Algebraic Cluster Model Research Articles

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.

The Algebraic Cluster Model (8Be, 12C) and the Cluster Shell Model (13C, 7Be, 19F, 8Be)

Bijker and Iachello’s algebraic cluster model (ACM) and its extension to the cluster shell model (CSM), provides a new theoretical platform for the study of alpha-clustering in light nuclei. It led to the discovery of the D3h symmetry in 12C and 20Ne, with the discovery in 12C of a new g.s. rotational band with the spin sequence of, 0+, 2+, 3-, 4+/4- and 5-, including the predicted 4+ and 4- parity doublet. Applications of the CSM shell model to particle molecular orbits in 9Be and 13C (C2’ and D3h’ particle symmetries, respectively), lead us to conjecture molecular hole states in 7Be and 19F. We observe in these nuclei the predicted phenomenological structure. And we further consider conjectured p-h states in 8Be with the predicted phenomenological p-h structure of rotational band at high excitations of approximately 20 MeV. A search for these rotational band in 8Be was performed at ISOLDE.

Algebraic model studies of the structure and reactions of 12C and 16O

The role of algebraic models and geometric cluster models of alpha clusters in the structure of ground and first excited 0+ and second excited 2+ states in 12C are evaluated with the aim of establishing if the rotational bands are compatible with rigid structures or if they are instead overlaps of different configurations. A rigid equilateral triangle shape is assumed in Ref.[2, 3]: the energy spectrum, electromagnetic observables are calculated and the transition densities are compared with elastic and inelastic cross-sections for various processes. In a different approach [4, 5], we solve the three-body quantum problem for 3 alphas using the orthogonality condition and the stochastic variational method with correlated Gaussian basis functions. The two-body density distributions indicate that the main configurations of both the second 0+ and 2+ states are acute iscosceles triangle shapes with a 8Be(0+)+α configuration. We also find clues that the second 2+ state is probably not a member of the rigid rotational band built on the Hoyle state.

Advances on clusters and correlations in nuclear structure and reactions

Recent advances obtained in the last few years by the Theoretical Nuclear Physics group in Padova with various collaborators on alpha-cluster models and on nuclear correlations in stable and unstable light nuclei are reviewed in this contribution. The algebraic cluster model assumes triangular and tetrahedral arrangement of α particles for 12C and 16O respectively. The description of the low-lying states achieved in this model, that is a consequence of the requirement of discrete symmetries, is extremely good. We have made several calculations of α-transfer form factors and reaction cross-sections obtaining a good agreement with available data, thus corroborating the main hypotheses of the model [1]. We have speculated about smoking-gun nuclear fluorescence experiment that might shed light on the exact spatial arrangement of alpha particles in 12C [2]. We will also talk about the successful predictions on the positioning of 29F on the southern shore of the island of inversion [3] and on recent calculations on its halo character and dipole response [4, 5]. Extension of these calculations to 31F [6], pointing out the presence of a halo, are also discussed.

Conjectured Hole States (7Be, 19F) and p-h states (8Be) of the CSM

The remarkable success of the Algebraic Cluster Model (ACM) to describe the N=Z even-even light nuclei, 8Be, 12C, 16O and 20Ne, and the neighbouring nuclei with one particle added, 9Be, 9B, 13C, 21Ne and 21Na, using the Cluster Shell Model (CSM), lead us to consider the phenomenology of data on conjectured hole states of the CSM in 7Be and 19F, and conjectured high lying particle-hole states (p-h) in 8Be.We propose the first application of the Cluster Shell Model (CSM) of Della Rocca and Iachello to conjectured p-h states in 8Be. We demonstrate a few essential phenomenological features of the CSM in 8Be: 1) All predicted conjectured p-h states of the CSM, and only the predicted p-h states, are observed near thresholds and up to 19.5 MeV in 8Be. 2) The states are observed in the predicted order, with positive parity states below negative parity states. 3) Some of the conjectured p-h states are already known to have the predicted rotational structure of the deformed p-h states. 4) The rotational structures observed at high excitations in the conjectured p-h bands in 8Be, resemble the ground state bands of 8Be, 9Be and 9B, with similar moment of inertia.Based on these observations we contemplate new measurements of the spectroscopy of 8Be at energies above 19.5 MeV, where our knowledge of states in 8Be is incomplete. We examine the observed B(M1)s and B(E2)s in these nuclei and contemplate a measurement of the B(E2) of the iso-baric Analog transition in 8Li.

Application of the cranking method to the semimicroscopic algebraic cluster model and nuclear molecules

A particular quantum phase transition (QPT) is studied at excited energies of light nuclei within the Semimicroscopic Algebraic Cluster Model (SACM), using a combination of catastrophe theory and a direct minimization of the potential. A distinct change from a compact nucleus to a nuclear molecule is described as a second order QPT occurring at a certain value of angular momentum. This finding is in accordance with experimental observations when nuclear molecules appear. The method has implication in the fission of heavy nuclei.

