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Using phenomenological fitting, Sood’s semi-empirical formula have been investigated the rotational structure of even-even rare-earth 170,172Hf nuclei. The calculated rotational energies were compared with their corresponding experimental data. It showed that in 172Hf and 170Hf isotopes, calculated energies fits the experimental values to a remarkable degree of accuracy with deviation less than 3 percent at Jπ = 18+ state and Sood’s semi-empirical formula predicted rotational energies.

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.

This paper has been explained to study the rotational arrangement of even-even 168Hf isotope using the phenomenological fitting, Sood’s semi-empirical formula. The rotational energies from the calculated values were compared to the experimental data. The result shows that in 168Hf, calculated energies fit the experimental values to a remarkable degree of accuracy.

Electric quadrupole $$\gamma$$ -ray transition energies $$E_{\gamma}$$ , rotational frequencies $$\hbar\omega$$ and dynamic $$J^{(2)}$$ and kinematic $$J^{(1)}$$ moments of inertia have been calculated for yrast and excited superdeformed (SD) bands in Hg, Tl, Pb isotonic ( $$N=113$$ ) nuclei by using two-parameter power-law formula. The model parameters and the bandhead spins were determined using a $$\chi^{2}$$ fit by random minimization via Monte Carlo simulation technique with a uniform distribution of random numbers. The calculated results of $$E_{\gamma}$$ agreed with the experimental data published in National Nuclear Data Center (nndc) very well, and present assigned spins of the studied superdeformed rotational bands (SDRB’s) are consistent with the spins suggested by other previous formulae. The majority of the studied SD bands displayed the same large, smooth increase in $$J^{(2)}$$ by increasing $$\hbar\omega$$ . A saturation of $$J^{(2)}$$ at higher values of $$\hbar\omega$$ was observed for the two bands (1, 2) in $${}^{195}$$ Pb due to the Pauli blocking of the quasineutron alignment in the presence of quasiproton alignment. The studied SD bands exhibited staggering in their transition energies. Five pairs of signature partner SDRBs exhibited an $$\Delta I=1$$ staggering effect in their transition energies, namely $${}^{193}$$ Hg (SD1, SD2), $${}^{193}$$ Hg (SD3, SD4), $${}^{194}$$ Tl (SD1, SD2), $${}^{195}$$ Pb (SD1, SD2), and $${}^{195}$$ Pb (SD3, SD4). To investigate this $$\Delta I=1$$ staggering, we extracted the difference between the average transition $$I+2\to I$$ and $$I\to I-2$$ energies in one band and the transition $$I+1\to I-1$$ energies in the signature partner. Also, a $$\Delta I=2$$ staggering exists in three SD bands of odd–odd $${}^{194}$$ Tl. This $$\Delta I=2$$ staggering effect was determined by calculating the energy difference between two consecutive $$\gamma$$ -ray transitions as a function of rotational frequency after subtracting a smooth reference expressed in the notation of Cederwall.

In this paper, even-even 104−106Ru isotopes have been studied the ground state bands using Matlab computer code (IBM-1.Mat). We apply the interacting boson model-1 (IBM-1) formula for O(6) symmetry in Ru isotopes with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12+ have been obtained for 104−106Ru isotopes. The yrast states, gamma band, beta band, and B(E2) values are calculated for these nuclei. The published experimental and calculated R4/2 values indicate that the even-even 104−106Ru isotopes have O(6) dynamic symmetry. The present results have been compared to the published experimental data and are found good harmony with each other. The outcome of our investigation of the potential energy surfaces (PES) of both isotopes belonging to O(6) character.

In this paper, Ruthenium Isotopes with neutron N = 60, 62 have been studied the ground state bands using Matlab computer code interacting boson model (IBM-1). We apply IBM-1 formula for O(6) symmetry in Ru isotopes with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12+ have been obtained for 104,106Ru isotopes. The yrast states, gamma band, beta band, and B(E2) values are calculated for those nuclei. The experimental and calculated R4/2 values indicate that the even-even 104-106Ru isotopes have O(6) dynamic symmetry. The calculated results are compared to the experimental data and are found in good harmony with each other. The plots of the potential energy surface of both nuclei are O(6) characters.

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.

