Nuclear Structure Study of Odd-Odd Yttrium Nuclei within Interacting-Boson Fermi-Fermion Model (IBFFM)

Afrah J. Mohaisen, Saad N. Abood

Physics Department, College of Science, Al-Nahrain University, Baghdad, Iraq.

DOI: 10.4236/jamp.2024.126123                

Abstract

Nuclear structure of odd-mass Yttrium nuclei is studied within the framework of the interacting boson-fermion model (IBFM) with parameters based on the optimization method. In addition to the energy surfaces, both single particle energies and occupation probabilities have been used as a microscopic input for building the IBFM Hamiltonian. Only three strength parameters for the particle-boson-core coupling are fitted to experimental spectra. The IBFM Hamiltonian is then used to compute the energy spectra and electromagnetic transition rates for selected odd-mass Sr nuclei. A reasonable agreement with the available experimental data is obtained for the considered odd-mass nuclei.

Keywords

Nuclear Structure, Yttrium Nuclei, Interacting Boson-Fermion Model, Energy Spectra, Electromagnetic Transition Rates

1. Introduction

The neutron deficient nuclei originating from the sector characterized by N and Z approximately equal to 38 to 40 have garnered sustained attention in recent years. Within this domain, the shell model orbitals exhibit discontinuities at significant deformations for nucleon numbers 38 and 40 [1] . Moreover, given that both neutrons and protons populate identical orbitals in this realm, the impacts of such discontinuities are notably pronounced. An “island” of nuclei characterized by a significant prolate deformation in their ground state (with a deformation parameter β₂ of approximately 0.4) has been identified at the proton and neutron numbers of 38 and 40, respectively [2] [3] . Furthermore, owing to the relatively low density of shell model orbitals in these nuclei, even minor variations in the number of nucleons lead to remarkable alterations in their shapes. In a similar manner, a correlation can be observed when the spin is enhanced, as various occurrences like alignment of particles and hindrance in their motion are linked with diverse modifications in shape. Furthermore, an additional area of significance within this vicinity adjacent to the N = Z boundary involves the examination of the impact stemming from the pairing of neutrons and protons, with a particular focus on the rivalry between its components characterized by isospin values of T = 0 and T = 1.

The nuclei within this isolated locality generally exhibit relatively small cross-sectional areas in heavy-ion-induced reactions achievable with stable targets and projectiles. For odd-odd nuclei, the challenge is further compounded by their intricate level structures. Hence, when investigating such nuclei, it becomes crucial to employ the IBFFM model.

In the current study, we provide findings from gamma-ray spectroscopy performed in-beam for the odd-odd nucleus ⁸⁰Y, situated closely to the highly deformed A ≈ 80 nuclei. Previous outcomes related to various rotational bands observed in this nucleus have been documented in Reference [4] . With the introduction of novel measurements, the high-spin level scheme has been fully explored and expanded. Recent supplementary investigations [5] [6] [7] [8] have yielded significant insights into the lower energy segment of the level scheme. The analysis of the detected formations in ⁸⁰Y is deliberated upon utilizing the interacting boson-fermion-fermion model (IBFFM) and the cranked shell model (CSM).

2. Interacting Boson Fermion-Fermion Model

The model of interacting boson-fermion-fermion is an expansion designed for odd-odd nuclei from the model of interacting boson (IBM) [9] (pertaining to even-even nuclei) and the model of interacting boson-fermion (IBFM) [10] (for odd-mass nuclei). The Hamiltonian of IBFFM is expressible as follows [11] .

$H_{IBFFM}^{^}=H_{IBFM}^{^}\left(\pi \right)+H_{IBFM}^{^}\left(\nu \right)+H_{RES}^{^}\left(\pi \nu \right)-H_{IBM}^{^}$ (1)

Here, the acronym H_IBM is utilized to denote the typical IBM Hamiltonian applicable to even-even core systems [9] . The symbol $H_{\pi \left(\nu \right)}$ is employed to characterize the interaction involving fermion single-particles, while $H_{B\pi \left(B\nu \right)}$ is used to elucidate interactions between bosons and fermions. The mathematical expression (1) can be restated as $H=H_{IBFM\left(\nu \right)}+H_{IBFM\left(\pi \right)}-H_{IBM}+H_{\pi \left(\nu \right)}$ , wherein $H_{IBFM\left(\nu ,\pi \right)}$ symbolizes the Hamiltonians in the Interacting Boson Fermion Model (IBFM) for nuclei with odd-mass configurations [11] . The final component accounts for a residual interaction between neutrons and protons.

