Shape Phase Transition Research Articles

Entanglement entropy approach for examining quantum phase transition in the framework of semiclassical approximation: Testing its validity in Casten triangle

The entanglement entropy of s and d bosons in the framework of Interacting Boson Model -1 (IBM-1) has been obtained using consistent-Q formalism and semiclassical approximation. This has been possible by using Schmidt decomposition and expressing s and d bosons entanglement entropy in terms of Schmidt numbers. In this research, a simple method in the framework of IBM-1 has been presented for deriving the entanglement entropy in the Casten triangle. The results indicated that the entanglement entropy is sensitive to the shape-phase transition in the various regions of the Casten triangle. It was demonstrated that the entanglement entropy of s and d bosons in the semiclassical approximation depends only on the values of the deformation parameter (β) and is independent of the angular parameter (γ). Also, the entanglement entropy between s and d bosons reaches its maximum value in the O(6) limit, while it decreases in the SU(3) limit, and reaches zero in the U(5) limit. Based on the results obtained via Schmidt decomposition, it is shown that the probability distribution functions of the number of s bosons in IBM-1 are the binomial distributions. For NB≫1, it was proved that the distribution function in the SU(3), O(6) and SU(3)‾ limits is the Gaussian, and in the U(5) limit is the Poissonian.

Collectivity at the prolate-oblate transition: The [formula omitted] lifetime of 190W

The neutron-rich rare isotope 190W is discussed as a candidate for a prolate-oblate transitional nucleus with maximum γ-softness. The collectivity of this isotope is assessed for the first time by the measurement of the reduced E2 transition probability of its first 2+ state to the ground state. The experiment employed the FAst TIming Array (FATIMA), comprised of 36 LaBr3(Ce) scintillators, which was part of the DESPEC setup at GSI, Darmstadt. The 41+ and 21+ states of 190W were populated subsequently to the decay of its 127(12) μs isomeric Jπ=10− state. The mean lifetime of the 21+ state was determined to be τ=274(28) ps, which corresponds to a B(E2;21+→01+) value of 95(10) W.u. The results motivated a revision of previous calculations within an energy-density functional-based interacting boson model-2 approach, yielding E2 transition properties and spectroscopic quadrupole moments for tungsten isotopes. From comparison to theory, the new data suggest that 190W is at the transition from prolate to oblate structure along the W isotopic chain, which had previously been discussed as a nuclear shape-phase transition.

Covariant density functional description of shape phase transitions and shape coexistence in heavy nuclei

Covariant density functional description of shape phase transitions and shape coexistence in heavy nuclei

Intertwined quantum phase transitions in the zirconium and niobium isotopes

Nuclei in the A ≈ 100 region exhibit intricate shape-evolution and configuration crossing signatures. Exploring both even–even and their adjacent odd-mass nuclei gives further insight on the emergence of deformation and shape-phase transitions. We employ the algebraic frameworks of the interacting boson model with configuration mixing and the new interacting boson-fermion model with configuration mixing in order to investigate the even–even zirconium with neutron number 52–70 (40Zr) and odd-mass niobium (41Nb) isotopes with 52–62. We compare between the evolution in energy levels, configuration and symmetry content of the wave functions, two neutron separation energies and E2 transition rates, alongside comparison with the experimental data. The comparisons between the two chains of isotopes denote the occurrence of two types of QPTs, a crossing of normal and intruder configurations (named Type II QPT), and a shape-evolution of the intruder configuration (named Type I QPT). The latter QPT begins from spherical shape to axially deformed rotor in the Zr isotopes and from weak to strong coupling scenario in the Nb isotopes. This occurrence, named intertwined quantum phase transitions (IQPTs), in the even–even Zr chain is thus demonstrated to persists when coupling a proton to the boson core for the odd-mass Nb chain, even as deformation increases.

DYNAMICS OF DEFORMATION FORCES IN SINGLE-PARTICLE SPECTRA OF ODD 2s1d-SHELL NUCLEI

With help of our newly-developed evolutionary model-independent approach we have found that with the increase in mass number of all studied odd 2s1d-shell nuclei 23Na, 27Al, 31P, and 35,37Cl the dominant deformation of the shape of these nuclei in low-laying single-particle states changes from quadrupole to hexadecapole and further to hexacontatetrapole. The single states and the continuous sets of states with abnormally weak deformation are found. The points of shape phase transitions with a change in the multipolity of deformation are found in the single-particle spectrum of 37Cl nucleus.

