Modified Surface Delta Interaction Research Articles

Particle-Particle Collective Excitations of Sn isotopes

In this paper, energy-level schemes and reduced electric transition strengths of neutron-rich Tin isotopes 102, 110, 116, 120, 122Sn (Z=50) are studied using collective models, that is, particle-particle Tamm-Dancoff Approximation and particle-particle Random Phase Approximation. According to these models, the excited states of closed-core A+2 systems with multipolarity J and isospin T can be described as a linear combination of particle-particle pairs. In our investigation, the low-lying states of the investigated isotopes 102, 110, 116, 120, 122Sn are described by acting two-particle operators on a correlated core 100Sn, 108Sn, 114Sn, 118Sn, and 120Sn, respectively. The Hamiltonian is diagonalized within the model space include {1g7/2, 2d5/2, 2d3/2, 3s1/2 and 1h11/2} orbits, using the matrix elements of neutron-neutron interaction and modified surface delta interaction. The calculated values are checked by using the resultant eigenvalues and eigenvectors to calculate the excitation energies and reduced electric transition strengths. Our calculated results are compared to the available experimental data, and these comparisons led to reasonable agreements. Effective charges are also used to account for the core polarization effect.

Hole-Hole Collective Excitations in 106, 112, 130Sn Isotopes

In this paper, energy-level schemes of neutron rich and reduced electric transition strengths of Tin isotopes 106Sn, 112Sn and 130Sn were studied using collective models, i.e., hole-hole Tamm-Dancoff Approximation (hh TDA) and hole-hole Random Phase Approximation (hh RPA). According to these models, the excited states of closed core A-2 systems with multipolarity J and isospin T can be described as a linear combination of hole-hole (hh) pairs. Therefore, in our approach, the low-lying states of the investigated isotopes 106Sn, 112Sn and 130Sn are obtained by acting two-hole operators on a correlated core 108Sn, 114Sn and 132Sn, respectively. The Hamiltonian is diagonalized within the model space include {1g7/2, 2d5/2, 2d3/2, 3s1/2 and 1h11/2} orbits, using the matrix elements of neutron-neutron (N-N) interaction and modified surface delta interaction (MSDI). The hh TDA and hh RPA are checked by using the resultant eigenvalues and eigenvectors to calculate the excitation energies and reduced electric transition strengths. A comparison had been made between our theoretical predictions and the recent available experimental data. Reasonable agreements were obtained from these comparisons.

Level Structure of 58Cu Within Modified Surface Delta-Interaction

The nuclear shell model with modified surface delta interaction MSDI was used to calculate the values of energy levels of the copper nucleus 58Cu. The neutron and proton in the model space 1p3/2 0f5/2 1p1/2 of the copper nucleus occur outside the closed core 56Ni. This research investigates the excitation energy and angular momentum. As a consequence, theoretical approaches are used to uncover a collection between excitation energies and classical coupling angles θa,b at various orbitals. Finally, we demonstrate that our results are supported by experimental evidence: Excitation energies have two major functions, both of which are influenced by classical coupling angles but are unaffected by angular momentum I.

Study the Nuclear Structure of Some Even-Even Ca Isotopes Using the Microscopic Theory

The root-mean square-radius of proton, neutron, matter and charge radii, energy level, inelastic longitudinal form factors, reduced transition probability from the ground state to first-excited 2+ state of even-even isotopes, quadrupole moments, quadrupole deformation parameter, and the occupation numbers for some calcium isotopes for A=42,44,46,48,50 are computed using fp-model space and FPBM interaction. 40Ca nucleus is regarded as the inert core for all isotopes under this model space with valence nucleons are moving throughout the fp-shell model space involving 1f7/2, 2p3/2, 1f5/2, and 2p1/2 orbits. Model space is used to present calculations using FPBM interaction, and with the effects of core-polarization are obtained by the first order core polarization through a microscopic theory is called modified surface delta interaction which allows all higher orbits are excited by particle-hole excitation from the core and model space orbits. Also, each isotope's effective charge is determined by using the collective model by Bohr and Mottelson formula. The current result corresponds to the experimental data by taking into account core polarization effects.

