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Nuclear Shape Coexistence and Quantum Phase Transitions in Nuclear Structure: Latest Research Survey
2026-05-01
Survey Date: 2026-05-01 Coverage: 2022–2026 key publications
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1. Experimental Discoveries of Shape Coexistence
1.1 Extreme Shape Coexistence in ⁷⁰Co (Nature Communications Physics, 2025)
| Item | Details |
|---|---|
| Title | Extreme shape coexistence observed in ⁷⁰Co |
| Authors | Cade Dembski, Artemis Spyrou, B. Alex Brown, Sean N. Liddick, et al. |
| Journal | *Communications Physics* 8, 77 (2025) |
| Date | 23 February 2025 |
| DOI | 10.1038/s42005-025-01998-2 |
Main Findings:
- First spectroscopic evidence for two Jπ = 1+ states in ⁷⁰Co — one spherical (weakly deformed) and one strongly deformed — separated by less than 275 keV
- The long-lived β-decaying state is strongly deformed (dominated by multiparticle-multihole excitations across the Z = 28 shell gap, i.e., 2p-3h configurations)
- Deformation driven by Type II Shell Evolution: as deformation increases, proton Nilsson orbitals cross, creating a new potential energy minimum at β ≈ +0.3
- ⁷⁰Co identified as a transition point between deformed ground states (Cr isotopes) and spherical configurations (Ni isotopes)
- Experiment conducted at NSCL/MSU using the SuN Total Absorption Spectroscopy detector
- Could influence astrophysical nucleosynthesis pathways far from stability
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1.2 Reassignment of 0⁺ State Shapes in ¹⁸⁶Pb (Nature Communications Physics, 2022)
| Item | Details |
|---|---|
| Title | Reassigning the shapes of the 0⁺ states in the ¹⁸⁶Pb nucleus |
| Authors | Joonas Ojala, Janne Pakarinen, Philippos Papadakis, et al. |
| Journal | *Communications Physics* 5, 213 (2022) |
| Date | 18 August 2022 |
| DOI | 10.1038/s42005-022-00990-4 |
Main Findings:
- Reverses the 2000 Nature assignment by Andreyev et al. for the ¹⁸⁶Pb 0⁺ triplet:
- 0₁⁺ (ground state) → Spherical (unchanged)
- 0₂⁺ → Predominantly prolate (was oblate)
- 0₃⁺ → Predominantly oblate (was prolate)
- Key evidence: B(E2; 2₁⁺ → 0₂⁺) = 190(80) W.u., indicating a collective transition within the prolate band
- First direct measurement of E0 transitions (0₂⁺ → 0₁⁺ and 0₃⁺ → 0₁⁺) via in-beam conversion electrons
- Used the SAGE spectrometer for simultaneous in-beam γ-ray and electron spectroscopy at University of Jyväskylä
- ¹⁸⁶Pb remains the only known system with a triplet of 0⁺ states with three different shapes within ~700 keV
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1.3 Shape Coexistence and Shape Isomers around ¹⁷²Hg (Nuclear Science and Techniques, 2025)
| Item | Details |
|---|---|
| Title | On shape coexistence and possible shape isomers of nuclei around ¹⁷²Hg |
| Authors | Xin Guan, Jing Guo, Qi-Wen Sun, Bożena Nerlo-Pomorska, Krzysztof Pomorski |
| Journal | *Nuclear Science and Techniques* 36, 128 (2025) |
| Date | 26 May 2025 |
| DOI | 10.1007/s41365-025-01737-w |
Main Findings:
- Used Lublin–Strasbourg Drop + Yukawa-Folded potential + pairing corrections (macro-microscopic approach)
- ¹⁷²Hg ground state transitions from triaxial to oblate with increasing pairing; nearly γ-unstable; three shape isomers identified (prolate, triaxial, oblate)
- ¹⁷⁰Pt–¹⁸⁰Pt chain: ground-state shape evolves from prolate → triaxial → γ-unstable → dominant prolate
- Pairing interactions are critical: can cause shape isomers to appear/disappear and drive transitions between deformation types
- BCS pairing tends to smooth out shape coexistence compared to exact pairing solutions
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2. Quantum Phase Transitions — Theoretical and Experimental Advances
2.1 Nuclear Shape Phase Transition via Ultra-Relativistic Collisions (PRL, 2024)
| Item | Details |
|---|---|
| Title | Exploring the Nuclear Shape Phase Transition in Ultra-Relativistic ¹²⁹Xe+¹²⁹Xe Collisions at the LHC |
| Authors | Shujun Zhao (赵沭钧), Hao-jie Xu (徐浩洁), You Zhou (周铀), Yu-Xin Liu (刘玉鑫), Huichao Song (宋慧超) |
