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.

Seniority and configurations in neutron-rich Nickel isotopes * *Supported by the Interdisciplinary Scientific and Educational School of Moscow University "Fundamental and Applied Space Research"

Excited states in low-energy spectra of Ni are considered. Accordingly, pairing forces in the form of surface delta interaction are employed to account for the formation of the ground state multiplet with seniority states. The multiplet splitting is described with mass relationships of masses of neighboring nuclei. Subsequently, the seniority model is adopted to reproduce or predict the states in odd-even isotopes and in even-even isotopes. The correct account of the state should allow for the description of the reversed order of states with and observed in experiments. The results obtained are compared with the structure of similar multiplets in isotones.

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.

Description of the energy levels and electric quadrupole transitions of the 92Nb and 92Mo nuclei using nuclear shell model

In current study ,92Nb and 92Mo isotopes have been determined for calculating energy levels and electric quadrupole transition probabilities. Two interaction have been applied in this study are surface delta and modified surface delta interactions. The calculations have been achieved by using appropriate effective charges for proton and neutron as well as parameter length of harmonic potential. Computed results have been compared with the experimental values. After this comparison, energy and the transition probability values have a good agreement with the experimental values, also there are values of the total angular momentum and parity are determined and confirmed for some of the experimental energies, undetermined and unconfirmed experimentally. Theoretically, new values of quadrupole electric transition probabilities have been explored which have not been known in the experimental data.

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.

Neutron–proton spin–spin correlations in the ground states of $$N=Z$$ nuclei

We present expressions for the matrix elements of the spin--spin operator $\vec S_{\rm n}\cdot\vec S_{\rm p}$ in a variety of coupling schemes. These results are then applied to calculate the expectation value $\langle\vec S_{\rm n}\cdot\vec S_{\rm p}\rangle$ in eigenstates of a schematic Hamiltonian describing neutrons and protons interacting in a single-$l$ shell through a Surface Delta Interaction. The model allows us to trace $\langle\vec S_{\rm n}\cdot\vec S_{\rm p}\rangle$ as a function of the competition between the isovector and isoscalar interaction strengths and the spin--orbit splitting of the $j=l\pm \frac{1}{2}$ shells. We find negative $\langle\vec S_{\rm n}\cdot\vec S_{\rm p}\rangle$ values in the ground state of all even--even $N=Z$ nuclei, contrary to what has been observed in hadronic inelastic scattering at medium energies. We discuss the possible origin of this discrepancy and indicate directions for future theoretical and experimental studies related to neutron--proton spin--spin correlations.

Application of the residual surface delta interaction in the /beta+ decay of (_^"40" )"Sc" nucleus

In this paper the beta decay of Sc40 has been investigated. Using the eigenfunctions of the spherical oscillator the eigenvalues and eigenfunctions of the Woods-Saxon potential with spin-orbit and Coulomb terms have been calculated. In order to obtain the more realistic nuclear wave functions, the surface delta interaction potential has been used as the residual interaction between the valence nucleons in the frame work of Tamm-Dancoff approximation (TDA) in the case of Ca40 and particle-hole Tamm-Dancoff approximation (pnTDA) in the case of Sc40 . The potential constants have been obtained by fitting the results with the experimental data. The half-life and transition probability of positive beta decay of Sc40 have been calculated. Good greement with the experimental data was observed.

Study of energy spectrum of 116Sn and 116Te nuclei by using surface delta and modified surface delta interactions

In this paper, surface delta and modified surface delta interactions have been used by applying the nuclear shell model to calculate values of excitation energies for isotopes of equal mass number containing two nucleons outside the closed core 114Sn, these nuclei are; The isotope (Tin) 116Sn contains two neutrons within the model space (3s1/2, 2d3/2, 1h11/2) and the other isotope is 116Te (Tellurium) contains two protons within the model space (1g7/2, 2d5/2, 3s1/2, 2d3/2, 1h11/2). The current results of the above isotopes are based on using the Fortran 95 program code to program the above two interactions equations to calculate the values of single particles energies, matrix elements in the state of mixing configurations of orbits for and through which it is possible to calculate the eigenvalues, which are necessary to calculate the excited energy of all isotopes used in this study. By comparing the current theoretical results with the practical results, the expected theoretical results appeared to be in good agreement with the practical values, as well as confirmed and determined of the total angular momentum and parities of some unconfirmed and undetermined practical values, in addition to obtaining a new theoretical energy levels were determined of unspecified practical energies and momentum and this increases the theoretical knowledge of the studied isotopes relative to the energy levels.

Study of excitation energies and transmission potential of the reduced quadruple B (E2) for the 70Zn isotope by applying the nuclear shell model

In this paper, the excited energy levels and reduced electric quadrupole transition probabilities (BE2 ↓) in both pure and mixed configuration s of 70Zn nucleus have been calculated by applying the nuclear shell model. In the present study, the surface delta and modified surface delta interactions within the model space (1p3/2,0f5/2, 1p1/2 and 0g9/2) have been used for two protons outside the closed nuclear core represented by the 68Ni nucleus. The equations are programmed for the above two interactions and the reduced electric quadrupole transition probabilities (BE2 ↓) equations using the Fortran version 95. Through the used Fortran version 95 program code, the elements of the matrix have been calculated for two particles which are necessary to calculate the eigenvalues, the eigenvectors and then the final energies of the nucleus that has been used in the current study. Through the calculations, theoretical results that have well agreement with the available practical values have been obtained. The total angular momentum and parity of some uncertain and unspecified energy levels have been confirmed and determined by the total angular momentum and the practical parity. Also, new values have been identified for both the excited energy levels and the reduced electric quadrupole transition probabilities (BE2 ↓), these values are considered as a suggestion that increases theoretical knowledge of each of the energy levels and the expected transition probabilities during this study.

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.

Formula omitted] transitions of 40Sc using Tamm–Dancoff approximation

Within a developed particle-hole approach we have studied the β+/EC transitions of 40Sc → 40Ca. The wave functions of nuclear excited states have been calculated using the Tamm-Dancoff approximation. The mean field phenomenological potential introduced recently by Salamon and Vertse has been used to find the single particle states of the nucleons. The surface delta interaction has been adopted as the residual interaction between valance nucleons. The excited states energies of 40Sc and 40Ca, branching ratios, partial decay half-lives, total decay half-life, and logft12 values have been calculated and reasonable agreement with experimental data have been achieved.

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.