Determining the excitation energies of 68Ni Nucleus a Function of the Coupling Angle ¸a,bBy Means of Modified Surface Delta-Interaction

In this paper, the nuclear shell model SM was applied to study the energy levels of the nucleus the effect of orbit on energy levels was studied,we consider the residual interaction to be modified surface delta interaction MSDI. We have studied the excitation energies of the 68Ni a nucleus, which contain two neutrons outside closed shell of the 66Ni. the computation of eigenvalues of 68Ni is performed in the frame of (1 p1/2 0g9/2) and (1 _p1/2 0g9/2 0d5/2), we search the effect of the orbital 0d5/2 on the energy levels. The energy levels and angular momentum of all possible cases were investigated. Thus, we have worked on a theoretical process to discover the relation between the energy levels and the classical coupling angle ¸a,b at different orbital within neutron - neutron interaction. Accordingly, We have noticed that the energy levels manifest to follow two inclusive functions, which depend on the classical coupling angles, but are not constrained of angular momentum I. In conclusion, we conclude that our results agree with the experimental data.

Energy Levels of 50Ca Nucleus as a Function of the Classical Coupling Angle within MSDI

In the construction of this kind of shell model, we take the residual interaction to be modified surface delta interaction MSDI. We have studied the excitation energies of the 50Ca a nucleus, which contain two neutrons outside closed shell of the 48Ca. Neutrons are in the model space pfpg. The energy levels and angular momentum of all possible cases were investigated. Thereby, we have effectively utilized a theoretical process to find link among the traditional coupling angle and energy levels at different orbital within neutron - neutron interaction. We observe the energy stages appear to follow two overall functions which depend on the classical coupling angles but are unconstrained of angular momentum I. We find out that our results agree with the experimental data.

Elastic and Inelastic Electron-Nucleus Scattering Form Factors for Be9 Nucleus

The computations of the elastic and inelastic Coulomb form factors for the electron-nucleus scattering of Beryllium nucleus Be9 have performed with Core Polarization (CP) effects including the realistic Michigan sum of Three Range Yukawa (M3Y) Interaction, and the other residual interaction which is Modified Surface Delta Interaction (MSDI). In addition to mean square root charge density and charge radii for the ground state. The perturbation theory was adopted to compute the Core Polarization by using the Harmonic Oscillators (HO) potential to calculate single-particle radial wave functions.
 In the comparison between the theoretical calculations of Coulomb form factors by (MSDI) interaction, realistic (M3Y) interaction, and the experimental results that measured before, it noticed that the Coulomb form factors for the (M3Y) interaction gave a reasonable depiction of the measured data.

Using MSDI and M3Y Core Polarization for the Coulomb Electron Scattering for some Ground state nuclei

The analysis of Coulomb electron scattering for the ground state-nuclei Li6 , C12 , O18 , F19 and Ne 20 are performed with core polarization effects for the Modified Surface Delta Interaction (MSDI) and realistic Michigan three-range Yukawa (M3Y) interaction. The basic calculations included the Coulomb form factor, charge density, and the charge radii of the ground states. The Coulomb form factors for the M3Y interaction gave the best fit with the experimental data.

Energy levels of nuclei 40Sc and 40K as a function of semi-classical coupling angle θ1,2 within the modified surface delta-interaction

Energy levels of nuclei 40Sc and 40K as a function of semi-classical coupling angle θ1,2 within the modified surface delta-interaction

Nuclear energy levels in 58Ni

Energy levels, total angular momenta and parity for two nucleons that present outside closed core 56Ni (Nickel) which occupied FP nuclear shell have been calculated using nuclear shell model application. We have used FP M3YE interaction to calculate energy levels of 58Ni, in addition, the results are compared with experimental data, modified surface delta interaction (MSDI), Reid's potential (RP) and non-zero pairing shell model (NZPSM).
 Harmonic Oscillator potential is used to generate single particle wave functions in FP shell and considering as an inert core.
 Oxford Buenos Aires Shell Model (OXBASH) code for nuclear shell model calculation has been utilized to carry out the calculations and comparison with experimental data had been done.

