Interacting Boson Approximation Model Research Articles

What can we learn from the Interacting Boson Model in the limit of large boson numbers?

Over the years, studies of collective properties of medium and heavy mass nuclei in the framework of the Interacting Boson Approximation (IBA) model have focused on finite boson numbers, corresponding to valence nucleon pairs in specific nuclei. Attention to large boson numbers has been motivated by the study of shape/phase transitions from one limiting symmetry of IBA to another, which become sharper in the large boson number limit, revealing in parallel regularities previously unnoticed, although they survive to a large extent for finite boson numbers as well. Several of these regularities will be discussed. It will be shown that in all of the three limiting symmetries of the IBA [U(5), SU(3), and O(6)], energies of 0+ states grow linearly with their ordinal number. Furthermore, it will be proved that the narrow transition region separating the symmetry triangle of the IBA into a spherical and a deformed region is described quite well by the degeneracies E(0^+_2 ) = E(6^+_1 ), E(0^+_3 ) = E(10^+_1 ), E(0^+_4 ) = E(14^+_1 ), the energy ratio E(6^+_1 )/E(0^+_2 ) turning out to be a simple, empirical, easy-to-measure effective order parameter, distinguishing between first- and second-order transitions. The energies of 0+ states near the point of the first order shape/phase transition between U(5) and SU(3) will be shown to grow as n(n+3), where n is their ordinal number, in agreement with the rule dictated by the relevant critical point symmetries studied in the framework of special solutions of the Bohr Hamiltonian. The underlying dynamical and quasi-dynamical symmetries are also discussed.

Group contractions and conformal maps in nuclear structure models

Group contraction is a procedure in which a symmetry group is reduced into a group of lower symmetry in a certain limiting case. Examples are provided in the large boson mumber limit of the Interacting Boson Approximation (IBA) model by a) the contraction of the SU(3) algebra into the [R5]SO(3) algebra of the rigid rotator, consisting of the angular momentum operators forming SO(3), plus 5 mutually commuting quantities, the quadrupole operators, b) the contraction of the O(6) algebra into the [R5]SO(5) algebra of the ∞-unstable rotator. We show how contrac- tions can be used for constructing symmetry lines in the interior of the symmetry triangle of the IBA model. In mathematics, a conformal map is a function which preserves angles. We show how this procedure can be used in the framework of the Bohr Hamiltonian, leading to a Hamiltonian in a curved space, in which the mass depends on the nuclear deformation Ø, while it remains independent of the collective variable ∞ and the three Euler angles. This Hamiltonian is proved to be equivalent to that obtained using techniques of Supersymmetric Quantum Mechanics.

Multipole Mixing Ratio, (E2/M1), and Electric Monopole Strength, (E0/E2), for γ-Transitions in 192 Pt

The low-lying collective states in 192Pt are investigated in the frame work of the interacting boson approximation model IBA-2. The contour plot of the potential energy surfaces, V(β, γ), shows an O(6) character to the nucleus. B(M1), B(E2), δ(E2/M1) and X(E0/E2) were calculated and reasonable agreement has obtained to the available experimental data. Also, sharp upper bending is observed at I+=10.

γ-softBa146and the role of nonaxial shapes atN≈90

Low-spin states in the neutron-rich, N = 90 nuclide $^{146}$Ba were populated following $\beta$-decay of $^{146}$Cs, with the goal of clarifying the development of deformation in Ba isotopes through delineation of their non-yrast structures. Fission fragments of $^{146}$Cs were extracted from a 1.7-Ci $^{252}$Cf source and mass-selected using the CARIBU facility. Low-energy ions were deposited at the center of a box of thin $\beta$ detectors, surrounded by a high-efficiency HPGe array. The new $^{146}$Ba decay scheme now contains 31 excited levels extending up to ~2.5 MeV excitation energy, double what was previously known. These data are compared to predictions from the Interacting Boson Approximation (IBA) model. It appears that the abrupt shape change found at N = 90 in Sm and Gd is much more gradual in Ba and Ce, due to an enhanced role of the $\gamma$ degree of freedom.

