Semirealistic shell-model interaction for neutron hole states inPb208

We perform 2\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta} and 0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta} calculations for $^{76}\mathrm{Ge}$, $^{128}\mathrm{Te}$, and $^{130}\mathrm{Te}$ using effective interactions derived from the Paris and Bonn-A potentials. Extended model spaces are employed in setting up the quasiparticle random phase approximation (QRPA) equations, on which our present calculations are based. For $^{76}\mathrm{Ge}$ the model space consists of nine orbits, the two major shells from 0${\mathit{f}}_{7/2}$ to 2${\mathit{s}}_{1/2}$, and for the tellurium case we include eleven orbits spanning three major shells from 1${\mathit{p}}_{3/2}$ to 1${\mathit{f}}_{7/2}$. The bare-G-matrix elements are first calculated, with the Pauli exclusion operator carefully treated with a matrix inversion method, so that double counting between the calculated effective interaction and the above model spaces is strictly avoided. We then calculate the renormalized effective interaction, including corrections from core polarizations and folded diagrams. The effect of core polarization is found to be highly significant, especially for $^{76}\mathrm{Ge}$. There appears to be a compensating effect from the folded diagrams; the net results with core polarizations and folded diagrams both included become rather close to the bare-G results. Unlike earlier QRPA calculations, our calculated ${\mathit{M}}_{\mathrm{GT}}$ matrix elements for 2\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta} do not seem to exhibit strong dependence on ${\mathit{g}}_{\mathrm{pp}}$, the particle-particle interaction strength parameter, in the vicinity of ${\mathit{g}}_{\mathrm{pp}}$=1.0.For 0\ensuremath{\nu}\ensuremath{\beta}\ensuremath{\beta} decays, our calculated values for ${\mathit{T}}_{1/2}^{0\ensuremath{\nu}}$〈${\mathit{m}}_{\ensuremath{\nu}}$${\mathrm{〉}}^{2}$ are typically 5\ifmmode\times\else\texttimes\fi{}${10}^{23}$ yr ${\mathrm{eV}}^{2}$ for $^{76}\mathrm{Ge}$, 2\ifmmode\times\else\texttimes\fi{}${10}^{24}$ yr ${\mathrm{eV}}^{2}$ for $^{128}\mathrm{Te}$, and 9\ifmmode\times\else\texttimes\fi{}${10}^{22}$ yr ${\mathrm{eV}}^{2}$ for $^{130}\mathrm{Te}$. Although the Bonn-A potential gives generally more pairing force, the final results for \ensuremath{\beta}\ensuremath{\beta} decays given by the Paris and Bonn-A potentials are rather close to each other.

Shell model calculations based on large basis has been conducted to study the nuclear structure of $^{20}Ne$, $^{22}Ne$ and $^{24}Mg$ nuclei. The energy levels, inelastic electron scattering form factors and transition probabilities are discussed by considering the contribution of configurations with high-energy beyond the model space of sd-shell model space which is denoted as the core polarization (CP) effects.~The Core polarization is considered by taking the excitations of nucleus from the $1s$ and $1p$ core orbits and also from the valence $2s$ $1d$ shell orbit in to higher shells with $4\hbar\omega$. The effective interactions $USDA$ and $USDB$ are employed with $sd$ shell model space to perform the calculation and the core polarization are calculated with $MSDI$ as residual interaction.~The calculated energy level schemes, form factors and transition probabilities were compared with the corresponding experimental data. The effect of core polarization is found very important for the calculation of $B(C2)$, $B(C4)$ and form factors, and gives excellent results in comparison with the experimental data without including any adjustable parameters.

The second order pseudopotential theory suggests the possibility of a break-up of the total energy of simple metals into a purely volume-dependent part and an effective central pairwise interaction between ions. In the present paper finite contributions for these two parts of the energy have been extracted in a form convenient for calculation. Using the local Heine-Abarenkov model potential, a reliable effective ion-ion interaction is generated and the volume-dependent energy is calculated for Al. The relative contributions of the effective interaction and the volume-dependent energy term to various metallic properties are also calculated. The importance of volume dependence on the effective interaction is also discussed. The interactomic force constants upto eighth neighbour are derived from the effective interaction and it is found that the force constants beyond the third neighbour are negligibly small. This result is also confirmed by the calculation of dispersion curves with force constants obtained from the effective interaction upto the third neighbour which is found to reproduce the results of the full pseudopotential calculations. The force constants obtained are also used to study some finite temperature properties of Al in the quasi-harmonic approximation and the limitations of the theory are pointed out.

Realistic microscopic calculations for the light nuclei A = 2 to 7

Higher order core polarization processes and effective interactions in 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.

An equation of motion method for deriving microscopic effective interactions in a correlated model space

Constrained spherical Hartree-Fock (CSHF) calculations under radial compression are presented for 90Zr in a model space consisting of nine major oscillator shells. An effective baryon-baryon interaction which includes the Δ resonances is used. The nucleon-nucleon (N-N) interaction is Reid Soft Core (RSC) potential. The sensitivity of the results to the choice model space is examined. It is found that the nuclear system becomes more compressible when the model space is increased. The radial density and the number of Δs are decreased by increasing model space. The results suggest that the behavior of single particle energies is independent of the model space.

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.

