Interacting Boson Model Hamiltonian Research Articles

Octupole correlations in collective excitations of neutron-rich N≈56 nuclei

Octupole correlations in the low-energy collective states of neutron-rich nuclei with the neutron number $N\approx56$ are studied within the interacting boson model (IBM) that is based on the nuclear density functional theory. The constrained self-consistent mean-field (SCMF) calculations using a universal energy density functional and a pairing interaction provide the potential energy surfaces in terms of the axially-symmetric quadrupole and octupole deformations for the even-even nuclei $^{86-94}$Se, $^{88-96}$Kr, $^{90-98}$Sr, $^{92-100}$Zr, and $^{94-102}$Mo. The SCMF energy surface is then mapped onto the energy expectation value of a version of the IBM in the boson condensate state, which consists of the neutron and proton monopole $s$, quadrupole $d$, and octupole $f$ bosons. This procedure determines the strength parameters of the IBM Hamiltonian, which is used to compute relevant spectroscopic properties. At the SCMF level, no octupole deformed ground state is obtained, while the energy surface is generally soft in the octupole deformation at $N\approx56$. The predicted negative-parity yrast bands with the bandhead state $3^-_1$ weakly depend on $N$ and become lowest in energy at $N=56$ in each of the considered isotopic chains. The model further predicts finite electric octupole transition rates between the lowest negative- and the positive-parity ground-state bands.

Two-neutrino double- β decay in the mapped interacting boson model

A calculation of two-neutrino double-$\beta$ ($2\nu\beta\beta$) decay matrix elements within the interacting boson model (IBM) that is based on the nuclear density functional theory is presented. The constrained self-consistent mean-field (SCMF) calculation using a universal energy density functional (EDF) and a pairing interaction provides potential energy surfaces with triaxial quadrupole degrees of freedom for even-even nuclei corresponding to the initial and final states of the $2\nu\beta\beta$ decays of interest. The SCMF energy surface is then mapped onto the bosonic one, and this procedure determines the IBM Hamiltonian for the even-even nuclei. The same SCMF calculation provides the essential ingredients of the interacting boson fermion-fermion model (IBFFM) for the intermediate odd-odd nuclei and the Gamow-Teller and Fermi transition operators. The EDF-based IBM and IBFFM provide a simultaneous description of excitation spectra and electromagnetic transition rates for each nucleus, and single-$\beta$ and $\beta\beta$ decay properties. The calculated $2\nu\beta\beta$-decay nuclear matrix elements are compared with experiment and with those from earlier theoretical calculations.

Linking Partial Dynamical Symmetry to Nuclear Energy Density Functional

We use self-consistent mean-field methods in combination with the interacting boson model (IBM) of nuclei, to establish a linkage between universal energy density functionals (EDFs) and partial dynamical symmetry (PDS). An application to $^{168}$Er shows that IBM Hamiltonians derived microscopically from known nonrelativistic and relativistic EDFs in this region, conform with SU(3)-PDS.

Evolution of octupole deformation and collectivity in neutron-rich lanthanides

The onset of octupole deformation and its impact on related spectroscopic properties is studied in even-even neutron-rich lanthanide isotopes Xe, Ba, Ce, and Nd with neutron number $86\leqslant N\leqslant 94$. Microscopic input comes from the Hartree-Fock-Bogoliubov approximation with constrains on the axially symmetric quadrupole and octupole operators using the Gogny-D1M interaction. At the mean-field level, reflection asymmetric ground states are predicted for isotopes with neutron number around $N=88$. Spectroscopic properties are studied by diagonalizing the interacting boson model Hamiltonian, with the parameters obtained via the mapping of the mean-field potential energy surface onto the expectation value of the Hamiltonian in the $s$, $d$, and $f$ boson condensate state. The results obtained for low-energy positive- and negative-parity excitation spectra as well as the electric dipole, quadrupole, and octupole transition probabilities indicate the onset of pronounced octupolarity for $Z\approx 56$ and $N\approx 88$ nuclei.

Octupole correlations in light actinides from the interacting boson model based on the Gogny energy density functional

The quadrupole-octupole coupling and the related spectroscopic properties have been studied for the even-even light actinides $^{218-238}$Ra and $^{220-240}$Th. The Hartree-Fock-Bogoliubov approximation, based on the Gogny-D1M energy density functional, has been employed as a microscopic input, i.e., to obtain (axially symmetric) mean-field potential energy surfaces as functions of the quadrupole and octupole deformation parameters. The mean-field potential energy surfaces have been mapped onto the corresponding bosonic potential energy surfaces using the expectation value of the $sdf$ Interacting Boson Model (IBM) Hamiltonian in the boson condensate state. The strength parameters of the $sdf$-IBM Hamiltonian have been determined via this mapping procedure. The diagonalization of the mapped IBM Hamiltonian provides energies for positive- and negative-parity states as well as wave functions which are employed to obtain transitional strengths. The results of the calculations compare well with available data from Coulomb excitation experiments and point towards a pronounced octupole collectivity around $^{224}$Ra and $^{226}$Th.