Studying shape phase transition in even-proton nuclei via mass measurements

The transition between the spherical and deformed shapes of the nuclei (shape phase transition) is an interesting property of nuclei that enriches our understanding of the nuclear structure. One method for studying it is through measurement of a wide range of atomic masses and derivation of the two-neutron separation energies (S2n). Using a multiple-reflection time-of-flight mass spectrometer at the FRS Ion Catcher, GSI, atomic masses in the region of A ∼ 150 were measured. This allowed an wide study of phase shape transitions in this region of the nuclear chart. In addition, comparison with the Skyrme Hartree–Fock plus BCS approach are done; showing a remarkable good agreement between theory and experiment.

Emerging γ-soft-like spectrum in 196Pt in the SU3-IBM (I)* *This research is supported by the Educational Department of Jilin Province, China (JJKH20210526KJ), C.-X. Z. gratefully acknowledges support from the Project Supported by Scientific Research Fund of Hunan University of Art and Science (23ZZ04)

Recently, studies have argued that a spherical-like spectrum emerges in the SU3-IBM, thus creating new approaches to understanding γ-softness in realistic nuclei. In a previous study, γ-softness with degeneracy of the ground and quasi-γ bands was observed. In this paper, another special point connected to the middle degenerate point is discussed. It is found to be related to the properties of 196Pt. This emergent γ-softness is also shown to be important for understanding the prolate-oblate asymmetric shape phase transition. The low-lying spectra, values, and quadrupole moments in 196Pt are discussed, and we show that the new model can account for several observed features. This is the first part of the discussion on the γ-soft-like spectrum of 196Pt.

Shape evolution of thorium isotopes in between the drip-lines

Commonly encountered shapes in nuclei are spherical, prolate and oblate. The shape of a nucleus is decided by the residual interactions among its nucleons. In our current investigation, we incorporate both axial and triaxial degrees of freedom to study how the shape of thorium nuclei changes within the boundaries of the drip lines. We employed the relativistic mean-field theory and utilized density-dependent meson-exchange as the functional throughout our calculations. We calculated the total binding energy as it varies with the deformation parameter [Formula: see text] and subsequently depicted the resulting binding energy curves. Furthermore, we examined the self-consistent energy surface for thorium isotopes with triaxial quadrupole deformations. The emergence of potential energy minima can be attributed to the presence of gaps or regions with reduced density of single-particle energy levels around the Fermi surface, as the nucleus undergoes deformation. The nuclear shape evolution of thorium isotopes ranging from [Formula: see text] to [Formula: see text] was investigated, [Formula: see text] and [Formula: see text] are spherical in shape and the nearby isotopes are less deformed and nearly spherical. Isotopes ranging from [Formula: see text] to [Formula: see text] exhibit a higher level of deformation. The most deformed nucleus is [Formula: see text] and is prolate. The weak triaxial behavior is observed within the region associated with shape phase transition.

Algebraic cluster model calculations for shape phase transitions of boson-fermion systems

The Algebraic Cluster Model (ACM) is an interacting boson model that gives the relative motion of the cluster configurations in which all vibrational and rotational degrees of freedom are present from the outset. We schemed a solvable extended transitional Hamiltonian based on affine [Formula: see text] Lie algebra within the framework for two-, three- and four-body algebraic cluster models that explains both regions [Formula: see text], [Formula: see text] and [Formula: see text], respectively. We offer that this method can be used to study [Formula: see text] nucleon structures with [Formula: see text] and [Formula: see text] in specific [Formula: see text] such as structures [Formula: see text], [Formula: see text], [Formula: see text]; [Formula: see text], [Formula: see text], [Formula: see text]; [Formula: see text], [Formula: see text]. Numerical extraction to the energy levels, the expectation value of the boson number operator, and the behavior of the overlap of the ground state wave function within the control parameters of this evaluated Hamiltonian are presented. The effect of the coupling of the odd particle to an even–even boson core is discussed along the shape transition and, in particular, at the critical point.

Quadrupole-octupole collective excitations in medium mass nuclei

A quadrupole-octupole axially symmetric collective model is used to describe the alternate parity bands observed in even–even medium mass nuclei. The nature of the octupole correlations in considered nuclei is ascertained from the phenomenology of the adopted model and the obtained parameters. It is found that the model parameters have a regular evolution as a function of neutron number, suggesting that the shape phase transition around N = 90 from low to well deformed shapes is also accompanied by the increase of the vibrational character for the octupole deformation. Model extrapolations are performed for various types of excited states and electromagnetic properties of measured energy levels.

