Carbon-12 and Beryllium-10 Quadrupole Transitions Demonstrate Isospin Symmetry, Revealing Anomalies

The subtle differences in nuclear structure between carbon-12 and beryllium-10 offer a unique window into the fundamental principles governing atomic nuclei. Takayuki Myo (Osaka Institute of Technology and Osaka University), Mengjiao Lyu (Nanjing University of Aeronautics and Astronautics), and Qing Zhao (Huzhou University), alongside their colleagues, explore the quadrupole properties of carbon-12 and compare them to its isospin symmetry partner, beryllium-10. Their research, utilising the antisymmetrized molecular dynamics method, reveals that while carbon-12 generally exhibits larger quadrupole transitions than beryllium-10, a notable exception exists in the transition between specific states. This anomaly aligns with experimental observations and arises from differing proton deformations in each nucleus, linked to their respective subclosed and two-cluster structures. Understanding these nuances in quadrupole transitions not only validates isospin symmetry but also provides valuable insights into the complex interplay of forces shaping nuclear behaviour.

Carbon and Beryllium-10 Quadrupole Properties Compared

Scientists demonstrate a detailed understanding of nuclear structure through a comparative study of carbon-10 and beryllium-10, focusing on their quadrupole properties. The research team achieved a breakthrough in describing these nuclei using an advanced theoretical approach, the variation of multiple bases of the antisymmetrized molecular dynamics (AMD) method. This innovative technique simultaneously optimizes multiple AMD bases within a total-energy variation, allowing for a more comprehensive exploration of nuclear configurations. Experiments show that while most quadrupole transitions exhibit larger values in carbon-10 compared to beryllium-10, the transition between the 2+1 and 0+1 states displays remarkably similar values in both nuclei.

The study reveals an anomaly in the quadrupole transition behavior between carbon-10 and beryllium-10, challenging conventional expectations for mirror nuclei. This unexpected similarity arises from the distinct proton configurations within each nucleus; carbon-10 exhibits a small proton deformation due to its subclosed shell structure, while beryllium-10 displays a large deformation driven by two-alpha clustering. Researchers confirmed this relationship by examining the quadrupole moments of both nuclei, providing further evidence for the influence of proton configuration on transition strengths. The work establishes a clear connection between nuclear deformation, clustering phenomena, and the observed quadrupole transition properties.

This research confirms isospin symmetry between carbon-10 and beryllium-10 through analysis of monopole transitions, achieved by exchanging proton and neutron configurations. Furthermore, the team successfully modeled large quadrupole transitions between elongated linear-chain states, demonstrating the versatility of the multicool AMD method. By optimizing numerous configurations simultaneously, the scientists were able to accurately represent both clustered and mean-field structures within the nuclei, a significant advancement over previous models. The theoretical framework employed in this study provides a robust platform for investigating the interplay between clustering and deformation in unstable nuclei.

The investigation extends beyond simply describing existing experimental data; it offers insights into the underlying mechanisms governing nuclear structure and transitions. The study unveils the importance of considering multiple configurations when modeling complex nuclei, particularly those near the threshold energies for cluster emission. This detailed analysis of carbon-10 and beryllium-10 opens avenues for future experimental work aimed at verifying the predicted characteristics of transitions within these nuclei, potentially leading to a more complete understanding of nuclear forces and the behavior of matter under extreme conditions.

AMD Multicool Optimisation of Light Nuclei

The study investigates the structures of carbon-10 and beryllium-10, focusing on their quadrupole properties and comparing them to their mirror nuclei. Researchers employed the antisymmetrized molecular dynamics (AMD) method, a technique well-suited to describing nuclei exhibiting both clustered and mean-field characteristics. This work pioneers an extension of the AMD method, introducing a “multicool” approach, a multiple cooling method, to simultaneously optimise numerous AMD configurations through total-energy variation. This innovative scheme bypasses the need for pre-defined physical constraints, allowing for a more flexible and potentially accurate determination of nuclear structure.

Scientists developed a theoretical framework where the nuclear wave function is defined as a Slater determinant constructed from single-nucleon Gaussian wave packets, each described by variational parameters including centroid and spin components. The energy variation, termed the ‘cooling method’, is achieved through imaginary-time evolution, iteratively refining these parameters to minimise the total intrinsic energy of each AMD basis state. Following this, angular-momentum projection is performed to obtain states with defined spin and parity. The team then implemented a crucial advancement: the superposition of multiple, projected AMD configurations, solving a generalized eigenvalue problem to determine the coefficients that minimise the total energy of the system.