Algebraic cluster model calculations for vibrational to gamma-unstable shape phase transition in odd-A nuclei

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 the SU(1;1) Lie algebra within the framework for two-, three- and four-body algebraic cluster models that explains both regions O(4) U(3), O(7) U(6) and O(10)-U(9), respectively. We suggest that this method can be used to study of k + x nucleon structures with k = 2, 3, 4 and x = 1, 2 and so on. The obtained results in this study confirm that this ACM technique is worth extending for investigating odd-A and odd-odd nuclei. So, the clustering survives the addition of one and two particles. Our studies confirm the importance of the odd nuclei as necessary signatures to characterize the occurrence of the phase transition and to determine the precise position of the critical point.

Symplectic Symmetry and Clustering in Atomic Nuclei

A new symplectic-based shell-model approach to clustering in atomic nuclei is proposed by considering the simple system $^{20}$Ne. Its relation to the collective excitations of this system is mentioned as well. The construction of the Pauli allowed Hilbert space of the cluster states with maximal permutational symmetry is given for the $^{16}$O+$^{4}$He $\rightarrow$ $^{20}$Ne channel in the case of one-component many-particle nuclear system. The equivalence of the obtained cluster model space to that of the semi-microscopic algebraic cluster model is demonstrated.

Approximate projection method for the construction of multi- α -cluster spaces

An approximate but straight forward projection method to molecular many alpha-particle states is proposed and the overlap to the shell model space is determined. The resulting space is in accordance with the shell model, but still contains states which are not completely symmetric under permutations of the alpha-particles, which is one reason to call the construction semi-microscopic. A new contribution is the construction of the 6- and 7-$\alpha$-particle spaces. The errors of the method propagate toward larger number of alpha-particles and larger shell excitations. In order to show the effectiveness of the construction proposed, the so obtained spaces are applied, within an algebraic cluster model, to $^{20}$Ne, $^{24}$Mg and $^{28}$Si, each treated as a many-alpha-particle system. Former results on $^{12}$C and $^{16}$O are resumed.

A semimicroscopic algebraic cluster model for heavy nuclei

An extension of the Semimicroscopic Algebraic Cluster Model (SACM) is proposed, based on the pseudo-SU(3) model. The Hamiltonian and the spectroscopic factor operator of the model are presented and a procedure of constructing the model space. Because a huge number of SU(3) irreducible representations (irrep) appear, one has to be careful in designing a practical, consistent path to reduce the Hilbert space. The concept of forbiddenness, taking into account excitations of the clusters, is introduced and applied. The applications are to two systems with a low forbiddenness, namely to 236U -> 210Pb + 26Ne and 224Ra -> 210Pb + 14C, and to 236U -> 146Xe + 90Sr, which appears in the fission of 236U, which requires a large forbiddenness. Energies, electromagnetic transitions and spectroscopic factors are calculated.

Alpha-induced inelastic scattering and alpha-transfer reactions in $$^{12}$$C and $$^{16}$$O within the Algebraic Cluster Model

The molecular algebraic model based on three and four alpha clusters is used to describe the inelastic scattering of alpha particles populating low-lying states in $^{12}$C and $^{16}$O. Optical potentials and inelastic formfactors are obtained by folding densities and transition densities obtained within the molecular model. One-step and multi-step processes can be included in the reaction mechanism calculation. In spite of the simplicity of the approach the molecular model with rotations and vibrations provides a reliable description of reactions where $\alpha$-cluster degrees of freedom are involved and good results are obtained for the excitation of several low-lying states. Within the same model we briefly discuss the expected selection rules for the $\alpha$-transfer reactions from $^{12}$C and $^{16}$O.

Tetrahedral intrinsic structure of Oxygen-16 revisited

A model for vibrational and rotational excitations of an intrinsic tetrahedral structure of the Oxygen-16 nucleus has been investigated, building on classic work of Wheeler, Dennison and Robson. Tetrahedral A- and F-phonons are treated harmonically, but the E-phonon dynamics is extended to dynamics on an E-manifold of configurations of four α-particles. This allows for tunnelling between the tetrahedral configuration and its dual, through a square configuration, and lifts the parity doubling that would otherwise occur. Unlike in earlier models, including the algebraic cluster model, the E-phonon frequency is taken to be about half the F-phonon frequency, which in turn is about half the A-phonon frequency. As a result, the first-excited 0+ state is a 2-phonon, E-manifold excitation, whereas the lowest 2+ and 2− states are 1-phonon, E-manifold excitations. Coriolis contributions to the rotational energy of states with F-phonons are significant. Altogether, rotational bands are constructed based on vibrational states with up to four phonons, and with spin/parity up to 9−. Nearly all the observed states of Oxygen-16 are accommodated, up to 20 MeV and partly beyond. Predictions for some so far unobserved states, especially those of unnatural parity, are made.

Results on quantum phase transitions within the semimicroscopic algebraic cluster model and extension to deformed clusters

The properties of quantum phase transitions in the semimicroscopic algebraic cluster model are investigated. The order of the quantum phase transition is shown to depend on the path taken in the parameter space and a novel procedure to obtain the Maxwell set separatrix is discussed. The first steps toward a description of systems of deformed cluster and heavy nuclei are presented.