In this particular model, the ⁸⁰Y nucleus is delineated as a composite system comprising two fermions: one proton fermion and one neutron-hole fermion, interacting with an ⁸⁰Sr core. The scope of this model is restricted to the depiction of states involving two quasiparticles. The initial phase involves the determination of the phenomenological parameters in the first five terms of Equation (1) through the characterization of the ⁸⁰Sr core nucleus and the two neighboring odd-mass nuclei, namely ⁸¹Y and ⁷⁹Sr. The Hamiltonian (Equation (1)) will subsequently undergo diagonalization within a limited space formed by linking proton and neutron states to the core space of bosons. The parameters from Ref. [12] were employed for the Sr core within the IBM framework. In the case of the positive-parity states in ⁸¹Y, the IBFM parameters established for the odd-mass Y isotopes in Ref. [13] were utilized for the analysis. In the present calculations, the eccentric proton was permitted to populate the orbitals 1g_9/2 and 2d_5/2 within the framework of the spherical shell model. Extensive information can be found in the references provided (Refs. [12] [13] ). Subsequently, both states of positive and negative parity have been computed for the nucleus ⁷⁹Sr, following a methodology akin to the one expounded in reference [13] . The odd fermion was permitted to inhabit the spherical orbitals 1f_5/2, 2p_3/2, and 2p_1/2 for the negative-parity states. The boson-fermion interaction terms parameters were A₀ = −0.11, Γ₀ = 0.29, and Γ₀ = 8.20 MeV2, while the values for the negative-parity states were −0.11, 0.055, and 8.20, respectively. These parameters determine the strengths of the boson-fermion interaction’s exchange, quadrupole-quadrupole, and monopole-monopole terms, respectively [9] . The codes PHINT and FBEM [14] , and ODDA and PBEM [15] , for even-even and odd-mass nuclei, respectively, have been used to calculate the energy levels and electromagnetic decay rates. Scholten IBFFM algorithm has been used to conduct the computations for the odd-odd nucleus [16] . The program’s description may be found in Ref. [17] , and for information on how the Hamiltonian (Equation (1)) is diagonalized, we refer to this work. The neutron-proton residual interaction in this code is made up of a spin-spin term and a quadrupole-quadrupole interaction:

$H_{\pi \nu }=V_q{Q}_{\pi }\cdot Q_{\nu }+V_s{\sigma }_{\pi }\cdot {\sigma }_{\nu }$ (2)

The IBFFM code includes the residual interactions spin-spin, multipole-multipole, spin-spin-delta, and tensor interaction, which are represented as [11] :

$H_{RES}^{^\pi \nu }=H_{\delta }^{^}+H_{\sigma \sigma \sigma }^{^}+H_{\sigma \sigma }^{^}+H_T^{^}+H_{MM}^{^}$ (3)

where

$H_{\delta }^{^}=4\pi V_{\delta }\left(r_{\pi }^{\to }-r_{\nu }^{\to }\right)\delta \left({}_{\pi }r-R_0\right)\delta \left(r_{\nu }-R_0\right)$ (4)

is a surface delta-function interaction between the proton and neutron,

$H_{\sigma \sigma \sigma }^{^}=4\pi V_{\sigma \sigma \sigma }\left(r_{\pi }^{\to }-r_{\nu }^{\to }\right)\left({\sigma }_{\pi }^{\to }\cdot {\sigma }_{\nu }^{\to }\right)\delta \left({}_{\pi }r-R_0\right)\delta \left(r_{\nu }-R_0\right)$ (5)

is a surface contact spin-spin interaction,

$H_{\sigma \sigma }^{^}=-\sqrt{3}{V}_{\sigma \sigma }{\sigma }_{\pi }^{\to }\cdot {\sigma }_{\nu }^{\to }$ (6)

is a spin-spin interaction,

$H_T^{^}=V_T\left[3\frac{\left({\sigma }_{\pi }^{\to }{r}_{\pi }^{\to }\right)\cdot \left({\sigma }_{\nu }^{\to }{r}_{\nu }^{\to }\right)}{r_{\pi \nu }^2}-{\sigma }_{\pi }^{\to }\cdot {\sigma }_{\nu }^{\to }\right]$ (7)

is a tensor interaction, and

$H_{MM}^{^}=4\pi \frac{\delta \left(r_{\pi }-r_{\nu }\right)}{r_{\pi }{r}_{\nu }}{\displaystyle \underset{km}{\sum }{V}_k{Y}_{km}^{*}}\left({\theta }_{\pi },{\varphi }_{\pi }\right)Y_{km}\left({\theta }_{\nu },{\varphi }_{\nu }\right)$ (8)

the residual interaction Hamiltonian was calculated as follows:

