Interacting Vector Boson Model Research Articles

The Evolution of 174Yb, 176Hf, 178W and 180Os Isotones

The characteristics of 174Yb, 176Hf, 178W, and 180Os isotones are determined, followed by the calculation of their energy states. The first energy states 𝐸21+ and the ratios of the second excited energy states to the first excited energy states𝑅4/2 = 𝐸41 +/𝐸21+ provide primary information regarding the characteristics of the nuclei at low levels. To know the properties of high-energy states of nuclei, the behavior of the ground states band (GSB) studied by determine the back-bending curves which may occur and determine the change in the moment of inertia 2𝜗/ℏ2 and the rotation energy of gamma emitted at different states 2𝜔2, the E-GOS curve which study the gamma energy emitted from each state 𝐸𝐵🁡nd the behavior of the successive energy states in comparison with those states close to them 𝑟 (𝐼+2🀩 are drawn and studied. The relationship between GSB and NPB (the negative parity band) which give the odd staggering ∆𝐼 = 1 curve of each nucleus have been studied and determine the probability of change of the property of 174Yb, 176Hf, 178W, 180Os isotones. We used all of the information from the previous study to calculate the energy states of GSB using the appropriate equations for the different limits of the interacting boson model IBM-1. The interacting vector boson model IVBM was also utilized to compute the GSB and NPB of the isotones 174Yb, 176Hf, 178W, 180Os. A modification of IVBM was introduced to improve the results of calculations.

Nuclear Gamma-soft Character in ¹²⁸Ba

We report the properties of gamma soft O(6) of 128Ba isotones with neutron N = 72 using Interacting Vector Boson Model (IVBM), interacting Boson Model (IBM-1), Bohr-Mottelson Model (BM), and Doma-El-Gendy (D-G) relation. The first energy level ( ) and ratio have been investigated which show that 128Ba has gamma-soft character. The curves Eγ/Vs.J of E-GOS of even 128Ba nucleus were compared with the standard curves of vibrational, gamma soft and rotational limits. The staggering factors were studied of available measured data of 128Ba nucleus. The yrast levels of this isotope are calculated by the model of VBMI, IBM-1, BM and D-G and they were compared by measured data. The negative parity band of 128Ba was calculated by IVBM and BM model and compared with experimental values.

Characteristics of Energy States and their Calculation Using IBM-1 and IVBM of Isotones 180HF, 182W, 184Os, 186Pt

The positive ground-state band (GSB) of isotones for N = 108 has been calculated using the Interacting Bosons Model (IBM-1), and Interacting Vector Boson Model (IVBM), while the negative-parity band (NPB) of , has been calculated using the (IVBM). since (IBM-1) have different calculation according to the characteristics of each nucleus, these characteristics were identified using different method, Plotting the relation of the moment of inertia ( ) against the rotational frequency of the emitted photons ( ) during nuclear transition between different energy states were determined, The calculation shows that nucleus do not have any back-bending effect which indicates that its characteristics have not changed, While have a back-bending which indicate that a change in their characteristics. Also using the relation between different excited states energies to their preceding values, one can determine the characteristic of each nucleus which shows a rotational characteristics for nucleus, while have transitional characteristics between the two vibrational and rotational challenges, To get to know the characteristics of the nucleus more accurately the relation of the gamma energy over spin ( ) as a function of the spin (I) E-GOS was drown, the calculation showed pure rotational characteristic for nucleus, While the nuclei have gamma soft O(6) characteristic, When staggering phenomena between (NPB) and (GSB) has been studied, The contour plot show that nucleus did not reach zero value which refer to the constancy of their characteristics.

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.

Dynamical Symmetries in Shell and Collective Models of Nuclear Structure

The dynamical symmetries in the algebraic shell model and the collective Interacting Vector Boson Model (IVBM) realized in terms of fermion and boson creation and annihilation operators are investigated. The obtained analytic eigen-energies in both models are compared with the experimental ones and the strengths of the corresponding terms of the model Hamiltonians are evaluated in both cases. In the algebraic realization of the Pairing plus Quadrupole Shell Model the correlations and transition between the quadrupole and pairing phases—dynamical symmetries are investigated in application to nuclear systems in the first few light shells. In the symplectic extension of the IVBM the spectra of heavy even-even nuclei with transitional between rotational and vibrational character is well reproduced. The algebraic connections between dynamical symmetries of the nuclear collective spectra and the ordering of the low-lying states with fixed angular momentum permits a reasonable and experimentally proved prediction of the position of the 0+ band heads of the collective bands. The models based on dynamical symmetries give an elegant and simple way to describe the complex spectra of nuclei with different shapes.