| Journal | *Physical Review Letters* 133, 192301 (2024) |
| Date | 4 November 2024 |
| DOI | 10.1103/PhysRevLett.133.192301 |
Main Findings:
- First proposal to use ultra-relativistic heavy-ion collisions to probe nuclear shape phase transitions
- The Xe isotope chain undergoes a second-order shape phase transition near ¹²⁸⁻¹³⁰Xe (γ-soft rotor → spherical vibrator)
- Previously reported correlators (ρ₂, Γ_{pT}) interpreted as evidence for rigid triaxial deformation of ¹²⁹Xe can be equally explained by γ-soft deformation
- New correlators ρ₄,₂ and ρ₂,₄ proposed to uniquely distinguish γ-soft from rigid triaxial deformation
- At LHC energies, the collision time is <10⁻²⁴ s, so nuclear ground-state structure is directly imprinted onto QGP initial conditions
- Bridges nuclear structure physics and high-energy nuclear collision physics — a completely new approach
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2.2 QPT Driven by Competition: Particle-Number-Conserved BCS (PRL, 2025)
| Item | Details |
|---|---|
| Title | Quantum phase transition in atomic nuclei driven by competition between pairing and quadrupole correlations |
| Authors | GuanJian Fu (傅冠健) et al. |
| Journal | *Physical Review Letters* (2025) |
| Date | 2 July 2025 |
| DOI | 10.1103/PhysRevLett.xxx (pending) |
Main Findings:
- Uses particle-number-conserved BCS theory + angular-momentum projection for the first consistent description of nuclear shape evolution and QPT
- Provides a unified framework bridging the shell model and deformed mean-field models
- Describes the competition between pairing and quadrupole correlations as the driving mechanism
- Covers pairing, shape coexistence, shape evolution, and quantum phase transitions in a comprehensive manner
- Developed at Tongji University by Prof. Fu GuanJian (国家优青基金获得者)
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2.3 E(5) Critical Point to Shape Coexistence: Catastrophe Theory (J. Phys. G, 2026)
| Item | Details |
|---|---|
| Title | From E(5) to shape coexistence: a catastrophe-theoretic phase diagram and Maxwell critical curve for the γ-unstable Bohr model |
| Authors | M. M. Hammad, M. M. Yahia, E. I. Saad, S. B. Doma |
| Journal | *Journal of Physics G: Nuclear and Particle Physics* 53, 035101 (2026) |
| Date | 12 March 2026 |
| DOI | 10.1088/1361-6471/ae49a7 |
Main Findings:
- Uses γ-unstable Bohr Hamiltonian with sextic polynomial potential (β² − aβ⁴ + bβ⁶)
- Derives analytic expressions for spinodal boundaries and the Maxwell set (equal-depth minima curve)
- E(5) critical-point symmetry identified as a cusp-adjacent case — not an isolated limit but a boundary within a broader QPT landscape
- Provides diagnostic fingerprints to distinguish first-order from second-order QPTs via wave-function morphologies and tunneling patterns
- Formulates an inverse-problem protocol to quantitatively assess a nucleus's distance to criticality and degree of shape mixing
- Validates against experimental data for ⁷²Kr, ¹⁸⁶Hg, and ¹⁰⁸Pd
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2.4 Shape Phase Transitions with Octupole Deformation in spdf-IBM (PRC, 2026)
| Item | Details |
|---|---|
| Title | Shape phase transitions and octupole deformation in the spdf interacting boson model |
| Authors | K. Nomura (Hokkaido University) |
| Journal | *Physical Review C* (2026) |
| Date | 12 January 2026 |
| DOI | 10.1103/PhysRevC.xxx |
Main Findings:
- Studies QPTs involving octupole degrees of freedom within the spdf-IBM
- Examines potential-energy surfaces in quadrupole-octupole (β₂, β₃) deformation space
- Maps self-consistent mean-field (Gogny D1M) energy surfaces onto IBM Hamiltonian parameters
- Describes transitions between different shape phases including stable octupole-deformed configurations
- Systematic calculations across multiple isotopic chains comparing with experimental E1, E2, E3 transition rates
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3. Unified Theoretical Frameworks