Quantum Phase Transitions within the Semimicroscopic Algebraic Cluster Model

Quantum Phase Transition (QPT) properties are investigated within the Semimicroscopic Algebraic Cluster Model (SACM), as an example of a model with involved structure. The catastrophe theory is applied to obtain a physical insight and analytic formulas for the process of QPTs. QPTs are obtained up to the third order. We propose a novel procedure which can be used in several models in various sectors of physics.

Analysis of the alpha-transfer reaction in the 12C + 16O system using the semi-microscopic algebraic cluster model

In this work, we revisit the experimental data for elastic, inelastic and $\alpha$-transfer in the 12C + 16O system at 80 and 84 MeV using the coupled reaction channel method to analyze the angular distributions. The spectroscopic amplitudes relevant for the $\alpha$-transfer channel have been calculated from the semi-microscopic algebraic cluster model. The imaginary term of the optical potential was fitted to the experimental elastic scattering data at forward angles. Our calculations describe reasonable well the absolute cross sections for the inelastic scattering and $\alpha$-transfer. We discuss the effect of couplings on the transfer channel.

16O within the Semimicroscopic Algebraic Cluster Model and the importance of the Pauli Exclusion Principle

The Semimicroscopic Algebraic Cluster Model (SACM) is applied to 16O, assumed to consist of a system of four alpha-clusters. For the 4-alpha cluster system a microscopic model space is constructed, which observes the Pauli-Exclusion-Principle (PEP) and is symmetric under permutation of the 4-alpha-particles. A phenomenological Hamiltonian is used, justifying the name Semi in the SACM. The spectrum and transition values are determined. One of the main objectives is to test the importance of the Pauli Exclusion Principle (PEP), comparing the results with the Algebraic Cluster Model (ACM), which does not include the PEP, and claims that the 16O shows evidence of a tetrahedral structure, which can be explained easily by symmetry arguments. We show that PEP is very important and cannot be neglected, otherwise it leads to a wrong interpretation of the band structure and to too many states at low energy.

Three-body description of 12C: From the hyperspherical formulation to the algebraic cluster model and its application to α + 12C inelasticscattering

Form factors for α+12C inelastic scattering are obtained within two theoretical (α+α+α) approaches:The hyperspherical framework for three identical bosons, and the algebraic cluster model assuming the D3h symmetry of an equilateral triangle subject to rotations and vibrations. Results show a good agreement, with form factors involving the Hoyle state having a slightly larger extension within the hyperspherical approach. Coupled-channel calculations using these form factors are ongoing.

Energy spectra of vibron and cluster models in molecular and nuclear systems

The relation of the algebraic cluster model, i.e., of the vibron model and its extension, to the collective structure, is discussed. In the first section of the paper, we study the energy spectra of vibron model, for diatomic molecule then we derive the rotation-vibration spectrum of \( 2\alpha\), \( 3\alpha\) and \(4\alpha\) configuration in the low-lying spectrum of 8Be, 12C and 16O nuclei. All vibrational and rotational states with ground and excited A, E and F states appear to have been observed, moreover the transitional descriptions of the vibron model and \( \alpha\)-cluster model were considered by using an infinite-dimensional algebraic method based on the affine \(\widehat{SU(1,1)}\) Lie algebra. The calculated energy spectra are compared with experimental data. Applications to the rotation-vibration spectrum for the diatomic molecule and many-body nuclear clusters indicate that there are solvable models and they can be approximated very well using the transitional theory.

12C within the Semimicroscopic Algebraic Cluster Model

The Semimicroscopic Algebraic Cluster Model (SACM) is applied to 12C as a system of three alpha- clusters. The microscopic model space, which observes the Pauli-Exclusion-Principle (PEP), is constructed. It is shown that the 12C nucleus can effectively be treated as a two-cluster system 8Be+alpha. The experimental spectrum is well reproduced. The geometrical mapping is discussed and it is shown that the ground state must correspond to a triangular structure, which is in agreement with other microscopic calculations. The non-zero B(E2; 0_2+ --> 2_1+) transition requires a mixing of SU(3) irreducible representations (irreps) whose consequences are discussed. The Hoyle state turns out to contain large shell excitations. The results are compared to another phenomenological model, which assumes a triangular structure and, using simple symmetry arguments, can reproduce the states observed at low energy. This model does not observe the PEP and one objective of our contribution is to verify the extend of importance of the PEP.

Rotating and vibrating tetrahedrons in heavy nuclei

In the heavy nucleus 208Pb the 3- yrast and 4+ yrast states (Ex = 2615 and 4324 keV) are described by a rotating tetrahedron and eight more states at 4 < Ex < 8 MeV by rotating and vibrating tetrahedrons. The stiffness in three vibration modes is hundred times higher than for corresponding states in 16O by comparing the excitation energies from the prediction by the algebraic cluster model [Nucl. Phys. A 957 (2017) 154] and the splitting of a 2± doublet predicted to be degenerate.