Two or more nuclei that have the same number of protons and different numbers of neutrons are referred to as isotopes. The model space for the IBM model is restricted to a smaller space. It gives a mathematical description of unpaired by coupling them in pairs, L = 0 or L = 2. The Hamiltonian for IBM1 is given in references (Iachello and Arima 1987; Casten and Warner 1988).

$H={\epsilon }_s(s^{†}\cdot \tilde{s})+{\epsilon }_d(d^{†}\cdot \tilde{d})$

$+\sum _{L=0,2,4}\frac{\sqrt{2L+1}}{2}{C}_L[{[{(d^{†}\times d^{†}]}^{(L)}\times [\tilde{d}\times \tilde{d}{]}^{(L)}]}^{(0)}$

$+\frac{1}{\sqrt{2}}{v}_2{[{[d^{†}\times d^{†}]}^{(2)}\times [\tilde{d}\times \tilde{s}{]}^{(2)}+{[d^{†}\times s^{†}]}^{(2)}\times [\tilde{d}\times \tilde{d}{]}^{(2)}]}^{(0)}$

$+\frac{1}{2}{v}_0{[{[d^{†}\times d^{†}]}^{(0)}\times [\tilde{s}\times \tilde{s}{]}^{(0)}+{[s^{†}\times s^{†}]}^{(0)}\times [\tilde{d}\times \tilde{d}{]}^{(0)}]}^{(0)}$

$+\frac{1}{2}{u}_0{[{[s^{†}\times s^{†}]}^{(0)}\times [\tilde{s}\times \tilde{s}{]}^{(0)}]}^{(0)}+u_2{[{[d^{†}\times s^{†}]}^{(2)}\times [\tilde{d}\times \tilde{s}{]}^{(2)}]}^{(0)}$ (1)

The Hamiltonian of the IBM1 model is made up of nine terms. Two of them correspond to one body contributions (L = 0 and L = 2), and the parameters ε_s and ε_d are the energies of s- and d- bosons, and the others are two body interactions for which the coefficients (C0, C1, C4, v0, v2, u0, u2) are written. The N, the number of bosons, is a conserved quantity. In total, the Hamiltonian for the IBM1 is in Eq. 1 (Arima and Iachello 1976, 1978):

$\hat{H}=\epsilon {\hat{n}}_d+a_0\hat{P}\cdot \hat{P}+a_1\hat{L}\cdot \hat{L}+a_2\hat{Q}\cdot \hat{Q}+a_3{\hat{T}}_3\cdot {\hat{T}}_3+a_4{\hat{T}}_4\cdot {\hat{T}}_4$ (2)

Boson energy ε = ε_d – ε_s, and the operators are as follows:

${\hat{n}}_d=d^{†}\cdot \tilde{d}$

$\hat{P}=0.5[(\tilde{d}\cdot \tilde{d})-(\tilde{s}\cdot \tilde{s})]$

$\hat{L}=\sqrt{10}{[d^{†}\times \tilde{d}]}^{(1)}$

$\hat{Q}={[d^{†}\times \tilde{s}+s^{†}\times \tilde{d}]}^{(2)}+\chi {[d^{†}\times \tilde{d}]}^{(2)}$

${\hat{T}}_r={[d^{†}\times \tilde{d}]}^{(r)}$ (3)

The four operators are indicated by the symbols n̂_d, P̂, L̂, Q̂ where, n̂_d (total amount of d – bosons) P̂ (pairing), L̂ (angular momentum), and Q̂ (quadruple) operator. The T̂_r denotes hexadecapole and octupole operator for r = 4 and 3. The representation χ denotes to the quadrupole creation limits 0 and ±√7/2 (Iachello 1980). The symbol a₀, a₁, a₂, a₃ and a₄ are limits for the designate the P̂, L̂, Q̂, and T̂_r connections between the bosons.