This methodology enables the simultaneous optimisation of many configurations, a significant advantage over traditional approaches. The researchers applied this multicool method to both carbon-10 and beryllium-10, examining monopole and quadrupole transitions to elucidate the isospin symmetry between the two nuclei. The AMD framework, coupled with the multicool optimisation, delivers energies for the ground and excited states of beryllium-10 that are lower than those obtained by other cluster models utilising the same Hamiltonian. This confirms the efficacy of the method in accurately describing complex nuclear structures.

Experiments employ a complex Hamiltonian and variational procedure to determine the optimal configurations for both nuclei. The study details the formulation of the variation of multiple AMD configurations, defining the nuclear wave function and outlining the cooling equation used for energy minimisation. By analysing the resulting quadrupole transitions, the research aims to resolve an anomaly observed in the experimental data, where the relationship between quadrupole transitions in mirror nuclei does not hold for carbon-10 and beryllium-10. The work anticipates that future experiments will investigate the characteristics of these transitions, validating the theoretical predictions generated by this advanced methodology.

Carbon and Beryllium-10 Quadrupole Moments and Isospin Symmetry

Scientists have achieved a detailed understanding of the nuclear structures of carbon-10 and beryllium-10, focusing on their quadrupole properties and isospin symmetry. The research team employed the antisymmetrized molecular dynamics (AMD) method, optimizing multiple bases simultaneously through total-energy variation to describe both nuclei. Experiments revealed that monopole transitions confirm isospin symmetry between carbon-10 and beryllium-10 via proton-neutron exchange, demonstrating a consistent relationship despite differing nucleon compositions. Data shows that, in most cases, carbon-10 exhibits larger quadrupole transition values than beryllium-10, with a notable exception in the 2+1 to 0+1 transition.

Measurements confirm this transition displays similar values in both nuclei, an anomaly that aligns with existing experimental observations. This unexpected similarity arises from the small proton deformation in carbon-10, attributed to its subclosed shell structure, contrasted with the larger proton deformation in beryllium-10, influenced by two-alpha clustering. This property is further substantiated by analysis of the quadrupole moments calculated for each nucleus. The study also investigated neutron deformations, confirming an opposite tendency to that of protons in both carbon-10 and beryllium-10, thereby reinforcing the isospin symmetry between the two mirror nuclei.

Scientists recorded large quadrupole transitions between elongated linear-chain states, providing insight into the collective behaviour of nucleons within these unstable nuclei. The breakthrough delivers a refined theoretical framework for understanding nuclear structure, particularly in unstable nuclei where cluster phenomena and mean-field effects coexist. Measurements confirm the electric quadrupole transition strength, B(E2, 2+1 → 0+1), for carbon-10 is 8.8(3) e2fm4, slightly smaller than the 9.2(3) e2fm4 observed in beryllium-10. The team’s multicool method, an extension of the AMD approach, successfully generated both cluster and mean-field states, providing the lowest energies for ground and excited states in beryllium-10. Future experiments are encouraged to further investigate the characteristics of these transitions in both carbon-10 and beryllium-10, potentially validating the theoretical predictions and deepening understanding of nuclear behaviour.

Isospin Symmetry in Carbon and Beryllium Nuclei

This work presents a detailed investigation into the quadrupole properties of carbon-10 and beryllium-10 nuclei, focusing on isospin symmetry between the two species. Through application of the antisymmetrized molecular dynamics model, with optimized multiple bases, researchers successfully reproduced the energy spectra of both nuclei, including the presence of elongated linear-chain shapes in excited states. Calculations of monopole and quadrupole transitions for protons and neutrons confirmed good isospin symmetry, with proton and neutron behaviours mirroring each other when comparing the two nuclei. A key finding concerns the electric quadrupole transitions between the 2+1 and 0+1 states, where the model replicates the experimentally observed anomaly of similar values despite differing proton numbers.

This is attributed to the subclosed nature of the proton p3/2 orbit in carbon-10, resulting in reduced proton deformation, contrasted with the significant proton deformation in beryllium-10 due to alpha-clustering. The authors acknowledge limitations inherent in the model, but suggest further experimental investigation of the observed transition characteristics would be beneficial. Future research could explore the applicability of this approach to other nuclei exhibiting similar structural features, potentially refining understanding of nuclear structure and isospin symmetry.