In the present research, the nuclear deformation of even–even rare-earth Er–Os isotopes with N = 102 has been calculated using a new empirical equation and an interacting boson model (IBM-1). The potential energy surface, reduced transition probabilities B(E2) and energy level have been calculated for these nuclei. The properties of gamma, beta and yrast band have been also calculated and compared with the available experimental data. The results of both models have been indicated the similarity or dissimilarity with the published experimental data. We have been studied the ratio $$E_{\gamma } (I + 2) /E_{\gamma } (I)$$ versus the spin (I) to determine the character of the ground-state band. From the outcome of our investigation, it is possible to conclude that the 170Er/172Yb/174Hf/176W nuclei show a rotational SU(3) character and 178Os shows X(5) character. Results of this calculation are in good agreement with the corresponding available experimental data.

In this paper, we have reviewed the calculation of ground states energy level up to spin 14+, electric quadrupole moments up to spin 12+, and reduced transition probabilities of Hafnium isotopes with even neutron N = 100-108 by Interacting Boson Model (IBM-1). The calculated results are compared with previous available experimental data and found good agreement for all nuclei. Moreover, we have studied potential energy surface of those nuclei. The systematic studies of quadrupole moments, reduced transition strength, yrast level and potential energy surface of those nuclei show an important property that they are deformed and have dynamical symmetry SU(3) characters.

Tin-Magnesium oxide (Sn1−xMgxO) thin films were prepared on glass substrates using the chemical spray pyrolysis technique, whereupon the samples were irradiated by gamma rays using a Co-60 radioactive source. X-ray diffraction showed that all prepared films were polycrystalline in nature with a tetragonal structure and a preferential growth of crystallites in the (110) plane. In general, the average crystallite size, lattice constants, dislocation density and crystallite density decreased with increasing Mg doping from 0% to 8%. Further, atomic force microscopy showed that the thin films were smooth and homogenous. The optical properties were obtained by ultraviolet–visible spectrophotometry, and the transmittance and absorbance spectra before and after gamma ray irradiation were compared for all samples, whereby the absorption and extinction coefficients and real and imaginary parts of the dielectric were studied before and after irradiation. It was found that the energy band gap values decreased from 3.94 eV to 3.72 eV with increasing Mg doping from 0% to 8% before irradiation, and from 3.92 eV to 3.59 eV after irradiation. All optical constants increased with doping percentage before and after irradiation. Energy-dispersive x-ray spectroscopy showed that all structures contained Sn and O elements in the undoped state, and contained SnO2 and Mg in the doped state.

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.

The properties of 220−230Th isotopes were studied and its energy states calculated. To identify the properties of each isotope, the values of the first excited states E 2 1 + and the ratio of the second excited state to the first excited state R 4 / 2 = E 4 1 + / E 2 1 + were adopted. The phenomenon of back (or up)-bending, the E-GOS curve, the relationship between successive excited states to those preceding them and the staggering in the difference of energies between the ground state band (GSB) and the negative party band (NPB) states were studied. After identifying the properties of each isotope, the appropriate limit in the interacting boson model IBM-1, the IVBM model and the Bohr and Mottelson model were used to calculate the energy states for each isotope and compared the results with the experimental values. Due to the inaccuracy of some calculations in the models IVBM and BM, an adjustment term was added for each model and recalculated the energy states and the results were improved. The contour plots of the potential energy surface (PES) for 220−230Th isotopes are studied using the simplified form of Interacting Boson Model with an intrinsic coherent state.

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 ( E γ / 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 ( E 2 ) 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.

The positive parity low-spin states of even-odd 129−133Ba isotopes were studied in this study using the Interacting Boson–Fermion Model (IBFM-1). The single fermion is predicted to be in one of three single-particle orbits: 2d5/2, 2d3/2, and 3s1/2. The Interacting Boson Model (IBM-1) was used to investigate the energy levels, electric quadrupole transition probabilities, and potential energy surface of even-even Barium isotopes (a core for even-odd nuclei). The measured positive parity low-state energy spectra and predicted energy levels, as well as the B(E2) transition probabilities, are reasonably consistent with the experimental data and previous research for Ba isotopes. The potential energy surface contour plot reveals that all interesting nuclei are deformed and have γ-unstable-like properties.

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 ( E γ / I ) as a function of the spin (I) have been drawn. The reduced transition probabilities B ( E 2 ) 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.

