Full Shell Model Calculations Research Articles

Triple enhancement of quasi-SU(3) quadrupole collectivity in Strontium-Zirconium N ≈ Z isotopes

It is notable that along the N=Z line in the nuclear chart, extremely large collectivity emerges suddenly in the mass-80 region. By applying the Monte Carlo shell model (MCSM) and the Hartree-Fock-Bogolyubov plus generator coordinate method (HFB+gcm), we study this problem to find the origin. On the basis that both calculations reproduce the experimental data of the N≈Z nuclei with A=64∼88, we identify the backbone from full shell-model calculations that can explain the strong prolate deformation. We find that inclusion of the 2d5/2 orbit in the model space to cooperate with 1g9/2 is the key ingredient to describe the rapid increase of collectivity from 70Se to 76Sr and to produce the observed large B(E2) values in 76Sr, 78Sr and 80Zr. The quadrupole-quadrupole (QQ) interaction acting between the quasi-SU(3) partner orbits, 1g9/2−2d5/2, is the driving force that changes the nuclear shape from oblate- to prolate-deformed. We further suggest that the quasi-SU(3) effect is particularly amplified in the N≈Z nuclei because these are the unique examples where quasi-SU(3) partner orbits can be formed, like the nuclear pairing, simultaneously in three different types: neutron-neutron (n-n), proton-proton (p-p), and neutron-proton (n-p), which respectively interact through the n-n, p-p, and n-p components of the QQ force to enhance the quadrupole collectivity coherently.

Union of rotational and vibrational modes in generator-coordinate-type calculations, with application to neutrinoless double- β decay

Good many-body methods for medium and heavy nuclei are important. Here we combine ideas from standard generator-coordinate methods (GCM) and the so-called Monte Carlo shell model, and set forth a novel approach: starting from a mean-field solution (Hartree-Fock-Bogoliubov), we create a set of non-orthogonal basis states, by using low-lying vibrational quasiparticleTamm-Dancoff modes, and then project onto states of good angular momentum and particle number. The results we benchmark against full shell model calculations. Even with just a few such modes we find improvement over standard GCM calculations in excitation spectra. We also find significant improvement in $0\nu\beta\beta$ nuclear matrix elements.

Neutrinoless double- β decay of Sn124 , Te130 , and Xe136 in the Hamiltonian-based generator-coordinate method

We present a generator-coordinate method for realistic shell-model Hamiltonians that closely approximates the full shell model calculations of the matrix elements for the neutrinoless double-$\beta$ decay of $^{124}$Sn, $^{130}$Te, and $^{136}$Xe. We treat not only quadrupole deformations but also the proton-neutron pairing amplitudes as generator coordinates. We validate this method by calculating and comparing spectroscopic quantities with the exact shell model results and experimental data. Our Hamiltonian-based generator-coordinate method produces $0\nu\beta\beta$ matrix elements much closer to the shell model ones, compared to the existing energy-density-functional-based generator-coordinate approaches. The remaining overestimation of $0\nu\beta\beta$ nuclear matrix element suggests that additional correlations may be needed to be taken into account for $^{124}$Sn, $^{130}$Te, and $^{136}$Xe when calculating with the Hamiltonian-based generator-coordinate method. The validation of this method may open the possibility of calculating $0\nu\beta\beta$ matrix element of $^{150}$Nd in a large shell-model space.

New algorithm in the variation after projection calculations for non-yrast nuclear states

We present a novel and simple algorithm in the variation after projection (VAP) approach for the non-yrast nuclear states. It is for the first time that the yrast state and non-yrast states can be varied on the same footing. The orthogonality among the calculated states is automatically fulfilled by solving the Hill-Wheeler equation. This avoids the complexity of the frequently used Gram-Schmidt orthogonalization, as adopted by the excited VAMPIR method. Thanks to the Cauchy's interlacing theorem in the matrix theory, the sum of the calculated lowest projected energies with the same quantum numbers can be safely minimized. Once such minimization is converged, all the calculated energies and the corresponding states can be obtained, simultaneously. The present VAP calculations are performed with time-odd Hartree-Fock Slater determinants. It is shown that the calculated VAP energies (both yrast and non-yrast) are very close to the corresponding ones from the full shell model calculations. It looks the present algorithm is not limited to the VAP, but should be universal, i.e., one can do the variation with different forms of the many-body wavefunctions to calculate the excited states in different quantum many-body systems.

Neutrinoless double- β decay matrix elements in large shell-model spaces with the generator-coordinate method

We use the generator-coordinate method with realistic shell-model interactions to closely approximate full shell-model calculations of the matrix elements for the neutrinoless double-beta decay of $^{48}$Ca, $^{76}$Ge, and $^{82}$Se. We work in one major shell for the first isotope, in the $f_{5/2}pg_{9/2}$ space for the second and third, and finally in two major shells for all three. Our coordinates include not only the usual axial deformation parameter $\beta$, but also the triaxiality angle $\gamma$ and neutron-proton pairing amplitudes. In the smaller model spaces our matrix elements agree well with those of full shell-model diagonalization, suggesting that our Hamiltonian-based GCM captures most of the important valence-space correlations. In two major shells, where exact diagonalization is not currently possible, our matrix elements are only slightly different from those in a single shell.