Pairing vibrations in the interacting boson model based on density functional theory

We propose a method to incorporate the coupling between shape and pairing collective degrees of freedom in the framework of the interacting boson model (IBM), based on the nuclear density functional theory. To account for pairing vibrations, a boson-number non-conserving IBM Hamiltonian is introduced. The Hamiltonian is constructed by using solutions of self-consistent mean-field calculations based on a universal energy density functional and pairing force, with constraints on the axially-symmetric quadrupole and pairing intrinsic deformations. By mapping the resulting quadrupole-pairing potential energy surface onto the expectation value of the bosonic Hamiltonian in the boson condensate state, the strength parameters of the boson Hamiltonian are determined. An illustrative calculation is performed for $^{122}$Xe, and the method is further explored in a more systematic study of rare-earth $N=92$ isotones. The inclusion of the dynamical pairing degree of freedom significantly lowers the energies of bands based on excited $0^+$ states. The results are in quantitative agreement with spectroscopic data, and are consistent with those obtained using the collective Hamiltonian approach.

Nuclear Structure and Energy Levels of 158Er, 160Yb and 162Hf Isotones

The properties of even-even158Er, 160Yb and 162Hf isotones are studied and its energy states calculated. To identify the properties of each isotone, the values of the first excited states and the ratio of the second to the first excited states were adopted. The phenomenon of back-bending, the E-GOS curve and the odd-even staggering were studied. Examining the algebraic structure of the Bohr model (BM) clarified the relationship with the interacting vector boson model (IVBM) and the interacting boson model (IBM) which it has related solvable limits and corresponding dynamical symmetries. BM, IVBM and IBM were used to calculate the energy states for each isotone and compared with the experimental data. The results showed that the BM and IVBM are more comfort than the calculation of IBM-1. The contour plots of the potential energy surface (PES) to the IBM Hamiltonian for Er–Hf with N = 90 have been obtained using the intrinsic coherent state and evolve from deformed shapes to γ- unstable with decreasing the boson number.

β decay of even- A nuclei within the interacting boson model with input based on nuclear density functional theory

We compute the $\beta$-decay $ft$-values within the frameworks of the energy density functional (EDF) and the interacting boson model (IBM). Based on the constrained mean-field calculation with the Gogny-D1M EDF, the IBM Hamiltonian for an even-even nucleus and essential ingredients of the interacting boson-fermion-fermion model (IBFFM) for describing the neighboring odd-odd nucleus are determined in a microscopic way. Only the boson-fermion and residual neutron-proton interaction strengths are determined empirically. The Gamow-Teller (GT) and Fermi (F) transition rates needed to compute the $\beta$-decay $ft$-values are obtained without any additional parameter or quenching of the $g_A$ factor. The observed $\ft$ values for the $\beta^+$ decays of the even-even Ba into odd-odd Cs nuclei, and of the odd-odd Cs to the even-even Xe nuclei, with mass $A\approx 130$ are reasonably well described. The predicted GT and F transition rates represent a sensitive test of the quality of the IBM and IBFFM wave functions.

Determine the 134–140Nd isotopes identity using IBM and NEF

The property of 134−140Neodymium nuclei have been studied in framework Interacting Boson Model (IBM) and a new method called New Empirical Formula (NEF). The energy positive parity bands of 134−140Nd have been calculated using (IBM) and (NEF) while the negative parity bands of 134−140Nd have been calculated using (NEF) only. The E-GOS curve (Eγ/I) as a function of the spin (I) has been drawn to determine the property of the positive parity yrast band. The parameters of the best fit to the measured data are determined. The reduced transition probabilities B(E2) of these nuclei was calculated. The critical point has been determined for 140Nd isotope. The potential energy surfaces (PESs) to the IBM Hamiltonian have been obtained using the intrinsic coherent state.

Structure of krypton isotopes within the interacting boson model derived from the Gogny energy density functional

The evolution and coexistence of the nuclear shapes as well as the corresponding low-lying collective states and electromagnetic transition rates are investigated along the Krypton isotopic chain within the framework of the interacting boson model (IBM). The IBM Hamiltonian is determined through mean-field calculations based on the several parametrizations of the Gogny energy density functional and the relativistic mean-field Lagrangian. The mean-field energy surfaces, as functions of the axial $\beta$ and triaxial $\gamma$ quadrupole deformations, are mapped onto the expectation value of the interacting-boson Hamiltonian that explicitly includes the particle-hole excitations. The resulting boson Hamiltonian is then used to compute low-energy excitation spectra as well as E2 and E0 transition probabilities for $^{70-100}$Kr. Our results point to a number of examples of the prolate-oblate shape transitions and coexistence both on the neutron-deficient and neutron-rich sides. A reasonable agreement with the available experimental data is obtained for the considered nuclear properties.