The energy levels and the electric quadrupole transition probabilities B(E2; Ji → Jf) for even–odd 191–199Hg have been investigated. The negative spin states of the even–odd 191–199Hg isotopes are studied within the framework of the interacting boson–fermion model (IBFM–1). The single fermion is assumed to be in one of the 2f5/2, 3p3/2, and 3p1/2 single-particle orbits. It is found that the calculated negative spin state energy spectra of the even–odd 191–199Hg isotopes agree quite well with the experimental data. The B(E2) values are also calculated and compared with the experimental data. In general, the results are in reasonably good agreement with the previous experimental values. Furthermore, the energy levels, electric quadrupole transition probabilities, and the wave function for even–even 1901–198Hg isotopes (as core for even–odd nuclei) have been calculated within the framework of the interacting boson model (IBM-1). The predicted energy levels and B(E2) transition probability results are reasonably consistent with the experimental data. These calculations have been compared with previous it and almost better than it. The study of the wave function’s structure shows that all interesting nuclei are deformed and have dynamical symmetry O(6) characters.

A description of the even-even Pd isotopes for A = 102 to 106 in the framework of the Interacting Boson Model (IBM) are carried out. The low-lying positive parity yrast bands, as well as the associated B(E2) values, have been calculated in good agreement with the experimental data. The contour plots of the potential energy surface are studied, by using the simplified form of IBM with an intrinsic coherent state. The critical points have been determined for 102Pd isotope, and O(6) symmetry is substantiated for 104,106Pd., Дано опис парно-парних iзотопiв паладiю для A вiд 102 до 106 в моделi взаємодiючих бозонiв (МВБ). Розрахованi низьколежачi iраст-смуги з позитивною парнiстю i вiдповiднi величини B(E2), що добре узгоджуються з експериментом. З використанням спрощеного варiанта МВБ з власним когерентним станом вивченi контурнi графiки поверхнi потенцiальної енергiї. Знайдено критичнi точки для 102Pd iзотопу i обґрунтовано O(6) симетрiю для 104,106Pd.

In this work, positive parity low-spin states in 120 – 127 Xe have been investigated using the Interacting Boson Model and the Interacting Boson–Fermion Model. The energy levels, electromagnetic transition B(E2) and B(M1) , mixing ratios, relative B(E2) values and the potential energy surface for the even–even 120 – 126 Xe isotopes have been calculated. The theoretical results are compared with the experimental data and it is seen that the obtained theoretical results are in good agreement with the experimental data. The single fermion is assumed to be in one of the 2 d 5/2 , 2 d 3/2 and 3 s 1/2 single-particle orbits. Furthermore, the energy levels, electromagnetic transition B(E2) and B(M1) and mixing ratios have been calculated and compared with the experimental data and previous study for even–odd 121 – 127 Xe isotope.

An effective two-body density operator for point nucleon system folded with the full two-body correlations (which include the tensor correlations and short-range correlations), which take account of the effect of the strong tensor force and the strong short-range repulsion in the nucleon–nucleon forces, is produced and used to derive an explicit form for ground-state two-body charge density distributions. The inelastic longitudinal form factors C2 are calculated using this transition charge density with excitation of the levels in 10B (J π , T):(1+, 0) at E x = 0.718 MeV, (1+, 0) at E x = 2.154 MeV, (2+, 0) at E x = 3.587 MeV, (3+, 0) at E x = 4.774 MeV, (2+, 0) at E x = 5.920 MeV, and (4+, 0) at E x = 6.025 MeV. In this work, the core-polarization transition density is evaluated by adopting the shape of Tassie model together with the derived form of the ground-state two-body charge density distributions with the effect of higher occupation probabilities η. It is noticed that the core-polarization effects which represent the collective modes are essential in obtaining a remarkable agreement between the calculated inelastic longitudinal F(q)s and those of the experimental data.

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 ( E 2 ) 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.

The low-lying nuclear structure of the even–even Hf isotopes for n = 106, 108, 110 in the framework of the Interacting Boson Model (IBM-1) is investigated. The energy levels, electromagnetic transition rates B(E2) and electric quadrupole moment are calculated. All calculated values are compared with the available experimental data and show reasonable agreement. A contour plot of the potential energy surfaces shows that three interesting nuclei are deformed and not exactly SU(3).