Large-scale shell-model calculations on the spectroscopy ofN<126Pb isotopes

Large-scale shell-model calculations are carried out in the model space including neutron-hole orbitals $2p_{1/2}$, $1f_{5/2}$, $2p_{3/2}$, $0i_{13/2}$, $1f_{7/2}$ and $0h_{9/2}$ to study the structure and electromagnetic properties of neutron deficient Pb isotopes. An optimized effective interaction is used. Good agreement between full shell-model calculations and experimental data is obtained for the spherical states in isotopes $^{194-206}$Pb. The lighter isotopes are calculated with an importance-truncation approach constructed based on the monopole Hamiltonian. The full shell-model results also agree well with our generalized seniority and nucleon-pair-approximation truncation calculations. The deviations between theory and experiment concerning the excitation energies and electromagnetic properties of low-lying $0^+$ and $2^+$ excited states and isomeric states may provide a constraint on our understanding of nuclear deformation and intruder configuration in this region.

Variation after projection with a triaxially deformed nuclear mean field

We implemented a variation after projection (VAP) algorithm based on a triaxially deformed Hartree-Fock-Bogoliubov vacuum state. This is the first projected mean field study that includes all the quantum numbers (except parity), i.e., spin ($J$), isospin ($T$) and mass number ($A$). Systematic VAP calculations with $JTA$-projection have been performed for the even-even $sd$-shell nuclei with the USDB Hamiltonian. All the VAP ground state energies are within 500 keV above the exact shell model values. Our VAP calculations show that the spin projection has two important effects: (1) the spin projection is crucial in achieving good approximation of the full shell model calculation. (2) the intrinsic shapes of the VAP wavefunctions with spin projection are always triaxial, while the Hartree-Fock-Bogoliubov methods likely provide axial intrinsic shapes. Finally, our analysis suggests that one may not be possible to associate an intrinsic shape to an exact shell model wave function.

Quasi- SU(3) truncation scheme for odd–even and odd–odd sd-shell nuclei

The quasi- SU(3) symmetry, as found in shell-model calculations, refers to the dominance of the single-particle plus quadrupole–quadrupole terms in the Hamiltonian used to describe well-deformed nuclei, and to the subspace relevant in its diagonalization. It provides a very efficient basis truncation scheme. It is shown that a small number of SU(3) coupled irreps, those with the largest C 2 values within the direct product of the proton and neutron SU(3) irreps with spin 0 and 1 (for even number of particles), and spin 1/2 and 3/2 for (for odd number of nucleons), are enough to describe the low-energy spectra, B(E2) transition strengths and band structure of 21Ne, 23Na and 25Mg odd-mass and 22Na, 24Na, 26Al and 28Al odd–odd nuclei. A simple but realistic Hamiltonian is employed. Results compare favorably both with experimental data and with full shell-model calculations. Limitations and possible improvements of the schematic Hamiltonian are discussed.

Independent nucleon-pair motion and homologous structure in nuclei withA∼150

Shell model calculations are carried out for nuclei with $148<~A<~152$ assuming ${}_{64}^{146}{\mathrm{Gd}}_{82}$ as an inert core. A weak coupling scheme is used to analyze the experimental data and the calculated results. The essential role of the proton intruder state $\ensuremath{\pi}{h}_{11/2},$ the large energy gaps between the neutron single-particle states $\ensuremath{\nu}{f}_{7/2}$ and the others, and the weak proton-neutron interaction provide a simple and interesting picture for the spectra and wave functions, i.e., the independent nucleon-pair motion and the corresponding homologous structure. The proton pairs in $\ensuremath{\pi}{h}_{11/2},$ $\ensuremath{\pi}{d}_{3/2},$ and $\ensuremath{\pi}{s}_{1/2}$ orbitals and the neutron pair in $\ensuremath{\nu}{f}_{7/2}$ or $\ensuremath{\nu}{f}_{7/2}{h}_{9/2}$ orbitals in nuclei ${}_{68}^{150,152}{\mathrm{Er}}_{82,84}$ and ${}_{66}^{150}{\mathrm{Dy}}_{84}$ are moving almost independently in the mean field provided by the core. Therefore, the yrast states below 5.3 MeV of these nuclei can be finely explained as the compositions of the excitations of these independent nucleon pairs. The negative parity levels of nuclei ${}_{66}^{149}{\mathrm{Dy}}_{83},$ ${}_{65}^{149}{\mathrm{Tb}}_{84},$ ${}_{67}^{151}{\mathrm{Ho}}_{84},$ and ${}_{69}^{153}{\mathrm{Tm}}_{84}$ can be well interpreted in terms of the concept of homologous states, i.e., the level structures in nuclei ${}_{66}^{149}{\mathrm{Dy}}_{83},$ ${}_{65}^{149}{\mathrm{Tb}}_{84},$ ${}_{67}^{151}{\mathrm{Ho}}_{84},$ and ${}_{69}^{153}{\mathrm{Tm}}_{84}$ are homologous to those of the parent structure in nuclei ${}_{66}^{148}{\mathrm{Dy}}_{82},$ ${}_{64}^{148}{\mathrm{Gd}}_{84},$ ${}_{66}^{150}{\mathrm{Dy}}_{84},$ and ${}_{68}^{152}{\mathrm{Er}}_{84},$ respectively. The level structure of the nucleus ${}_{68}^{151}{\mathrm{Er}}_{83},$ homologous to that of parent states in the nucleus ${}_{68}^{150}{\mathrm{Er}}_{82},$ is predicted and quite impressive in our full shell model calculation. This desires experiments to verify. The conditions for the onset of the independent nucleon-pair motion and the homologous structure are discussed in detail.