Low-lying (K π = 0+) states of gadolinium isotopes

The sd-interacting boson approximation (sd-IBA) and the df-interacting boson approximation (df-IBA) can be related to each other and the states of the interacting boson approximation model can be identified with the fully symmetric states in the sdf interacting boson approximation model. A systematic study of the sdf-IBA model showed that the constructed Hamiltonian can successfully describe the strong octupole correlations in the deformed nuclei. We showed that the interacting boson approximation may account for many of these K π = 0+ states. It was found that the calculated energy spectra of the gadolinium isotopes agree quite well with the experimental data. The observed B(E2) values were also calculated and compared with the experimental data.

Group contractions and conformal maps in nuclear structure

Group contraction is a procedure in which a symmetry group is reduced into a group of lower symmetry in a certain limiting case. Examples are provided in the large boson number limit of the Interacting Boson Approximation (IBA) model by a) the contraction of the SU(3) algebra into the [R5]SO(3) algebra of the rigid rotator, consisting of the angular momentum operators forming SO(3), plus 5 mutually commuting quantities, the quadrupole operators, b) the contraction of the 0(6) algebra into the [R5]SO(5) algebra of the γ-unstable rotator. We show how contractions can be used for constructing symmetry lines in the interior of the symmetry triangle of the IBA model.In mathematics, a conformal map is a function which preserves angles. We show how this procedure can be used in the framework of the Bohr Hamiltonian, leading to a Hamiltonian in a curved space, in which the mass depends on the nuclear deformation β, while it remains independent of the collective variable 7 and the three Euler angles. This Hamiltonian is proved to be equivalent to that obtained using techniques of Supersymmetric Quantum Mechanics.

Analytic derivation of an approximate SU(3) symmetry inside the symmetry triangle of the interacting boson approximation model

Using the contraction of the SU(3) algebra to the algebra of the rigid rotator in the large boson number limit of the Interacting Boson Approximation (IBA) model, a line is found inside the symmetry triangle of the IBA, along which the SU(3) symmetry is preserved. The line extends from the SU(3) vertex to near the critical line of the first order shape/phase transition separating the spherical and prolate deformed phases, and lies within the Alhassid--Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices amidst chaotic regions. In addition to providing an explanation for the existence of the arc of regularity, the present line represents the first example of an analytically determined approximate symmetry in the interior of the symmetry triangle of the IBA. The method is applicable to algebraic models possessing subalgebras amenable to contraction. This condition is equivalent to algebras in which the equilibrium ground state (and its rotational band) become energetically isolated from intrinsic excitations, as typified by deformed solutions to the IBA for large numbers of valence nucleons.

Structural evolution of theZ=52–62neutron-deficient nuclei in the interacting boson approximation framework

The interacting boson approximation (IBA) is employed in the present article to follow the structural evolution of the neutron-deficient nuclei from the $Z=52--62$ region. The IBA model parameters are determined to reproduce the properties of the low-lying positive parity excitations for a wide range of even-even collective nuclei. The parameters aim to describe simultaneously the existing electromagnetic data (energy levels, transition matrix elements, etc.) and hadronic ones (two-nucleon transfer intensities). It is shown that a simple Hamiltonian with only two terms is not adequate to describe the properties across this region. It is found that the octupole term plays an important role in reproducing the properties of the ${2}_{\ensuremath{\gamma}}^{+}$ and ${0}_{2}^{+}$ states, as well as in the description of the two-neutron transfer intensities patterns. A mapping of these parameters in the IBA symmetry triangle allows the comparison of representative trajectories for different isotopic chains.

SU(3) Quasidynamical Symmetry Underlying the Alhassid-Whelan Arc of Regularity

The first example of an empirically manifested quasidynamical symmetry trajectory in the interior of the symmetry triangle of the interacting boson approximation model is identified for large boson numbers. Along this curve, extending from SU(3) to near the critical line of the first order phase transition, spectra exhibit nearly the same degeneracies that characterize the low energy levels of SU(3). This trajectory also lies close to the Alhassid-Whelan arc of regularity, the unique interior region of regular behavior connecting the SU(3) and U(5) vertices, thus offering a possible symmetry-based interpretation of that narrow zone of regularity amidst regions of more chaotic spectra.