Properties of some Nuclei with Mass Number A=140

The positive ground-state band (GSB) of isobars was calculated using the Bohr – Mottelson (BM) model, interacting vector boson model (IVBM), and interacting boson model (IBM-1), while the negative-parity band (NPB) of was calculated using the BM model and IVBM. To determine the properties of the ground-state band, the back-bending curve, the E-GOS curve (Eγ/I), the ratio (r(I +2)/I) and the staggering have been plotted as a function of the spin (I). The contour plots show that all nuclei under consideration have complicate properties.

The collective Properties of Even-Even164-174 W Isotopes

Ground states bands (GSB) and negative parity band (NPB) of isotopes have been calculated. Bohr-Mottelson (BM), Interacting boson approximation-1(IBM-1) and Interacting vector boson model (IVBM) were used for this purpose. The principal excited state ( ) and the proportion of the second to the primary excited state provide primary information about the properties of the nucleus. The ratio of the gamma energy, over spin of each state as a function of the angular momentum (E-GOS), has been assessed to decide the ground states property of each nucleus. The ratio of energies of ( ) and ( ), states as a function of the angular momentum , have been attracted to decide numerically the properties of the ground states band of all states and all nuclei. The staggering between octupole band and ground state band is found to display a beat pattern as a function of angular momentum . The methods which were used, showed the transitional U(5)-O(6) properties of , the general properties that U(5)-O(6)-SU(3) of , and the transitional properties O(6)-SU(3) of ; while showed the SU(3) properties.

Investigation of energy staggering effects in Thorium isotopes in framework of interacting vector Boson model

The energies of the positive parity ground state band with Lπ=0+,2+,4+,… and the negative parity octupole band with Lπ=1−,3−,5−,… in normally deformed Thorium (Th) even-even nuclei are described in the framework of the rotational limit of the symplectic interacting vector boson model (IVBM) based on the algebra of Sp(12,R) group. The experimental data best fit to adopt the best model parameters are determined by utilizing a simulated search program in order to achieve minimum root mean square deviation between the experimental and calculated excitation energies. The agreement between experimental and calculated energies is very good. The odd-even ΔL=1 staggering effects between ground and octupole bands are investigating by calculating staggering index depending on the dipole transitions linking the two bands and the quadruple transitions within each band. Also the presence of ΔL=2 energy staggering (commonly called ΔL=4 bifurcation) in ground state band of 232Th has been investigated, by calculating staggering index representing the finite difference approximation to the forth order derivative of the quadruple transition energies at a given spin.

Critical Point of the 152Sm, 154Gd, and 156Dy Isotones

The even—even rare-earth nuclei in U(5)—SU(3) region at neutron number (N) = 90, have been systematically studied using the Bohr—Mottelson Model (BM), Interacting Vector Boson Model (IVBM), and Interacting Boson Model (IBM). The positive ground-state band (GSB) of 152Sm, 154Gd, and 156Dy nuclei has been calculated by using BM, IVBM, and IBM, while the negative-parity band (NPB) of those nuclei are calculated by BM and IVBM only. To determine the intermediate structure, the ratio (r(I + 2)/I) and E-GOS curve (E-Gamma Over Spin) as a function of the spin (I) have been drawn. In the IBM, the calculated reduced B(E2) transition probabilities of the GSB in the 152Sm, 154Gd, and 156Dy nuclei are analyzed and compared to the prediction of vibrational U(5) and rotational SU(3) limits. In the Sm—Dy nuclei with N = 90, the U(5)—SU(3) IBM potential energy surfaces (PES’s) are analyzed and evolve from spherical to deformed shapes with increasing the boson number. The critical phase transition points are identified in the space of model parameters and 152Sm,154Gd, and 156Dy nuclei have been found to be close to critical points.

Description of Energy Levels for 84,86,96,98Sr Isotopes

The positive ground-state band (GSB) of 84,86,96,98Sr has been calculated using the Bohr–Mottelson (BM) model, interacting vector boson model (IVBM), and interacting boson model (IBM-1), while the negative-parity band (NPB) of 84,86,96,98Sr has been calculated using the BM model and IVBM. IBM-1 has been used to calculate the energy of the gamma and beta bands. To determine the properties of the ground-state band, the ratio (r(I +2)/I) and the E-GOScurve (Eγ/I) have been plotted as a function of the spin (I). The potential energy surface (PES) has been calculated using IBM-1. The contour plots of the PESs show that 84,86,96Sr nuclei have vibrational properties and 98Sr has rotational characteristics.