$\begin{array}{c}{H}_{RES}^{^}=4\pi V_{\delta }\delta \left(r_{\pi }-r_{\nu }\right)\delta \left(r_{\pi }-R_0\right)-\sqrt{3}{V}_{\sigma \sigma }\left({\sigma }_{\pi }\cdot {\sigma }_{\nu }\right)\\ +V_T\left[3\frac{\left({\sigma }_{\pi }^{\to }{r}_{\pi }^{\to }\right)\cdot \left({\sigma }_{\nu }^{\to }{r}_{\nu }^{\to }\right)}{r_{\pi \nu }^2}-{\sigma }_{\pi }^{\to }\cdot {\sigma }_{\nu }^{\to }\right]\end{array}$ (9)

where $R_0=1.2A^{1/3}$ , the parameters $V_{\delta },V_{\sigma \sigma \sigma },V_{\sigma \sigma },V_T$ and $V_R$ are constants, and determined the strengths of the interactions.

3. Results and Discussion

We took into consideration negative-parity configurations in ⁷⁹Sr and positive-parity configurations in ⁸¹Y for the negative-parity ones we were able to calculate the positive-parity states in ⁸⁰Y by permitting positive-parity configurations in both linked odd-mass nuclei.

The other two potential combinations, which consist of configurations with negative parity in ⁸¹Y, were not examined, as it is anticipated that they would contribute to elevated excitation levels. Calculation of B(E2) transition probabilities employed an operator that was deliberately selected to be identical to the one utilized in characterizing the E2 characteristics of the even-even core and the two corresponding odd-mass isotopes. Conversely, the M1 characteristics were assessed using an operator that bore resemblance to the one employed for the odd-mass isotopes [17] .

Having established the IB(F)M parameters for the core and odd-mass nuclei, the computations pertaining to the odd-odd nucleus were uncomplicated, necessitating only the selection of the neutron-proton interaction parameters. The impact of the quadrupole-quadrupole interaction was deemed minimal on the outcomes, ultimately leading to its exclusion. It is comprehensible, given that the majority of quadrupole interactions have already been addressed by the interaction between bosons and fermions. The interaction between spins has implications on the arrangement of the lowest energy states; however, a trade-off was necessary to describe both positive and negative states effectively, requiring very low values for V_s. Consequently, the final outcomes were minimally affected by this parameter. Here, we present the results obtained using V_s = 0.03 MeV.

Figure 1 and Figure 2 depict the outcomes of the computations for the positive-parity states, reaching spin 14, and the negative-parity states, extending up to spin 15, respectively. The derived states have been organized into groupings that are tentatively linked to those observed in experimental settings. The correlation between the computed states and the ones witnessed has been established by examining their decay mechanisms. The computed branching ratios are juxtaposed with the empirical ones in Table 1, revealing that the computations satisfactorily depict the data. Evident from Table 1 is the assignment of the 5⁺ state

Figure 1. Comparison of the experiment [18] for the positive-parity states with the predictions from IBFFM.

Figure 2. Comparison of the experiment [18] for the negative-parity states with the predictions from IBFFM.

Table 1. A comparison is made between the IBFFM and experimental [18] branching ratios in ⁸⁰Y. The initial section of the table illustrates the in-band transitions, while the subsequent part delineates the interrelations among the lowest states of the positive-parity bands. In the case of each particular level, the most robust branch is standardized to 100.

emerging from band 5 as the second computed ( $5_2^{+}$ ) state. The calculated $5_1^{+}$ state could possibly correspond to the level situated immediately below the 6⁺ state of band 3. The three states positioned beneath the 6⁺ state are influenced by the feeble transitions of 0.383, 0.232, and 0.186 MeV, thus complicating their proper characterization, with uncertainties even surrounding their sequential arrangement. However, supposing that the state at 0.675 MeV (which experiences decay from the 1.059 MeV, 6⁺ state through the 383 keV transition) represents the $5_1^{+}$ state, it becomes apparent that the $6_1^{+}$ state primarily decays into this particular state, whereas the $7_1^{+}$ and $6_2^{+}$ states (belonging to band 5) exhibit minimal connections to this state, a phenomenon corroborated by experimental observations.