3.1 Unified Description of Phase Transition, Coexistence, and Mixing (EPJ Web Conf., 2025)
| Item | Details |
|---|---|
| Title | A unified description of the shape phase transition, shape coexistence and mixing phenomena in nuclei |
| Authors | R. Budaca, P. Buganu, A. I. Budaca, A. Ait Ben Mennana, R. Benjedi, Y. El Bassem, A. Lahbas, M. Oulne |
| Journal | *EPJ Web of Conferences* 324, 00026 (2025) |
| Conference | V International Conference on Nuclear Structure and Dynamics (NSD2024) |
| Date | 11 April 2025 |
| DOI | 10.1051/epjconf/202532400026 |
Main Findings:
- Provides a single theoretical framework (Bohr Hamiltonian with sextic potential) that simultaneously describes:
- Shape phase transitions (spherical-to-deformed)
- Shape coexistence (two distinct minima at different deformations)
- Shape mixing (quantum mechanical interaction between coexisting configurations)
- Uses quasi-exactly solvable (QES) approach with sextic oscillator
- Extends to triaxial nuclei via Davydov-Chaban Hamiltonian at γ = 30°
- Applications to Se, Mo, Ge, Kr, Ca, Pt isotope chains
- Addresses the controversial case of ⁸⁰Ge (conflicting experimental results)
- Analyzes E2 and E0 transition probabilities as experimental signatures
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3.2 Shape and Multiple Shape Coexistence Across the Nuclear Chart (PRC, 2023)
| Item | Details |
|---|---|
| Title | Shape and multiple shape coexistence of nuclei within covariant density functional theory |
| Authors | Y. L. Yang, P. W. Zhao, Z. P. Li |
| Journal | *Physical Review C* 107, 024308 (2023) |
| Date | February 2023 |
| arXiv | 2302.07417 |
Main Findings:
- Systematic calculation of low-lying spectra and quadrupole shape invariants for even-even nuclei with 10 ≤ Z ≤ 104 (proton drip line to neutron drip line)
- Uses five-dimensional collective Hamiltonian based on PC-PK1 covariant density functional
- Predicts a wealth of nuclei with shape/multiple shape coexistence in neutron-rich regions where experimental data is scarce
- Strong E0 transition strength generally correlates with shape coexistence, but not conversely
- Results serve as guidelines for future experimental searches
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4. Review Articles
4.1 Multifaceted Character of Shape Coexistence (Prog. Part. Nucl. Phys., 2024)
| Item | Details |
|---|---|
| Title | Multifaceted character of shape coexistence phenomena in atomic nuclei |
| Authors | K. Heyde et al. (continuation of the seminal Heyde & Wood review tradition) |
| Journal | *Progress in Particle and Nuclear Physics* (2024) |
| Date | November 2024 |
| DOI | 10.1016/j.ppnp.2024.104112 |
Key Content:
- Comprehensive review extending the classic Heyde & Wood (*Rev. Mod. Phys.* 2011) framework
- Covers E0 transitions as unique probes of shape coexistence type (extreme vs. mixed-shape)
- Discusses the "macroscopic" picture of multiple nuclear configurations with different intrinsic shapes
- Addresses the still-open theoretical challenges in describing shape coexistence microscopically
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4.2 Physics of Exotic Nuclei (Nature Reviews Physics, 2024)
| Item | Details |
|---|---|
| Title | Physics of exotic nuclei |
| Authors | Yanlin Ye (叶沿林), Xiaofei Yang, Hiroyoshi Sakurai, Baishan Hu |
| Journal | *Nature Reviews Physics* 7, 21–37 (2025) |
| Date | 25 November 2024 |
| DOI | 10.1038/s42254-024-00782-5 |
Key Content on Shape Coexistence:
- Islands of inversion: N = 20 and N = 40 regions where deformed intruder configurations compete with spherical normal configurations
- ⁷⁸Ni region: predicted as a "fifth island of inversion" with shape coexistence near the doubly magic ⁷⁸Ni
- ⁷⁹Zn: isomer shift and magnetic moment measurements provide signature of shape coexistence near ⁷⁸Ni
- ⁷⁴Zn: first evidence of axial shape asymmetry and configuration coexistence (northern extension of N = 40 island)
- Shell evolution driven by tensor force and proton-neutron interaction (Otsuka et al.)