The IBM1 makes three sorts of active symmetry and eigenvalues are given by equation 4 (Iachello 1980).

$$\begin{gathered} E(n_d,v,L)=\epsilon n_d+\frac{a_1}{12}{n}_d(n_d+4)+(\frac{a_3}{7}-\frac{a_1}{10}-\frac{3a_4}{70})v(v+3)+\frac{1}{14}(a_4-a_3)L(L+1) \\ ,\mathrm{U}(5) \end{gathered}$$

$E(\lambda ,\mu ,L)=\frac{a_2}{2}({\lambda }^2+{\mu }^2+\lambda \mu +3(\lambda +\mu ))+(a_1-\frac{2a_2}{8})L(L+1),$

$E(\sigma ,\tau ,L)=\frac{a_0}{4}(N-\sigma )(N+\sigma +4)+\frac{a_3}{2}\tau (\tau +3)+(a_1-\frac{a_3}{10})(L(L+1)$ (4)

The symbol ε, a₀, and a₂ indicate to the limits for U(5), O(6), and SU(3), respectively. The Hamiltonian (Arima and Iachello 1976; Iachello and Arima 1987) is familiar for the designs that break rendering to equations 5:

$\hat{H}=a_0\hat{P}\cdot \hat{P}+a_1\hat{L}\cdot \hat{L}+a_2\hat{Q}\cdot \hat{Q}$ (5)

IBM2 (Arima et al. 1977; Otsuka et al. 1978) is a variant of the interacting-boson model that considers the nucleus to consist of protons and neutrons, provides for the differentiation of the two kinds of bosons, and counts their number for which the nucleons are outside the major closed shells. Each boson has a nonzero value of the angular momentum, L = 0 or L = 2, and therefore can exist in one of two states, the s-boson (L = 0) or the d-boson (L = 2). In the case of nuclei whose protons number and neutrons number is even, these bosons that represent protons and neutrons are grouped into collective states with L = 0 and L = 2. The symbol s and d represent these values of angular momentum. Bosons of protons and neutrons are denoted by p and n, respectively; therefore, s_π and s_υ are protons and neutrons s bosons (L = 0), and d_π and d_v are protons, and neutrons d bosons (L = 2).

In IBM2, the equation (6) gives Hamiltonian as:

$\mathrm{H}={\mathrm{H}}_v+{\mathrm{H}}_{\pi }+{\mathrm{V}}_{\pi v}$ (6)

where H_π and H_v are the Hamiltonians of proton and neutron boson, while V_πv is the interaction of proton-neutron.

A basic Hamiltonian (Puddu et al. 1980):

$H=\epsilon ({\hat{n}}_{d\pi }+{\hat{n}}_{dv})+k{Q}_{\pi }\cdot Q_v+V_{\pi \pi }+V_{vv}+M_{\pi v}$ (7)

Where ε_v, ε_π are the neutron and proton energies, respectively, assumed that (ε_v = ε_π = ε), and the quadrupole operator is

$Q_{\rho }={(d^{†}\times s+s^{†}\times \tilde{d})}_{\rho }^2+{\chi }_{\rho }{(d^{†}\times \tilde{d})}_{\rho }^2\rho =\pi ,v$ (8)

where χ_p is a parameter to calculate the structure of the boson quadruple operators. The relations V_ππ + V_vv suggest d – bosons preserving residual n-n and p-p interactions. We can get equation (9).

${\hat{V}}_{\rho \rho }=\sum _{k=1.24}\frac{1}{2}(2L+1{)}^{\frac{1}{2}}{C}_L^{\rho }{[{(d_{\rho }^{†}\times d_{\rho }^{†})}^{(L)}\cdot {({\tilde{d}}_{\rho }\times {\tilde{d}}_{\rho })}^{(L)}]}^{(0)}$ (9)

The last term, Majorana interactions M_πv, is written as equation (10).

$$\begin{gathered} M_{\pi v}={\xi }_2{(s_v^{†}\times d_{\pi }^{†}-d_v^{†}\times s_{\pi }^{†})}^2\cdot {(s_v\times d_{\pi }^{†}-d_v^{†}\times s_{\pi })}^2-2\sum _{k=1,3}{\xi }_k{(d_v^{†}\times d_{\pi }^{†})}^{(k)}\cdot { \\ ({\tilde{d}}_v\times {\tilde{d}}_{\pi })}^{(k)} \end{gathered}$$ (10)

If there is investigational confirmation for mixed symmetry states formerly the Majorana parameters are diverse to fix the site of these states in the spectrum. The energy level are calculated using computer code NPBOS (Otsuka and Yoshida 1985). We diagonalizable Hamiltonian of equation (7) and agree the parameters ε, k, x_π, x_v and C_L for the best acceptable measured data.