The IBFFM wave-functions for the positive-parity states are dominated by the g_9/2 proton and neutron orbitals, while in the negative-parity states the dominating orbitals are πg_9/2 and νf_5/2. In some cases, this confirms the Nilsson assignments made on the basis of the two-qp-plus-rotor model. On the other hand, the present calculations cannot provide a strong contribution of the d_5/2 orbital to the wave-functions. The d_5/2 orbital in a spherical shape is positioned at a considerable distance in comparison to the g_9/2 orbital. Consequently, it becomes involved in the wave-functions of odd-A isotopes with contributions that are less than 10%. As a result, we anticipate that the depiction of the well deformed scenario will not be very accurate, particularly when the [431] 1/2 orbital (originating from d_5/2) significantly competes.

Figure 3 illustrates the prediction of the B(M1)/B(E2) ratio for the designated bands. The estimation indicates an overestimation by a factor of 3 - 4 at the lower spins within band 5, and an underestimation for band 6. Notably, the contribution of the [431] 1/2 orbital is deemed significant for both bands. In addition,

Figure 3. IBFFM calculations and experimental data [18] for B(M1)/B(E2) for the bands in ⁸⁰Y. The dashed-dotted line represents the IBFFM forecasts for each band, and the cranked shell model (CSM) is used to compute the other curves for the given combinations.

the B(M1) strengths are also projected to be underestimated for the odd-spin members of the positive-parity band 3, a trend reflected in the predicted branching ratios presented in Table 1.

The limitations of these computations seem to lie in their incapacity to precisely delineate the relative positioning of band-heads and the spacings of the lowest levels within the bands. Enhanced positioning accuracy of negative and positive-parity band-heads would have necessitated adjustments in the parameter Vs. Conversely, while there could have been some enhancement in the spacing of the lowest states, its impact on the computed decay characteristics was minimal. These deficiencies may be ascribed to the insufficiency of the neutron-proton interaction model employed. The significance of this interaction in elucidating intricacies of the observed level structure was underscored in both the two-qp-plus-rotor model [6] and in other IBFFM computations. Particularly in the IBFFM calculations, superior outcomes have been achieved by incorporating a surface-δ interaction [11] . Due to the limitation of the utilized code to only incorporate interaction, the examination of such effects was precluded. Further scrutiny was conducted on our IBFFM computations pertaining to the ${\text{g}}_{\text{9}/\text{2}}\otimes {\text{g}}_{\text{9}/\text{2}}$ band, which was also identified in the more massive odd-odd Y isotopes, ⁸²Y [19] [20] [21] and ⁸⁴Y [22] [23] . Illustrated in Figure 4 is the manner in which this band is delineated in ⁸²Y and ⁸⁴Y, employing calculations akin to those expounded upon earlier for ⁸⁰Y. The fundamental characteristics of these bands are effectively expounded upon until the point of backbending (interaction with other bands, as elaborated below). While not perfectly in phase with the observed data, the computations also exhibit an inversion of the signature splitting at lower angular momenta, and at higher angular momenta (exceeding 9ħ for the lighter isotopes, and 11ħ for ⁸⁴Y), the signature splitting is accurately depicted.

4. Conclusion

High-spin states have been identified in the neutron-deficient nucleus ⁸⁰Y through IBFFM. The unambiguous association of prompt gamma rays with this nucleus has been achieved through their calculations within this model. The existing level arrangement [4] has been expanded by the inclusion of fresh bands and transitions. Numerous band configurations exhibiting rotational properties have been detected up to angular momenta of at least 17, suggesting a nucleus characterized by significant quadrupole deformation. The scrutinized band structures are deliberated based on the interacting boson-fermion-fermion model,

Figure 4. IBFFM predictions for the positive-parity band 5 and its equivalents in ⁸²Y [19] [20] [21] [22] and ⁸⁴Y [22] [23] are compared in an experiment.

thereby facilitating identifications of their principal configuration. The determination of absolute electromagnetic decay transition probabilities (level lifetimes), which can now be achieved using the most extensive Ge detector arrays, would be crucial in enhancing the characterization of this nucleus structure and the progression of its shapes as a function of rotational frequency.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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