Structural evolution in germanium and selenium nuclei within the mapped interacting boson model based on the Gogny energy density functional

The shape transitions and shape coexistence in the Ge and Se isotopes are studied within the interacting boson model (IBM) with the microscopic input from the self-consistent mean-field calculation based on the Gogny-D1M energy density functional. The mean-field energy surface as a function of the quadrupole shape variables $\beta$ and $\gamma$, obtained from the constrained Hartree-Fock-Bogoliubov method, is mapped onto the expectation value of the IBM Hamiltonian with configuration mixing in the boson condensate state. The resultant Hamiltonian is used to compute excitation energies and electromagnetic properties of the selected nuclei $^{66-94}$Ge and $^{68-96}$Se. Our calculation suggests that many nuclei exhibit $\gamma$ softness. Coexistence between prolate and oblate, as well as between spherical and $\gamma$-soft, shapes is also observed. The method provides a reasonable description of the observed systematics of the excitation energy of the low-lying energy levels and transition strengths for nuclei below the neutron shell closure $N=50$, and provides predictions on the spectroscopy of neutron-rich Ge and Se isotopes with $52\leq N\leq 62$, where data are scarce or not available.

Structural evolution inA≈100nuclei within the mapped interacting boson model based on the Gogny energy density functional

The structure of even-even neutron-rich Ru, Mo, Zr and Sr nuclei in the $A\approx 100$ mass region is studied within the interacting boson model (IBM) with microscopic input from the self-consistent mean-field approximation based on the Gogny-D1M energy density functional. The deformation energy surface in the quadrupole deformation space $(\beta,\gamma)$, computed within the constrained Hartree-Fock-Bogoliubov framework, is mapped onto the expectation value of the appropriately chosen IBM Hamiltonian with configuration mixing in the boson condensate state. The mapped IBM Hamiltonian is used to study the spectroscopic properties of $^{98-114}$Ru, $^{96-112}$Mo, $^{94-110}$Zr and $^{92-108}$Sr. Several cases of $\gamma$-soft behavior are predicted in Ru and Mo nuclei while a pronounced coexistence between strongly-prolate and weakly-oblate deformed shapes is found for Zr and Sr nuclei. The method describes well the evolution of experimental yrast and non-yrast states as well as selected $B$(E2) transition probabilities.

Phase diagram of the two-fluid Lipkin model: A “butterfly” catastrophe

Background: In the last few decades quantum phase transitions have been of great interest in Nuclear Physics. In this context, two-fluid algebraic models are ideal systems to study how the concept of quantum phase transition evolves when moving into more complex systems, but the number of publications along this line has been scarce up to now. Purpose: We intend to determine the phase diagram of a two-fluid Lipkin model, that resembles the nuclear proton-neutron interacting boson model Hamiltonian, using both numerical results and analytic tools, i.e., catastrophe theory, and to compare the mean-field results with exact diagonalizations for large systems. Method: The mean-field energy surface of a consistent-Q-like two-fluid Lipkin Hamiltonian is studied and compared with exact results coming from a direct diagonalization. The mean-field results are analyzed using the framework of catastrophe theory. Results: The phase diagram of the model is obtained and the order of the different phase-transition lines and surfaces is determined using a catastrophe theory analysis. Conclusions: There are two first order surfaces in the phase diagram, one separating the spherical and the deformed shapes, while the other separates two different deformed phases. A second order line, where the later surfaces merge, is found. This line finishes in a transition point with a divergence in the second order derivative of the energy that corresponds to a tricritical point in the language of the Ginzburg-Landau theory for phase transitions.

Spectroscopy of quadrupole and octupole states in rare-earth nuclei from a Gogny force

Collective quadrupole and octupole states are described in a series of Sm and Gd isotopes within the framework of the interacting boson model (IBM), whose Hamiltonian parameters are deduced from mean-field calculations with the Gogny energy density functional. The link between both frameworks is the (${\ensuremath{\beta}}_{2}{\ensuremath{\beta}}_{3}$) potential energy surface computed within the Hartree-Fock-Bogoliubov framework in the case of the Gogny force. The diagonalization of the IBM Hamiltonian provides excitation energies and transition strengths of an assorted set of states including both positive- and negative-parity states. The resultant spectroscopic properties are compared with the available experimental data and also with the results of the configuration mixing calculations with the Gogny force within the generator coordinate method (GCM). The structure of excited ${0}^{+}$ states and its connection with double-octupole phonons is also addressed. The model is shown to describe the empirical trend of the low-energy quadrupole and octupole collective structure fairly well and turns out to be consistent with GCM results obtained with the Gogny force.

