT strength to lower energies, especially in heavier Zn where high-$l$ transitions contribute. The approach provides a computationally efficient tool for global deformation studies in relativistic DFT and supports improved modeling of beta-decay rates and related weak-interaction processes; future enhancements include incorporating quasi-particle-vibration coupling (QPVC) to capture damping effects.

Abstract

The quasiparticle finite amplitude method (QFAM) is extended to describe charge-exchange transitions based on the relativistic Hartree-Bogoliubov model, adopting the point-coupling energy density functional DD-PC1 and a finite-range separable pairing force. After validation through comparison with relativistic quasiparticle random-phase approximation (QRPA) results in spherical nuclei, the deformation effects on isobaric analog resonances (IAR) and Gamow-Teller (GT) transitions in Zn isotopes are investigated. The GT strength exhibits significant fragmentation in deformed nuclei. The analysis of summed strengths and centroid energies in GT resonance region between the $K=0$ and $K=1$ components reveals that prolate configurations exhibit stronger $K=1$ strength and lower $K=1$ centroid energy, while oblate shapes show an opposite behavior, with stronger $K=0$ strength and lower $K=0$ energy. The effects of isoscalar pairing on GT strength distributions for different shape configurations are also examined.

Figures (10)

• Figure 1: Gamow-Teller (GT) strength distributions in $^{74}\text{Zn}$ calculated using different harmonic oscillator (HO) basis truncations ($N_\text{HO}=16$, 18, 20), and the corresponding difference compared to the reference case ($N_{\text{HO}}^\text{max}=22$). Panels (a) and (b) show the GT strength distributions for $K=0$ and $K=1$ modes, respectively, while panels (c) and (d) present the corresponding normalized differences relative to the reference case, scaled by the Ikeda sum rule ($N-Z$) for each $K$ component. The truncations $N_\text{HO}=16$ (black solid), 18 (red dashed), and 20 (green dotted) are shown with distinct line styles. The gray horizontal line marks the 1% level of $\delta S/(N-Z)$.
• Figure 2: Comparisons of strength distributions between QFAM with $\beta_2=0.0$ (open black circles) and spherical QRPA (solid green lines) calculations. The upper panels display the isobaric analog state (IAS) strength distributions, while the lower panels show the GT strength distributions. The left and right panels correspond to nuclei $^{70}\text{Zn}$ and $^{72}\text{Ge}$, respectively.
• Figure 3: Comparisons of IAS and GT strengths for different shapes of $^{64}\text{Zn}$ . The upper panel (a) displays the IAS strengths, while the lower panel (b) shows the GT strengths. Results based on spherical, prolate, and oblate configurations are indicated by solid black, dashed red, and dotted green lines, respectively.
• Figure 4: Left Column: Potential energy curves for Zn isotopes, showing the total energy (black solid lines) as a function of the quadrupole deformation parameter $\beta_2$. Global and local minima are indicated by black spheres and open circles, respectively. The middle and right Columns: GT strengths based on the configurations marked in the left column. The total GT strength is shown by black solid lines, with the $K=0$ and $K=1$ components represented by red dashed and blue dotted lines, respectively. For $^{64}\text{Zn}$, the experimental $B(\text{GT})$ is smeared with a width of $0.25\text{ MeV}$ and scaled by a factor of $4.67$ (gray shaded area) according to the ratio of theoretical and experimental summed strengths FDiel_2019_PRC. The vertical gray dashed line indicates the lower boundary of the GT resonance (GTR) region.
• Figure 5: Non-energy-weighted summed strength $m_0$ [panels (a) and (d)] energy-weighted summed strength $m_1$ [panels (b) and (e)], and centroid energies $E_\text{cent.} = m_1/m_0$ [panels (c) and (f)] in the resonance region for the $K=0$ and $K=1$ strengths in Zn isotopes. For spherical nuclei, the $K=0$ and $K=1$ components are degenerated and represented by solid black circles. For deformed nuclei, open symbols denote the $K=0$ components, while solid symbols correspond to the $K=1$ components. Blue upward triangles and red downward triangles represent results obtained on top of prolate (left column) and oblate configurations (right column) , respectively.