The effects of core polarisation on some even–even sd-shell nuclei using Michigan three-range Yukawa and modified surface delta interactions

Elastic and inelastic electron scattering from even–even \(Z=N\) sd-shell (\(^{28}\hbox {Si}, {}^{32}\hbox {S}\) and \(^{36}\hbox {Ar}\)) nuclei has been studied using the nuclear shell-model configurations. The transition rates \(B\left( {C2\uparrow } \right) \) from the ground 0\(^{+}\) state to the first excited \(2_{1}^{+}\) state, the electric quadrupole moments Q, the elastic longitudinal C0 and inelastic longitudinal C2 form factors are calculated. SDBA and USDA model spaces have been used. The radial wave functions of the single-particle matrix elements have been calculated in terms of the harmonic oscillator (HO) and Skyrme–Hartree–Fock (SHF) potentials. The configurations higher than the core and the model space are taken into account within a microscopic theory that includes one particle–one hole excitations from the core and model space orbits to higher allowed orbits with 2\(\hbar \omega \) excitations. These effects are defined as core polarisation (CP) effects. Two-body Michigan three-range Yukawa (M3Y) effective nucleon–nucleon interaction and the modified surface delta interaction (MSDI) have been used as residual interactions for the CP matrix elements. The calculations are performed using the shell-model code Nushell@MSU, where the deduced results, including CP, are more compatible with the available experimental and theoretical results.

Energy Levels of Isobaric Nuclei (16N, 16F) within the Modified Surface Delta-Interaction Model

The modified surface delta-interaction model is applied to calculate the energy levels of 16N and 16F) nuclei. The good agreement between theoretical and experimental data is attained for excitation energies. This indicates that the shell model describes properly the structure of these nuclei.

Shell Model Calculations on Positive and Negative Parity States for 90Sr,90Y,92Nb and 92Zr Nuclei

The excitation energies for both the positive and negative parities of 90Sr,90Y,92Nb and 92Zr isotopes have been calculated by applying nuclear shell model with in the 2d5/2,1g7/2, 2d3/2, 3s1/2 and 1h11/2 model space for neutrons in 90Sr and 92Zr nuclei, also the 2p1/2, 1g9/2 for proton and 2d5/2,3s1/2 for neutron specified as model space in 90Y nucleus. While, model space for 92Nb nucleus was comprised of 1g9/2 for proton and 2d5/2 for neutron. Modified surface delta interaction with Matlab program have been used to calculate the present theoretical results. The agreement between theoretical prediction and experiment values were good. we determine and confirm the angular momentum and parity of energy levels that are not determined and unconfirmed experimentally. In addition to that, new energy levels have been predicted for all the isotopes in this present work. This will add more information which increases the theoretical Knowledge for all isotopes with respect to energy levels.

Calculation of the energy levels for nucleusby applying the nuclear shell model using modified surface delta interaction

In this work, we applied the nuclear shell model to study the energy levels for nucleus which contain two holes outside core and occupy the shell .In these calculations used computer program MatLab 2016 to calculate Modified Surface Delta Interaction (MSDI) is employed as a residual interaction. Comparison with experimental values and found to be rather good agreement.

Contribution of high energy configurations to longitudinal and transverse form factors in p- and sd-shell nuclei

Longitudinal and transverse electron scattering form factors for some selected positive and negative parity states of stable odd-A nuclei ( 7 Li, 13 C and 17 O) in the p- and sd-shells are investigated by considering the higher energy configurations outside the p- and sd-shells. The higher energy configurations, referred to as core polarization effects, are considered by means of a microscopic theory that includes excitations from the core 1s-1p ,2 s-1d orbits to the higher allowed orbits with 4�x excitations. The calculations are performed in the p- and sd-shell model spaces by employing Cohen-Kurath, Reehal and Wildenthal interactions respectively, while the core polarization effects are calculated with the modified surface delta interaction as residual interaction. The predicted form factors are compared with the available experimental data and it is shown that the inclusion of the higher excited configurations is very essential in the calculations of the form factors and the reduced transition probabilities to obtain reasonable description of the data with no adjustable parameters.