Structure investigation with the (p,t) reaction onBa132,134nuclei

The low-lying excited states in $^{132,134}\mathrm{Ba}$ isotopes have been studied with high-resolution ($p,t$) reactions. The experiments were performed at the Munich Q3D spectrograph with a 25-MeV proton beam and the 1.5-m-long focal plane detector. The high-resolution triton spectra allowed the observation of levels up to $~4$ MeV. The experimental results revealed 75 excited states in $^{134}\mathrm{Ba}$ and 79 in $^{132}\mathrm{Ba}$, many of them observed for the first time. The measured angular distributions compared with distorted-wave Born approximation calculations allowed spin assignments for these levels in most cases. The systematics of the monopole and quadrupole two-neutron transfer strengths is compared with the prediction of the interacting boson approximation model. The results indicate a transitional structure in $^{132}\mathrm{Ba}$ and $^{134}\mathrm{Ba}$ and contribute additional evidence in favor of a description between the U(5) and O(6) symmetries of the model.

Simple Empirical Order Parameter for a First-Order Quantum Phase Transition in Atomic Nuclei

A simple, empirical, easy-to-measure effective order parameter of a first-order phase transition in atomic nuclei is presented, namely, the ratio of the energies of the first excited 6+ and 0+ states, distinguishing between first- and second-order transitions, and taking on a special value in the critical region, as data in Nd-Dy show. In the large NB limit of the interacting boson approximation model, a repeating degeneracy between alternate yrast and successive 0+ states is found in the critical region around the line of a first-order phase transition, pointing to a possible underlying symmetry.

192Pt: A piece in the Pt intruder state puzzle

A common interpretation of the light Pt isotopes involves the coexistence and mixing with proton intruder states from above the Z = 82 shell gap. An alternative description of the Pt isotopes using a simple, single configuration, the two-parameter interacting boson approximation model Hamiltonian, is discussed. To test these predictions, experiments on 192Pt were performed. New coincidence data found no support for several low-energy, low-spin states previously proposed in β decay. Results from an (α, 2n) experiment identified states of the γ band up to spin 9+. Results of these experiments are discussed in terms of IBA calculations.

Octupole collectivity in the Sm isotopes

Microscopic models suggest the occurrence of strong octupole correlations in nuclei with $N\ensuremath{\approx}88$. To examine the signatures of octupole correlations in this region, the spdf interacting boson approximation model is applied to Sm isotopes with $N=86\text{\ensuremath{-}}92$. The effects of including multiple negative-parity bosons in this basis are compared with more standard one negative-parity boson calculations and are analyzed in terms of signatures for strong octupole correlations. It is found that multiple negative-parity bosons are needed to describe properties at medium spin. Bands with strong octupole correlations (multiple negative-parity bosons) become yrast at medium spin in $^{148,150}\mathrm{Sm}$. This region shares some similarities with the light actinides, where strong octupole correlations were also found at medium spin.

Analytic interpretation of universal anharmonic vibrator behavior in the interacting boson approximation model.

The recently discovered nearly universal anharmonic vibrator behavior (with {ital constant} anharmonicity) of nuclei with {ital E}(4{sub 1}{sup +})/{ital E}(2{sub 1}{sup +}) between 2.0 and 3.15 has been shown to be a natural and nearly automatic outcome of numerical interacting boson approximation (IBA) calculations. Here, we present an approximate analytic derivation and discussion of this based on the idea of the {ital Q} phonon.

Description of the Even Tin Isotopes in a Mixing of Shell Model and IBA Model

124 Sn isotopes are studied in terms of a mixing of shell model and interacting boson approximation model. In the calculation, the basis states consist of a pure boson configuration and Ns-1 bosons plus a fermion pair configuration, where N8 is the valence boson number outside the closed shell. The fermions are allowed to excite to the s 112, d312 , d512, [/712 and hw, single particle orbitals. The interaction Hamiltonian contains three parts: pure boson Hamiltonian, pure fermion Hamiltonian and boson-fermion mixing Hamiltonian. The calcu­ lated energy levels are compared with the observed data. The wavefunctions are used to calculate B(£2) values. It is found that the calculated energy spectra and B(£2) values agree reasonably well with the experimental values.