Theoretical description of the deformation properties for 154–164Gd isotopes

The 154–164Gd isotopes in SU(3) region were investigated. For these nuclei, the positive ground-state band of 154–164Gd has been calculated using Bohr–Mottelson Model, Interacting Vector Boson Model and Interacting Boson Model, while the negative-parity band of 154–164Gd has been calculated using Bohr–Mottelson Model and Interacting Vector Boson Model only. To determine the property of the ground-state band, the ratio (r(I+2)/I) and E-GOS curve (Eγ/I) as a function of the spin (I) have been drawn. The reduced transition probabilities B(E2) of these nuclei were calculated. The results were compared with the experimental data and the previous calculations. It is shown that there is a good agreement between the results found and especially with the experimental ones. The potential energy surface (PES) has been calculated within the framework using interacting boson model (IBM-1). The contour plot of the potential energy surfaces shows that the interested nuclei are deformed and have rotational characters.

Investigation of Energy Levels and Electromagnetic Transitions for Yb–Pt Nuclei with N=108 Using IBM, IVBM, and BMM

The interacting boson and vector boson models, as well as the Bohr–Mottelson one, are employed to describe the energy levels and electromagnetic transitions of the 178Yb–186Pt (N = 108) nuclei. For the purpose of determining the evolution of the ground state, both r((I + 2)/I) and E-GOS ratios have been calculated as functions of the spin I. Based on the interacting vector boson model and Bohr–Mottelson model, the negative-parity and GSB bands have been calculated, while the interacting boson model is only employed to calculate GSB, y, and b. The interacting boson model is also used to calculate the reduced transition probabilities B(E2). The obtained findings show a very well agreement with experimentally obtained results elsewhere. We also used the intrinsic coherent state to obtain the potential energy surfaces. These results indicate that these nuclei have a rotational property SU(3), while 186Pt has property O(6).

Microscopic description of the even-even 140-148Ba isotopes using BM, IBM and IVBM

A description of the even-even Ba isotopes for A = 140 to 148 in framework of Bohr-Mottelson model, interacting boson model and interacting vector boson model are carried out. The E-GOS curve (E $\gamma$ /I and the ratio between the energies of the (I + 2) and (I) states (r(I + 2)/I) as a function of the spin (I have been drawn to determine the property of the ground-state band. The positive ground-state band of 140-148Ba has been calculated using Bohr-Mottelson model, interacting boson model and interacting vector boson model, while the negative-parity band of 140-148Ba has been calculated using Bohr-Mottelson model and interacting vector boson model only. The reduced transition probabilities B(E2) of these nuclei were calculated. The parameters of the best fit to the measured data are determined. The potential energy surfaces (PESs) to the IBM Hamiltonian have been obtained using the intrinsic coherent state.

The Evolution of 146Nd– 156Nd Isotope States under the Framework of BM, IBM-1, IVBM and D-G

The properties of the ground, beta and negative parity state bands of 146Nd–156Nd isotopes have been studied. The ratio of gamma energy over spin has been calculated and drawn as a function of spin (I) to show the ground-state evolution for each isotope. The ratio between the excitation energies of successive (I+2) and (I) states as a function of spin (I) has been calculated and drawn to determine the property of the ground-state bands of 146Nd–156Nd isotopes. The odd-even staggering has been drawn to determine the energy difference of the states of the ground-state bands and the negative-parity bands. The energy levels for the ground-state band of 146Nd–156Nd have been calculated using the Bohr-Mottelson Model, the Interacting Boson Model, the Interacting Vector Boson Model and the Doma–El-Gendy Relation. The energy levels of the beta band for the measured experimentally levels of 148Nd, 150Nd and 152Nd have been calculated using the Interacting Boson Model. The energy levels of the negative parity band of the 146Nd, 148Nd, 150Nd, 152Nd and 156Nd isotopes have also been calculated using the Bohr-Mottelson Model and the Interacting Vector Boson Model. The calculated energy levels compared with the experimental data indicate the quality of the fit presented in this paper.