Longitudinal electron scattering form factors for 54,56Fe

In this paper, inelastic longitudinal electron scattering form factors for C2 transition have been studied in 54 Fe and 56 Fe with the aid of shell model calculations. The GX1 effective interaction for the fp-shell is used with the nucleon–nucleon realistic interaction Michigan three-range Yukawa and Modified surface delta interaction as a two-body interactions. The core polarization effects is taken into account through the first-order perturbation theory with the effective charge, which is taken to the proton and the neutron. The effective charge along with the core effects up to 6 ℏw enhanced the calculation very well and improving good agreement with the experimental data.

The electron scattering form factor of 10B, 32S and 48Ca nuclei

Longitudinal and transverse electron scattering form factors and transition probabilities have been calculated for different states in 10B, 32Sc and 48Ca nuclei, where a microscopic theory is employed to include the effect of high configuration outside the model space, which is called the core-polarization (CP) effect. The Cohen–Kurath interaction for p-shell, the universal sd of Wildenthal interaction for sd-shell and the GXFP1 interaction for fp-shell are used with the Michigan three-range Yakawa (M3Y) effective NN interaction and modified surface delta interaction as a residual interaction for the CP calculations. The calculation is performed using the shell model codes CPM3Y and NuShell. The radial wave function for the single-particle matrix elements have been calculated with the Skyrme–Hartree Fock, harmonic oscillator potential and Wood Saxson potential to calculate the single-particle wave function. The inclusion of CP effects modifies the form factors markedly and describes the experimental data very well in the momentum transfer dependence.

The effect of core polarization on longitudinal form factors in 10B

Electron scattering Coulomb form factors for the single-particle quadrupole transitions in the p-shell 10B nucleus have been studied. Core polarization effects are included through a microscopic theory that includes excitations from the core orbits up to higher orbits with 2ℏω excitations. The modified surface delta interaction is employed as a residual interaction. The effect of core polarization is found essential in both the transition strengths and momentum transfer dependence of form factors, and gives remarkably good agreement with the measured data with no adjustable parameters.

Single-particle quadrupole electromagnetic transitions in p-shell and sd-shell nuclei

Electron scattering Coulomb form factors for the single-particle quadrupole transitions in p-shell and sd-shell nuclei have been studied. Core polarization effects are included through a microscopic theory that includes excitations from the core orbits up to higher orbits with 2ω excitations. The modified surface delta interaction is adopted as a residual interaction. The results are discussed for the ( 1p1/2-1↦1p3/2-1) proton transition in 15N, ( 1d5/2↦2s1/2) neutron transition in 17O and ( 1d3/2-1↦2s1/2-1) proton transition in 39K. The inclusion of core polarization effects modifies the form factors markedly and describes the experimental data very well in both the absolute strength and the momentum transfer dependence.

Core polarization effects on C4 form factors of sd-shell nuclei

Coulomb form factors of C4 transitions in even-even N = Z sd-shell nuclei ( 20Ne, 24Mg, 28Si and 32S) are discussed taking into account higher-energy configurations outside the sd-shell model space which are called core polarization effects. Higher configurations are taken into account through a microscopic theory, which allows particle-hole excitations from the 1s and 1p shells core orbits and also from the 2s1d-shell orbits to the higher allowed orbits with excitations up to 4 ω. The effect of core polarization is found essential in both the transition strengths and momentum transfer dependence of form factors, and gives a remarkably good agreement with the measured data with no adjustable parameters. The calculations are based on the Wildenthal interaction for the sd-shell model space and on the modified surface delta interaction (MSDI) for the core polarization effects.