Correlation between the shape coexistence and stability in Mo and Ru isotopes

In a rapidly changing shape phase region, the presence of shape coexistence and its possible impact on the decay modes and half-lives, has been explored in astrophysically interesting Mo and Ru isotopes, in an extensive study within the microscopic theoretical framework using Nilsson Strutinsky Method and Relativistic Mean Field Model. The isotopic chains of Mo and Ru exhibit rapid shape phase transitions, triaxial γ softness, shape instability along with many coexisting states mostly with oblate and triaxial shapes. Proton and neutron separation energies have been calculated and compared with the available data. Results obtained from both the formalisms are in good agreement with each other as well as the available experimental data. Our computed β-decay half-lives and separation energy for nuclei exhibiting shape coexistence were examined for the decay mode from second minima state of the parent nuclei to the ground or excited state of the daughter nuclei. The second minima state of the coexisting shapes in Mo and Ru isotopes, were seen to impact the structural properties, β-decay half-lives, separation energy and hence the stability of the nuclei.

Quantum entanglement of SO(6)-U(5) transitional nuclei in the interacting boson model-2 (IBM-2)

The quantum shape phase transition between the spherical and deformed γ-unstable (U(5)−O(6)) even-even nuclei within the frameworks of Interacting Boson Model-1 and 2 (IBM-1,2) for low-lying states, using the “entanglement entropy” (S) has been studied. In both frameworks, the theoretical results showed that there exist minimum and maximum entanglement values between s bosons in the U(5) and O(6) limits, respectively. In order to confirmation of the theoretical results, we have calculated and analyzed the entanglement entropy of s−d bosons and proton (π) - neutron (ν) bosons in Cerium (C58122−136e) isotopes. The results indicate that the entanglement entropy correctly describes the transition from U(5) to O(6), for ground state (01+), but it cannot accurately determine the transitional nucleus.

Algebraic model analysis of shape phase transitions in odd-odd nuclei

Algebraic model analysis of shape phase transitions in odd-odd nuclei

Nuclear Shape-Phase Transition From Spherical U(5) to Deformed gamma-Unstable O(6) Dynamical Symmetries of Interacting Boson Model Applied to Ru, Pd, and Xe Isotopic Chains

The shape transition from spherical vibrator U(5) to gamma-unstable deformed rotor O(6) in even-even Ru, Pd, and Xe isotopic chains are studied in framework of sd interacting boson model (IBM1) using the coherent state formalism to obtain the potential energy surfaces (PES’s). The location of critical points in the transition are identified by analysis the PES’s in terms of the deformation parameter beta and by using the catastrophe theory in terms of the two essential parameters (r_1, r_2). By using the most general IBM1 Hamiltonian in Casimir form with neglecting the U(5) and SU(3) quadratic Casimir operators and introducing only one control parameter the PES leads to the same energy surface as the Q-consistent IBM Hamiltonian at gamma =0. For the studied isotopic chains, the chi ^2-test is used to perform the fitting between the experimental and the corresponding calculated IBM for some selected energy levels and electric quadrupole transition probabilities B(E2) values using a simulated search program. A good agreement is produced for both energies and B(E2) transition rates. The present model calculations suggest that ^{100}Ru, ^{102}Pd and ^{130}Xe nuclei are good candidates for the E(5) critical point symmetry.

Deformation of Odd Nuclei 27Al, 31P and 35Cl in Single-Particle States

Using the evolutionary approach recently developed by us, the shapes of odd s-d-shell 27Al, 31P and 35Cl nuc­lei in the ground and single-particle excited states have been extracted from the experimental data on the energies, spins, and parities of these states, as well as the measured probabilities of electromagnetic transitions between them. The key ingredient of our procedure is the evolutionary algorithm that evolves the population of the bad-quality data-fitting nuclear shapes to the high-quality data-fitting nuclear shapes. We have found that the studied nuclei in the ground states are abnormally weakly deformed, which is not expected for the nuclei in the shell middle. Even in their low-laying single-particle excited states, the nuclei 27Al and 31P are found to be weakly deformed, too. With the increase of the single-particle excitation energy, the change of the state of the only one nucleon – the valence proton the spin and parity of which determine the spin and parity of the 35Cl nucleus – causes the shape phase transition from the high-symmetry phase – spherical ground state – to the low-symmetry phase – deformed excited states. The angular part of the 27Al and 31P nuclei shape is de­scribed by two harmonics – quadrupole and hexadecapole. The angular part of the 35Cl nucleus shape is de­scribed by three harmonics – quadrupole, hexadecapole, and hexacontatetrapole, but the contribution of hexadecapole deformation is not independent. At present, there are no fundamental nuclear models that account for or predict the dominant hexacontatetrapole deformation, especially for light and medium nuclei. We have found that the spin and parity of the 27Al, 31P and 35Cl nuc­lei are determined by the spin and parity of the last odd (valence) proton. At the same time, some of the nucleons of the nucleus core change their characteristics, too. Thus, the electromagnetic transitions between the single-particle states of the 27Al, 31P and 35Cl nuc­lei are the multi-particle processes.