- New facilities (FRIB, SPIRAL2, RIBF, FAIR, HIAF) will expand exploration of terra incognita
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4.3 Nuclear Supersymmetry and Quantum Shape-Phase Transition (Indian J. Phys., 2021)
| Item | Details |
|---|---|
| Title | Nuclear supersymmetry and quantum shape-phase transition |
| Authors | Not extracted |
| Journal | *Indian Journal of Physics* (2021) |
| DOI | 10.1007/s12648-021-02152-8 |
Key Content:
- Experimental evidence for vibrational to γ-unstable transition in Ru–Rh and Zn–Cu supermultiplets
- Demonstrates the second-order nature of the U(5)→O(6) transition
- Connects nuclear supersymmetry to shape-phase transitions
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5. Chinese Research Contributions
5.1 原子核形状相变的玻色子代数模型理论 (Sci. Engine, 2024)
| Item | Details |
|---|---|
| Title | 原子核形状相变的玻色子代数模型理论 |
| Journal | Sci. Engine (2024) |
| Date | 16 August 2024 |
- 系统总结了基于玻色子代数模型框架的核形状相变基本概念和核心问题
5.2 中等质量核形状共存与壳效应研究 (Acta Phys. Sin., 2025)
| Item | Details |
|---|---|
| Title | 中等质量核形状共存与壳效应研究 |
| Journal | *Acta Physica Sinica* (物理学报) 74 (2025) |
| DOI | 10.7498/aps.74.20250095 |
- 在 Zn, Ge, Se, Kr 同位素中发现显著的形状共存现象与刚性三轴性特征
5.3 傅冠健 (同济大学): Angular-Momentum-Projected PNC-BCS
- 上海交通大学李政道研究所 INPAC 特别报告 (2026年4月9日)
- 开发了角动量投影粒子数守恒BCS理论，桥接壳模型和变形平均场模型
- 首次一致描述了原子核的形状演化和量子相变
- 获国家优青基金、中国核物理学会"胡济民奖"
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6. Key Experimental Signatures and Methodologies
| Signature | What It Probes | Key References |
|---|---|---|
| E0 transitions (electric monopole) | Shape coexistence type (extreme vs. mixed); nuclear volume changes | ¹⁸⁶Pb (2022), Heyde review (2024) |
| B(E2) values | Collectivity and deformation of specific bands | ⁷⁰Co (2025), ¹⁸⁶Pb (2022) |
| E0/E2 ratios | Distinguish continuous from discontinuous shape transitions | Hammad et al. (2026) |
| Two-neutron transfer reactions | Interplay between intruder and normal configurations | IBM-CM framework |
| Ultra-relativistic collision correlators (ρ₄,₂, ρ₂,₄) | γ-soft vs. rigid triaxial; second-order QPT | Zhao et al., PRL (2024) |
| Quadrupole shape invariants | Systematic identification of coexistence regions | Yang, Zhao, Li, PRC (2023) |
| Total Absorption Spectroscopy | β-decay strength distributions revealing deformed configurations | ⁷⁰Co (2025) |
| In-beam electron spectroscopy | Direct E0 transition measurement | ¹⁸⁶Pb (2022) |
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7. Key Research Groups and Institutions
| Institution | Key Researchers | Focus |
|---|---|---|
| Michigan State University / FRIB | Dembski, Spyrou, Brown, Liddick | Exotic nuclei β-decay, shape coexistence |
| University of Jyväskylä | Ojala, Pakarinen, Papadakis, Julin | In-beam spectroscopy, E0 transitions |
| Peking University (北京大学) | 宋慧超, 刘玉鑫 | Ultra-relativistic collisions → nuclear shape QPT |
| Tongji University (同济大学) | 傅冠健 | PNC-BCS theory, shell model truncation, QPT |
| Hokkaido University | K. Nomura | spdf-IBM, octupole deformation |
| Damanhour University (Egypt) | Hammad, Yahia, Saad, Doma | Catastrophe theory, E(5), phase diagrams |
| IFIN-HH Bucharest | Budaca, Buganu | Sextic potential, unified framework |
| Peking University / CAS | 叶沿林 | Exotic nuclei physics, shell evolution |
| Lanzhou University / IMP CAS | Z.P. Li, P.W. Zhao | Covariant DFT, shape invariants |
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8. Summary of Key Trends (2022–2026)
- New experimental probes: Total Absorption Spectroscopy (⁷⁰Co), in-beam electron spectroscopy (¹⁸⁶Pb), ultra-relativistic collision correlators (¹²⁹Xe) are expanding the toolkit for detecting shape coexistence and QPTs
- Shape reassignments: The ¹⁸⁶Pb result (2022) overturning a 22-year-old Nature assignment shows that even "established" shape coexistence cases require careful re-examination
- Type II Shell Evolution: Recognized as a key mechanism for extreme shape coexistence near shell closures (⁷⁰Co), going beyond traditional particle-hole excitation pictures
- Unified frameworks: Multiple groups are developing frameworks that simultaneously treat phase transitions, coexistence, and mixing — moving beyond treating these as separate phenomena
- Cross-disciplinary bridges: The PRL 2024 paper (Zhao et al.) creates an entirely new connection between high-energy nuclear collisions and low-energy nuclear structure QPTs
- Neutron-rich frontier: Systematic predictions (Yang, Zhao, Li 2023) and the Nature Reviews Physics review (2024) highlight that the most exciting shape coexistence discoveries are likely in unexplored neutron-rich regions
- Octupole joins quadrupole: The spdf-IBM work (Nomura 2026) extends QPT studies to include octupole degrees of freedom, important for parity-splitting patterns and reflection-asymmetric shapes
- Catastrophe theory revival: The Hammad et al. (2026) paper reconnects nuclear QPTs to the mathematical framework of catastrophe theory, providing analytic phase diagrams and diagnostic tools