Spectral distribution methods in effective interaction theory

Model-space effective interactions Veff derived from free-space nucleon–nucleon interactions VNN are reviewed. We employ a double decimation approach: first we extract a low-momentum interaction from VNN using a T-matrix equivalence decimation method. Then Veff is obtained from by way of a folded-diagram effective interaction method. For decimation momentum , the interactions derived from different realistic VNN models are nearly model independent, and so are the resulting shell-model effective interactions. For nucleons in a low-density nuclear medium like valence nucleons near the nuclear surface, such effective interactions derived from free-space VNN are satisfactory in reproducing experimental nuclear properties. But it is not so for nucleons in a nuclear medium with density near or beyond nuclear matter saturation density. In this case it may be necessary to include the effects from Brown–Rho (BR) scaling of hadrons and/or three-nucleon forces V3N, effectively changing the free-space VNN into a density-dependent one. The density-dependent effects from BR scaling and V3N are compared with those from empirical Skyrme effective interactions.

The in-medium feature of nuclear force, which includes both nucleon-nucleon () and hyperon-nucleon () interactions, impacts the description of single-Λ hypernuclei. With the alternated mass number or isospin of hypernuclei, such effects may be unveiled by analyzing the systematic evolution of the bulk and single-particle properties. From a density-dependent meson-nucleon/hyperon coupling perspective, a new effective interaction in the covariant density functional (CDF) theory, namely, DD-LZ1-, is obtained by fitting the experimental data of Λ separation energies for several single-Λ hypernuclei. It is then used to study the structure and transition properties of single-Λ hypernuclei in oxygen hyperisotopes, in comparison with those determined using several selected CDF Lagrangians. A discrepancy is explicitly observed in the isospin evolution of spin-orbit splitting with various effective interactions, which is attributed to the divergence of the meson-hyperon coupling strengths with increasing density. In particular, the density-dependent CDFs introduce an extra contribution to reduce the value but enhance the isospin dependence of the splitting, which originates from the rearrangement terms of Λ self-energies. In addition, the characteristics of hypernuclear radii are studied along the isotopic chain. Owing to the impurity effect of the Λ hyperon, a size shrinkage is observed in the matter radii of hypernuclei compared with the cores of normal nuclei, and its magnitude is further elucidated to correlate with the incompressibility of nuclear matter. Moreover, there is a sizable model-dependent trend in which the Λ hyperon radii evolve with neutron number, which is decided partly by the in-medium interactions and core polarization effects.

The configuration interaction (CI) approach to solving the nuclear many-body problem, also known as the interacting shell model, has proven to be powerful tool in understanding the structure of nuclei. The principal criticism of past applications of the shell model is the reliance on empirical tuning to interaction matrix elements. If an accurate description of nuclei far from the valley of stability, where little or no data is available, a more fundamental approach is needed. This starts with recent ab initio approaches with effective interactions in the no-core shell model (NCSM). Using effective-field theory for guidance, fully ab initio descriptions of nuclei up to 16O with QCD based NN, NNN, and NNNN interactions will be possible within the next five years. An important task is then to determine how to use these NCSM results to develop effective interactions to describe heavier nuclei without the need to resort to an empirical retuning with every model space. Thus, it is likely that more traditional CI applications utilizing direct diagonalization and more fundamental interactions will be applicable to nuclei with perhaps up to one hundred constituents. But, these direct diagonalization CI applications will always be computationally limited due to the rapid increase in the number of configurations with particle number. Very recently, the shifted-contour method has been applied to the Auxiliary-field Monte Carlo approach to the Shell Model (AFMCSM), and preliminary applications exhibit a remarkable taming of the notorious sign problem. If the mitigation of the sign problem holds true, the AFMCSM will offer a method to compute quantum correlations to mean-field applications for just about all nuclei; giving exact results for CI model spaces that can approach 1020-25. In these lectures, I will discuss modern applications of CI to the nuclear many-body problem that have the potential to guide nuclear structure theory into the next decade.

In many shell model interactions, the tensor force monopole matrix elements often retain systematic trends originating in the bare tensor force. However, in this work, we find that Isospin [Formula: see text] tensor force monopole matrix elements of widely used p-shell effective interaction CK(8–16) do not share these systematic. We correct these discrepancies by modifying [Formula: see text] tensor force two-body matrix elements (TBMEs) of CK(8–16) by the analytically calculated tensor force TBMEs. With some additional modification of single-particle energies and TBMEs, the revised effective interaction is named as CKN. The effective interaction CKN has been tested for the calculations of p-shell nuclei of normal parity states from various physics viewpoints such as excitation spectra, electromagnetic moments, and electromagnetic and Gamow–Teller (GT) transitions. The obtained results are found to be satisfactory with respect to the experimental results.

In the present work, the inelastic electron scattering for longitudinal and transverse form factors of 65Cu and 71Ga nuclei lies in the fp-shell region are studied in the framework of the shell model. The calculation is performed in the (1f5/2, 2p3/2, 2p1/2, 1g9/2) model space using jun45 effective interaction. The wavefunctions employed to conduct the shell model calculations are extracted from the jun45 effective interaction for these nuclei with the jj44 shell model space and (Sk35−Skzs ∗) residual interaction to evaluate the interactions matrix element between initial and final states. The effective charges used to account for the core-polarization (CP) effect are created using calculations of microscopic perturbations that include intermediate one-particle, one-hole excitation from the core and the model space (MS) orbits into all upper orbits with \(n\hslash \omega\) excitations following the same approach done in [Radhi et al. in Euro. Phys. J. A 50:1–9, 2014]. To account for the (CP) effects contribution, the inelastic form factor is obtained by employing the shape of Tassie and Bohr–Mottelson models with appropriate proton and neutron effective charges. The calculated form factors were compared with available experimental data.