Algebraic solutions for two-level pairing model in IBM-2 and IVBM

In this paper the affine $ SU(1,1)$ approach is applied to numerically solve two pairing problems. A dynamical symmetry limit of the two-fluid interacting boson model-2 (IBM-2) and of the interacting vector boson model (IVBM) defined through the chains $ U_{\pi}(6) \otimes U_{\nu}(6) \supset SO_{\pi}(5)\otimes SO_{\nu}(5) \supset SO_{\pi}(3) \otimes SO_{\nu}(3) \supset SO(3)$ and $ U(6) \supset U_{\pi}(3) \otimes U_{\nu}(3) \supset SO_{\pi}(3) \otimes SO_{\nu}(3) \supset SO(3)$ are introduced, respectively. The quantum phase transition between spherical and $ \gamma$ -soft shapes in medium-mass nuclei is analyzed using $ U(5) \leftrightarrow SO(6)$ transitional nuclei in IBM-2 and one case $ U_{\pi}(3) \otimes U_{\nu}(3) \leftrightarrow SO(6)$ transitional nuclei in IVBM found by using an infinite dimensional algebraic method based on affine $ SU(1,1)$ Lie algebra. The calculated energy spectra, energy ratio and energy staggering of Mo isotopes are compared with experimental results. The interplay between phase transitions and configuration mixing of intruder excitations between spherical vibrations and the $ \gamma$ -soft shapes in Mo isotopes is succinctly addressed and displays fingerprints of the transitional dynamical symmetry E(5).

Nuclear structure of even 120–136Ba under the framework of IBM, IVBM and new method (SEF)

The positive and negative-state bands of even–even 120–136Ba nuclei have been studied. The energy levels for the ground-state band of 120–136Ba have been calculated using Interacting Boson Model (IBM), Interacting Vector Boson Model (IVBM) and a new method called Semi-Empirical Formula (SEF). The negative parity band of even–even 120–136Ba isotopes has been also calculated using (IVBM) and (SEF). The γ and β-bands of even–even 120–136Ba isotopes have been calculated using IBM and SEF. 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 points have been determined for 134,136Ba isotopes.The potential energy surfaces for even–even Ba isotopes are studied using the simplified form of Interacting Boson Model (IBM) with an intrinsic coherent state.

Energy Staggering in Ground Bands of 232Th and 236,238U Nuclei Using the Interacting Vector Boson Model

The ΔI = 2 and ΔI = 4 staggering parameters of transition energies Eγ for normally deformed positive parity ground bands in 232Th and 236,238U nuclei are studied in framework of the symplectic extension of the interacting vector boson model. The model parameters are obtained from the fitting procedure between the calculated excitation energies and the corresponding experimental ones. The staggering parameters represent the finite difference approximations to higher order derivatives dnEγ/dIn of the γ-ray transition energies in a ΔI = 2 and ΔI = 4 bands, which yielding multipoint formulae. The first order derivative (two-point formula) provides us with information about the dynamical moment of inertia. The staggering oscillation for the fourth order derivative (five-point formula) is about 0.5 KeV and is even larger than that in superdeformed bands. The quite similarity in dynamical moments of inertia of the isotopes 236,238U up to high spin states indicate that the phenomenon of identical bands is not restricted to superdeformed bands.

The rotational–vibrational properties of 178−188Os isotopes

The properties of the ground and excited-state bands of 178−188Os nuclei have been studied. The ratio Eγ/I has been calculated as a function of the spin (I) to determine the ground-state evolution. The ratio between the energies of the (I + 2) and (I) states as a function of the spin (I) has been drawn to determine the property of the ground-state band. The energy levels for the ground-state band of 178−188Os have been calculated using Bohr–Mottelson model, Interacting Boson Model, interacting vector boson model and Doma–El-Gendy relation. The negative parity band of 178−188Os has been also calculated using above models. The γ-band of 178−188Os isotopes and β-band states of 178,182,184Os isotopes have been calculated using Interacting Boson Model. The parameters of the best fit to the measured data are determined.

Analysis of Low Lying Collective States in the Symplectic Interacting Vector Bosons Model

Successful description of both positive and negative parity band configurations and their corresponding band-head's structure is obtained within the framework of the algebraic Interacting Vector Boson Model (IVBM). The analysis of the collective structure of the low lying excited states possessing different parity is carried out together with the results obtained from parabolic distributions of the energies in the space of integer and positive classification parameter (supposedly number of bosons) for large amount of even-even nuclei.

Negative parity states in the IVBM

The low-lying spectra including the first few excited positive and negative parity bands of some heavy even-even nuclei from the rare earth and actinide mass regions are investigated within the framework of the symplectic Interacting Vector Boson Model with Sp(12,R) dynamical symmetry group. Symplectic dynamical symmetries allow the change of the number of excitation quanta or phonons building the collective states providing for larger representation spaces and richer subalgebraic structures to incorporate more complex nuclear spectra. The theoretical predictions for the energy levels and transition probabilities between the collective states are compared with experiment and some other collective models incorporating octupole degrees of freedom.