Prolate-oblate asymmetric shape phase transition in the interacting boson model with SU(3) higher-order interactions

Prolate-oblate asymmetric shape phase transition in the interacting boson model with SU(3) higher-order interactions

DEFORMATION OF ODD NUCLEI 23Na, 25Mg, AND 25Al IN SINGLE-PARTICLE STATES

Using the evolutionary approach recently developed by us, the shapes of odd 2s1d-shell 23Na, 25Mg and 25Al nuclei in the ground and single-particle excited states have been extracted from the experimental data on the energies, spins, and parities of these states, as well as the measured probabilities of electromagnetic transitions between them. We have found that the single-particle spectra of the nuclei studied contain the single states and the continuous sets of states with abnormally weak deformation. This indicates the existence of the shape phase transitions from the spherical state of the nucleus into a deformed state.

Probing DDM and ML quantum concepts in shape phase transitions of γ-unstable nuclei

In a recent paper (S. Ait El Korchi et al. (2020) [32]), we explored, inside the context of Critical Point Symmetries (CPSs) X(3) and Z(4), a correlation between two exceedingly known quantum concepts, the Minimal Length (ML) and the Deformation-Dependent Mass (DDM), that are commonly applied in various areas of physics. Such a correlation has been strongly identified in transition nuclei by calculating some physical observables of that quantum system, like as energy spectra, moments of inertia and transition probabilities. In this paper we extend that study to E(5) dynamical symmetry corresponding to the shape phase transition U(5)↔O(6). The experimental realization of the models was found to occur in some nuclei, using the existing phenomenological potentials: Infinite Square Well, Davidson and Kratzer, whose models fits provide the best agreement. Importantly the calculations performed in this work using these potentials, corroborate the fact that the revealed correlation between both quantum concepts is not destructively affected by the presence of other model's parameters and hence its existence is independent of the form or type of the used potential. Undoubtedly, the present work will open the way for more investigations of this correlation in the limits of other critical points symmetries in nuclear shape phase transitions which play today a major role in nuclear structure research from theoretical as well as experimental point of view.

Microscopic analysis of octupole shape phase transitions and critical points in neutron rich actinides

Octupole constrained energy surfaces, and spectroscopic observables of four isotopic chains of: Cm, Cf, Fm and No with neutron numbers 186 N200 are analysed using a collective quadrupole - octupole Hamiltonian (QOCH). The parameters of the Hamiltonian are determined by axially reflection-asymmetric relativistic Hartree-Bogoliubov calculations based on the energy density functional DD-PC1, and a finite-range pairing interaction. The theoretical results suggest quantum phase transitions from non-octupole to octupole deformed shapes and to octupole vibrations with increasing neutron number. 288Cm is possibly close to the critical point of a simultaneous phase transition from spherical to prolate deformed and from non-octupole to stable octupole deformed configurations.

Investigation of giant dipole resonance in Mo isotopes within TDHF theory

The isovector giant dipole resonance (IVGDR) in the chain of even-even Mo isotopes is investigated within the time-dependent Hartree-Fock (TDHF) using the Skyrme force Sly6. The GDR calculated in Mo92−108 are presented, and compared with the available experimental data. An overall agreement between them is obtained. Moreover, the dipole strength in Mo102−108 is predicted. Shape phase transition from spherical to oblate as well as shape coexistence (A ∼ 100) in Mo isotopes are also investigated in this work. In addition, the correlation between the deformation splitting ΔE and the quadrupole deformation parameter β2 is studied. The results confirm that ΔE is proportional to the deformation of nucleus. We also discuss the dependence of GDR strength on some nuclear properties of Skyrme forces, such as the asymmetry energy as. We find that as=32 MeV, corresponding to SLy6, is quite consistent with experimental data.