The properties of 170 Er, 172Yb, 174Hf, and 176Wisotones have been studied and their energy states calculated. To identify the properties of each isotone, the values of the first excited states, E 2 1 + and the ratio of the second excited states to the first excited states, R 4 / 2 = E 4 1 + / E 2 1 + for all nuclei under consideration were adopted. To determine the properties of each nucleus, the relationship between the moment of inertia 2𝜗/ℏ 2 and the square of the angular frequency, ℏ 2 𝜔 2, the relationship between successive excited states to those preceding them r ( I + 2 ) I ) and the ΔI = 1 staggering between the GSB and the NPB states were studied for all states of 170Er, 172Yb, 174Hf, and 176W isotones. After identifying the properties of each isotone, the rotational limit in the interacting boson model IBM-1 and the IVBM model was used to calculate the energy states for each isotone and the results were compared with the experimental values. and good agreement was observed with some exception. The inaccuracy of some calculations in the IBM-1 results from the lying of some high states out the range of the rotational properties that were used.

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 E 2 1 + and the ratio of the second to the first excited states R 4 / 2 = E 4 1 + / E 2 1 + 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.

The phenomenological description of Barium nuclei which have proton number (56) and neutron numbers 74 and 75 have been calculated by Interacting Boson – Fermion Model. Yrast and excited bad, probability of electric transition B(E2) and potential energy surface have been estimated. The input parameters which are used in the present study the best approximation that has been carried out so far. These calculations show that a good agreement with those of experimental data for Barium nuclei.

In this paper, the nuclear structure of Barium nuclei has been studied by interacting boson model. The properties of eigenstates and reduce probability of electric transition and potential energy surface have been calculated. The nuclei understudy has proton number equal 56 and neutron numbers 75 and 76. The results of energy states and B(E2) values have been compared with the experimental data. These comparisons show that acceptable between the calculations and experimental data. The parameters which used in the calculation have been estimated and give best approximation. The shape of potential energy surface show that the even-even nucleus have O(6) dynamical symmetry.

In this study, the Palladium nuclei have been studied by interacting boson fermion model for even-odd nuclei. The isotopes under study have proton number 46 and neutron numbers 63-67. The calculations involved the eigenstates and reduce electric probability. The calculations show that the energy levels are reasonable with experimental data. The B(E2) values compared with those of experimental and are in good agreements. All calculations compared with experimental and give more data for reader.

The some properties, Energy levels, B(E2) values and potential energy surface, for even-even 122,124Cd isotopes have been studied using the interacting boson model. The predicted levels (energies, spins and parities) and B(E2) values results were reasonably consistent with the experimental data. The contour plot of the potential energy surfaces shows all interest nuclei were deformed and have O(6)-U(5) transition symmetry.

In this study, mixed Nickel- Copper oxide (NiO)1-x(CuO)x thin films were deposited on glass substrates using chemical spray pyrolysis technique at substrate temperature of 400 °C and thickness of 200 nm. UV-Visible spectroscopy and X- ray diffraction (XRD) were used to study the optical and structural properties of films. The optical energy gap for the thin films was estimated using Tauc's function. It is observed that the band gap decreases with the increase of the mixing ratio of copper oxide. The constants of optical properties contain absorption coefficient as a function of photon energy. Extinction coefficient for all samples was estimated as a wavelength function and the refractive index increased as (CuO) contents increased. The results of XRD showed that these films were polycrystalline in nature with orientation along (111) plane. The average values of crystallite size was estimated by using Scherrer's method and W-H questions are found to increase as the (CuO) contents increased.

In this paper, the yrast bands of the even 96,98Ru isotopes are studied within the framework of the interacting boson model-1 (IBM-1), using the MATLAB computer code (IBM-1.Mat). The theoretical energy levels are obtained for the 96,98Ru isotopes, with neutron numbers 52 and 54, up to spin-parity 16+ and 12+, respectively. The ratio of the excitation energies of the first 4+ to the first 2+ excited states (R 4/2), the backbending curves and the potential energy surfaces are also calculated. The calculated and experimental R4/2 values show that the 96,98Ru nuclei have U(5) dynamic symmetry. The calculated energies of the yrast states are compared with experimental results and they are shown to be in good agreement with the data. The contour plots of the potential energy surfaces show two interesting nuclei having a slightly oblate but almost spherical vibrator-like character.