Description of the Even Tin Isotopes in a Mixing of Shell Model and IBA Model

The positive parity energy levels of even 108-124Sn isotopes are studied in terms of a mixing of shell model and interacting boson approximation model. In the calculation, the basis states consist of a pure boson configuration and NB - 1 bosons plus a fermion pair configuration, where NB is the valence boson number outside the closed shell. The fermions are allowed to excite to the s1/2, d3/2, d5/2, g7/2 and h11/2 single particle orbitals. The interaction Hamiltonian contains three parts: pure boson Hamiltonian, pure fermion Hamiltonian and boson-fermion mixing Hamiltonian. The calculated energy levels are compared with the observed data. The wavefunctions are used to calculate B(E2) values. It is found that the calculated energy spectra and B(E2) values agree reasonably well with the experimental values.

The Interacting Boson Approximation Model of the Deformed Nuclei158–162Dy,182–186W, and158Gd

Abstract The low-lying energy levels of the deformed 158–162Dy, 182–186W and 158Gd nuclei were obtained by using the IBA-1 model. In these nuclei, it is necessary to have the symmetry breaking terms for the energy levels calculated according to the SU(3) limit to be in better agreement with the experimental data. Therefore, a small symmetry breaking term was added to SU(3) Hamiltonian in the direction of O(6) limit. It was seen that these newly obtained energy levels were in good agreement with the experimental data. For these nuclei, B(E2) values were also calculated and compared with the experimental data and the previous studies. It was concluded that the agreement was remarkably good.

Interacting boson approximation studies of the negative parity states of 172-180Os isotopes.

The negative parity state of [sup 172[minus]180]Os are studied in the framework of the interacting boson approximation model. The negative parity states are formed by including one [ital f] boson or allowing one boson to break into two fermions which are placed in orbits with opposite parity. The calculated level energies and [ital B]([ital E]2)'s are in reasonable agreements with experiments. From the comparison of intrinsic structure it suggests that the relative energy of the quasiparticles may be the most important factor in the calculations.

Robust predictions of the interacting boson approximation model.

The Interacting Boson Approximation Model (IBA) has existed for nearly twenty years and has now entered the lexicon of standard nuclear models used to interpret data on low-energy nuclear structure. While most recognized for its symmetries and algebraic structure, the model has a number of other less-well-known but equally intrinsic properties which gave unavoidable, parameter-free predictions. Since these predictions cannot be altered and since they concern central aspects of low-energy nuclear collective structure, they provide a set of basic tests of the inherent framework of the model. This paper outlines these ``robust'' predictions and compares them with the data.

Negative parity states and octupole collectivity of even Ge isotopes.

The negative parity energy levels of the even-even Ge isotopes with mass number between 64 and 76 are studied systematically by enlarging the model space of the interacting boson approximation model to include both collective and noncollective basis states. The basis states consist of ${\mathit{N}}_{\mathit{B}}$-1 sd-boson plus a f-boson configuration and ${\mathit{N}}_{\mathit{B}}$-1 sd boson plus a fermion pair configuration. The fermions are allowed to occupy the ${\mathit{f}}_{5/2}$ and ${\mathit{g}}_{9/2}$ single-particle orbitals, respectively. It was found that the negative parity energy levels of $^{64\mathrm{\ensuremath{-}}76}\mathrm{Ge}$ nuclei can be described reasonably well. The intensities of the collective configuration in ${3}^{\mathrm{\ensuremath{-}}}$ states increase when going from nucleus $^{64}\mathrm{Ge}$ to nucleus $^{72}\mathrm{Ge}$ and decrease from nucleus $^{72}\mathrm{Ge}$ to nucleus $^{74}\mathrm{Ge}$. The B(E3;${3}_{1}^{\mathrm{\ensuremath{-}}}$\ensuremath{\rightarrow}${0}_{1}^{+}$) values are calculated and compared with the available observed data.