The collective energies of even-even nuclei are consists of three groups based on the ratio R_4/2:

$R_{4/2}=\frac{E\left(4_1^{+}\right)}{E\left(2_1^{+}\right)}$ (11)

where E(4₁⁺) the energy level at 4₁⁺ and E(2₁⁺) is the energy level at 2₁⁺. Even-even nuclei can be classified according to ratios R_4/2 (Klein 1980; Singh and Chen 2020; NNDC 2026). The R_4/2 = 2.00 indicate a harmonic vibrator U(5), R_4/2 = 2.50 indicates g-unstable O(6) symmetry and R_4/2 = 3.33 indicate an axially symmetric rotor SU(3)(Iachello and Arima 1987). We recognized O(6) symmetry for ¹⁹⁰Hg nucleus since R_4/2 values of nucleus is 2.5 (Singh and Chen 2020; NNDC 2026) presented in Table 1. The ratio of E(4₁⁺)/ E(2₁⁺) values as a function of even neutron numbers of ¹⁹⁰Hg isotope is shown in Fig. 1. The solid line shows U(5), O(6) and SU(3) symmetry (Iachello and Arima 1987). The solid line O(6) is near to the data ratios of R_4/2 values of ¹⁹⁰Hg isotope (Iachello and Arima 1987; Klein 1980; Singh and Chen 2020; NNDC 2026).

A boson, e.g., s(L = 0) or d(L = 2) boson, describes a pair of valence nucleons, i.e., a proton and a neutron. The boson number is determined by counting the number of pairs, of course the valence nucleons together. Valence nucleons are worked out with the help of double - mirror nuclei. In the present study, valence nucleons are counted with respect to the doubly magic nucleus ²⁰⁸Pb (Z = 82, N = 126). The number of valence protons Np and valence neutrons Nn would give a total boson number N = Np+Nn/2 bosons. For this purpose, ²⁰⁸Pb is considered an inert core for the boson counting. For ¹⁹⁰Hg nucleus, the number of valence protons and neutrons are 2 and 16, respectively. Thus the total boson number for ¹⁹⁰Hg is (2/2+ 16/2) = 9 as in Table 2.

We have analyzed the excitation spectra of the ground-band, gamma-band and beta band of the ¹⁹⁰Hg isotope. In both of IBM models within O(6) limit energy levels of the g-, gamma and beta band was calculated. The energy ratio R = E4₁⁺ / E2₁⁺ is taken as an indicator of nuclear symmetry. For ¹⁹⁰Hg, a level ratio R = 2.50 (Singh and Chen 2020; NNDC 2026) is measured which points to a deformed, gamma soft O(6) symmetry (Iachello and Arima 1987).

Tables 2, 3 shows parameters used in the calculation of IBM1 and IBM2 for the ¹⁹⁰Hg isotope, respectively. In Table 2, all the parameters are expressed in MeV and consist of the expected N and CHQ(x) for the case of IBM1. In Table 3, the parameters are also given in MeV for IBM2. Table 4 contains comparative study of ground state band, gamma-band and beta-band of ¹⁹⁰Hg nucleus. The calculated data from IBM1 and 2 are in good agreement with earlier experimental measurements. Fig. 2 presents a systematic comparison of the three types of bands, which were calculated with IBM1, IBM2 and with the formerly measured data for the ¹⁹⁰Hg nucleus. The results of the calculations of the g - band in the ¹⁹⁰Hg nucleus with IBM1 and IBM2 are in satisfy to the measured data. The precision of the calculations in the IBM2 band is higher than the precision of the calculations in the IBM1 band.

The energy levels of excited states of even-even nuclei are given as L_i⁺ = 2₁⁺, 4₁⁺, 6₁⁺, 8₁⁺.......) generally decay to the lower-lying state L_f⁺ = L_i⁺-2 through E2 transition. Study of strength of the reduced transition probability B(E2) is one of very important parameter to under the decay properties of a nucleus. At present work the B(E2) data were calculated using the computer code PHINT (Scholten 1991), which requires values for effective charge (e_B) from Eq. (12).