In this paper, the negative parity low-spin states of even–odd 187–197Pt isotopes have been studied within the framework of the Interacting Boson–Fermion Model (IBFM-1). The single fermion is assumed to be in one of the 2 f 5 / 2 , 3 p 3 / 2 and 3 p 1 / 2 single-particle orbits. The calculated negative parity low-states energy spectra agree quite well with the experimental data. The B ( E 2 ) values have been also calculated and compared with the experimental data. The calculated energy levels and B ( E 2 ) are in good agreement with experimental data than that in the previous study for 195Pt isotope. Furthermore, the energy levels, electric quadrupole transition probabilities and the potential energy surface for even–even platinum isotopes (as core for even–odd nuclei) have been calculated within framework of the Interacting Boson Model (IBM-1). The predicted energy levels and B ( E 2 ) transition probabilities results are reasonably consistent with the experimental data. The contour plot of the potential energy surfaces shows all interesting nuclei are deformed and have γ-unstable-like characters.

The Hamiltonian is determined in the Interacting Boson Model-1 (IBM-1) frame work for 80Se and 82Kr nuclei. Fit values of parameters are used to construct the Hamiltonian, energy levels, electromagnetic transitions (B(E2) and B(M1)) and multipole mixing ratios (E2/M1) for some even-even Se and Kr nuclei. These properties are calculated. The potential energy surfaces (PESs) to the IBM Hamiltonian have been obtained by using the intrinsic coherent state.

The interacting boson model has been applied to calculate the low-lying positive parity yrast bands in Gd, Dy and Er nuclei for N = 96 neutrons. The results closed to the SU(3) limit in this model. Reasonable agreement with available energies and B(E2) transition rates. The potential energy surfaces (PESs) to the IBM Hamiltonian have been obtained using the intrinsic coherent state.

Energy levels and B(E2) transitions for palladium isotopes with proton number Z=46 and neutron numbers (n) between 57 and 67 have been calculated through the interacting boson-fermion model. The set of parameters used in this calculation is the best approximation that has been carried out so far. Good agreement was found from comparison between the calculated energy levels and the transition probabilities B(E2) with those of experimental.

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.

In this work, the positive parity low-spin states for 122Xe and 123Xe have been investigated using the interacting boson mode-l and the interacting boson-fermion mode-l, respectively. The energy levels and reduced transition probabilities B(E2) have been calculated. The theoretical results are compared with the experimental data. It is in good agreement with the experimental data. The contour plot of the potential energy surface shows that the 122Xe nucleus is deformed and has γ-unstable-like characters.In this work, the positive parity low-spin states for 122Xe and 123Xe have been investigated using the interacting boson mode-l and the interacting boson-fermion mode-l, respectively. The energy levels and reduced transition probabilities B(E2) have been calculated. The theoretical results are compared with the experimental data. It is in good agreement with the experimental data. The contour plot of the potential energy surface shows that the 122Xe nucleus is deformed and has γ-unstable-like characters.

The energies of the ground, [Formula: see text] and [Formula: see text] bands as well as the associated [Formula: see text] values have been calculated using interacting boson model (IBM). We have developed a new method called “new empirical equation” for each even–even rare-earth Er–Os for [Formula: see text]. Also, the relation of the gamma energy over spin as a function of the spin [Formula: see text] (E-GOS) has been drawn. The ratio between the energies of the [Formula: see text] and [Formula: see text] states as a function of [Formula: see text] has been drawn to determine the property of the ground-state band, these curves indicated that these isotopes have a rotational property SU(3), with the [Formula: see text] property for [Formula: see text]Os isotopes. The contour plots of the potential energy surface for Er–Os for [Formula: see text] are studied using the simplified form of interacting boson model with an intrinsic coherent state. The critical points have been determined for [Formula: see text]Os isotopes. The behaviors of energy and [Formula: see text] ratios in the ground-state band are examined.

A description of the even-even Hf isotopes for A = 172 ‒ 176 is carried out in the framework of the interacting boson model (IBM-1). The energy levels, B(E2) transition probabilities, electric quadrupole moment QL, and potential energy surface of those nuclei have been calculated. The calculated results are compared with the most recent experimental data. A good agreement for low lying energy states is obtained between experimental results and theoretical calculations. The contour plots of the potential energy surfaces show that the interested nuclei have rotational characters.

The electric and magnetic transitions in the [Formula: see text]Mg nucleus are studied based on the calculations of the longitudinal and the transverse electron scattering form factors. The universal sd-shell model Hamiltonian (USDA) is used for calculations. The wave functions of radial single-particle matrix elements are calculated using the Skyrme potential. For the longitudinal form factors, a good agreement is obtained between the calculations and the experimental data. For the transverse form factors, the effective [Formula: see text] factors are made as adjustable parameters in order to describe the experimental data.