Nuclear Shape Transition Between Spherical U(5) and γ-Unstable O(6) Limits of the Interacting Boson Model

The interacting boson model (IBM) with intrinsic coherent state (characterized by and ) is used to describe the nuclear second order shape phase transition (denoted E(5)) between the spherical oscillator U(5) and the -soft rotor O(6) structural limits. The potential energy surfaces (PES's) have been derived and the critical points of the phase transition have been determined . The model is examined for the spectra of even-even neutron rich xenon isotopic chain. The best adopted parameters in the IBM Hamiltonian for each nucleus have been adjusted to reproduce as closely as possible the experimental selected numbers of excitation energies of the yrast band, by using computer simulated search program.Using the best fitted parameters , the  energy ratios for the levels are calculated and compared to those of the O(6) and U(5) dynamical symmetry limits.122Xe and 132Xe are considered as examples for the two O(6) and U(5) dynamical symmetry limits

Evolution of shapes in even–even nuclei using the standard interacting boson model

The sd-version of the interacting boson model (IBM) is used to establish the shape phase transitional structure. A simplified Hamiltonian is used which is intermediate between the three dynamical symmetries of U(6), namely the spherical U(5), the prolate and oblate deformed SU(3) and the \(\gamma \)-unstable O(6) limits. The potential energy surfaces (PESs) to the IBM Hamiltonian have been obtained using the intrinsic state formalism which introduces the shape variables \(\beta \) and \(\gamma \). The Gadolinium (Gd) and Ruthenium (Ru) isotopic chains have been taken as examples in illustrating the U(5)–SU(3) and U(5)–O(6) shape phase transitions, respectively. We used the standard \(\chi ^2\) test to get the IBM Hamiltonian parameters. The fit is performed by minimizing the \(\chi ^2\) function for some selected experimental low-lying energy levels, the two neutron separation energies and B(E2) transition rates.

Partial dynamical symmetry and odd-even staggering in deformed nuclei

Partial dynamical symmetry (PDS) is shown to be relevant for describing the odd-even staggering in the $\gamma$-band of $^{156}$Gd while retaining solvability and good SU(3) symmetry for the ground and $\beta$ bands. Several classes of interacting boson model Hamiltonians with SU(3) PDS are surveyed.

Microscopic description of octupole shape-phase transitions in light actinide and rare-earth nuclei

A systematic analysis of low-lying quadrupole and octupole collective states is presented, based on the microscopic energy density functional framework. By mapping the deformation constrained self-consistent axially symmetric mean-field energy surfaces onto the equivalent Hamiltonian of the $sdf$ interacting boson model (IBM), that is, onto the energy expectation value in the boson condensate state, the Hamiltonian parameters are determined. The study is based on the global relativistic energy density functional DD-PC1. The resulting IBM Hamiltonian is used to calculate excitation spectra and transition rates for the positive- and negative-parity collective states in four isotopic chains characteristic for two regions of octupole deformation and collectivity: Th, Ra, Sm and Ba. Consistent with the empirical trend, the microscopic calculation based on the systematics of $\beta_{2}$-$\beta_{3}$ energy maps, the resulting low-lying negative-parity bands and transition rates show evidence of a shape transition between stable octupole deformation and octupole vibrations characteristic for $\beta_{3}$-soft potentials.

Test of the coherent state approach in the axially deformed region

Abstract Coherent State Approach on the Interacting Boson Model (IBM) is tested between the axially deformed and γ -soft transitional regions. Excitation energies of the bands are obtained for the simple IBM Hamiltonian written for this region. Deformation parameter which minimizes the energy of the state is found in a closed form as a function of boson number for each band. Then energy of state within the band is defined by using the moment of inertia obtained from the solution of cranking problem. Matrix elements of quadrupole moment operator and reduced electric quadrupole transitions are given. Then the Coherent State Approach is tested in the case of some selected Hf isotopes.

Interacting boson model with energy density functionals

A brief overview of the recent advancement in the microscopic study of the interacting boson model (IBM) is given. A new nucleon-boson mapping method has been proposed recently, that derives the IBM Hamiltonian based on the self-consistent mean-field model with the microscopic energy density functional (EDF). The mean-field total energy surface computed with a given EDF is mapped onto the analogous energy expectation value in the boson condensate, thereby the energy spectrum and the transition rates are generated. Since the EDF framework allows an accurate global description of nuclear intrinsic properties, the IBM is derived in a unified way, basically for any situation of low-energy quadrupole collective states of nuclei. The basic notion of the new mapping technique is sketched.