$B(E2:2_1^{+}\to 0_1^{+})=\frac{{\alpha }_2^2}{5}N(N+4)=\frac{e_B^2}{5}N(N+4)$ (12)

In order to obtain reduced transition probability values, we have close-fitting the calculated absolute strengths B(E2) of transitions within ground state band to experimental ones. Value of effective charge (a₂) of IBMI has been determined by normalizing experimental data B(E2;2₁⁺ → 0₁⁺) of isotope using Eq. (12). The value of parameter a₂² for isotope has been calculated from the given measured value of transitions (2₁⁺ → 0₁⁺) and used to calculate B(E2) values for transitions 4₁⁺ → 2₁⁺, 6₁⁺ → 4₁⁺ and 8₁⁺ → 6₁⁺.The model wave functions were found by diagonalization of the IBM2 Hamiltonian and the program NPBEM (Otsuka and Yoshida 1985) were used to estimate the electromagnetic transition.

The E2 transition operator (Lange et al. 1982):

$T^{(E2)}=e_{\pi }{Q}_{\pi }+e_v{Q}_v$ (13)

Where, Q_ρ is the quadrupole operator and has the same definition as in Hamiltonian (7). ℮_π and ℮_v are boson effective charges contingent on the boson number N and they can be obtained by fitting B(E2;2₁⁺ → 0₁⁺) to the measured data. The effective charge a in e.b and e_b in e.b and B(E2) value in e²b² by IBM1, IBM2 and previous experimental data is presented in Table 5. The B(E2) data of IBM1 and IBM2 are consistent with the measured data. The reduced transition probability B(E2) is a microscopic interpretation of nuclear structure.

The potential energy surface (PES) gives good indication to the infinitesimal and symmetric shapes of nuclei. IBM Hamiltonian in the Refs (Bentley and Frauendorf 2011; Nomura et al. 2011; Kassim et al. 2020; Waheed et al. 2020) is used to create the PES plots using the Skyrme mean-field method. For IBM1, the energy surface is given as the expectation value of the IBM1 Hamiltonian (eq. 1) in the coherent state |N, ꞵ, γ⟩ (Siciliano et al. 2020). The creation operators b_c^† act on the boson vacuum |0⟩ in order to create this coherent state as follows:

$"N,\beta ,\gamma ⟩=\frac{1}{\sqrt{N!}}{\left(b_c^{†}\right)}^N"0⟩$ (14)

where

$b_c^{†}=\frac{1}{\sqrt{1+{\beta }^2}}\{s^{†}+\beta [\cos \gamma \left(d_o^{†}\right)+\sqrt{1/2}\sin \gamma (d_2^{†}+d_{-2}^{†})]\}$ (15)

then, the PES can be written in terms of β and γ as:

$E(N,\beta ,\gamma )=\frac{N{\epsilon }_d{\beta }^2}{(1+{\beta }^2)}+\frac{N(N+1)}{{(1+{\beta }^2)}^2}[{\alpha }_1{\beta }^4+{\alpha }_2{\beta }^3\cos 3\gamma +{\alpha }_3{\beta }^2+{\alpha }_4]$ (16)

Alpha’s parameters are the coefficients CL, v2, vo and uo in Eq. (1). The factor (attribute) of beta is related to the total deformation of the nucleus. A nucleus can be sphere shaped or distorted depending on whether the value of beta is equal to zero or not. Additionally, the term gamma describes the symmetry of the nucleus, i.e., when gamma = 0, the nucleus is prolate and when gamma = 60 deg, the nucleus is oblate. Fig. 3 shows the PES contour plot for ¹⁹⁰Hg. The PES values in MeV are indicated on the color panel.

The authors have declared that no competing interests exist.

No ethical statement was reported.

The authors accept full responsibility for the content of the manuscript, including the disclosure of any use of AI.

No AI tools were used in the preparation of this manuscript.

No funding was reported.

Alyaa A. Abd-Aljbar contributed to the research concept, manuscript writing and manuscript formatting. Mahdi J. S. Al Musawi contributed to the literature review and data organization. Huda H. Kassim contributed to the theoretical calculations and data interpretation. Asmaa A. Elbndag contributed to the data analysis, preparation of figures and tables. Mushtaq A. Al-Jubbori contributed to the methodology and scientific review of the manuscript. I. Hossain contributed to the validation of results and proofreading. Fadhil I. Sharrad supervised the research, contributed to the interpretation of results, and revised the manuscript. N. Aldahan contributed to data checking and final manuscript review. All authors have read and approved the final version of the manuscript.

All of the data that support the findings of this study are available in the main text.