The effects of the spin-orbit and tensor interactions in nuclei

We devise a nucleon-nucleon interaction V c + xV so + yV t to study the effects of the spin-orbit and tensor interactions by varying x and y. For negative parity 1 p−1 h states in 16O, we find that in a TDA calculation, the weighted sum of the eigenenergies Σ J (2 J + 1) Σ i E i ( J, T) is zero for a pure two-body spin-orbit interaction or a pure tensor interaction. If a minor closed shell nucleus like 12C is taken as the core, the tensor interaction lowers the p 1 2 single-particle energy relative to p 3 2 . This is precisely opposite to what the two-body spin-orbit interaction does. For the 1 + states in 12C, a 1 p−1 h diagonalization leads to a near collapse with E ∗(T = 0) ∼ 0.9 MeV and E ∗(T = 1) ∼ 3.7 MeV as compared with experiment (12.7 MeV for T = 0 and 15.1 MeV for T = 1). Only a full scalar shell model calculation brings these states up to a respectable energy. We find that in a full shell model calculation, the excitation energies of these 1 + states are surprisingly insensitive to the spin-orbit interaction over a rather wide range of the parameter x and there does not appear to be any phase transition as we vary x. We reconsider the old problem of the effect of the spin-orbit and tensor interactions on the nearly vanishing Gamow-Teller matrix element 14C( J = 0, T = 1) → 14N( J = 1, T = 0). We calculate the single-particle energies with the same interaction that is used for the residual particle-particle or particle-hole matrix elements.

Comparisons between shell-model calculations, seniority truncation, and quasiparticle approximations: Application to the odd Ni isotopes and odd N=82 isotones.

A detailed study of the results of correcting BCS approximation for the effects of particle-number projection and blocking has been carried out. A low-seniority shell-model approximation was used as the frame of reference for investigating the mixing of one- and three-quasiparticle states in odd-mass Ni isotopes and in odd-mass N=82 isotones. We discuss the results obtained for the energy spectra and electromagnetic decay properties. Effects of seniority-five configurations on the low-lying states have also been studied through the comparison of the low-seniority shell-model results with those which arose from the corresponding full shell-model calculations.

The quasiparticle rpa as a model for gamow-teller ß-decay

The validity of the quasiparticle random-phase approximation is studied by comparing it with full shell-model calculations in the sd-shell. Phenomenological interactions relevant for the sd-shell have been applied, and Gamow-Teller ß + strengths are calculated. It is found that in the QRPA, the ß + strength is a factor of two higher than the shell-model result. The shell-model results are less sensitive to the particle-particle interaction strength than the QRPA. With u, v factors from the shell-model wavefunctions, the ß + strength in the QRPA reproduces the shell-model result. This suggests that the QRPA does not contain enough ground-state correlations.

Low-lying Negative-parity Levels of 17N and 18N

On the basis of a weak-coupling model, adjustments are made to the interactions used in the full shell model calculations of Millener in order to fit the experimental energies of the low-lying negativeparity levels of 16N and of the low-lying positive-parity levels of 180 and 190 . The predicted energies of the low-lying negative-parity levels of 17N then agree better with experiment, while those for 18N lead to suggested spin assignments for the observed levels.

Truncation of the nuclear shell-model space using spectral strength distribution

A procedure, based on spectral distribution methods, is introduced to truncate a large shell-model matrix and to renormalize the matrix for the effects of the truncation. The basis states are selected depending on their contribution in the ground-state region and the renormalization is achieved by requiring that the strength distributions of each configuration in the truncated space to have the same widths as those in the full space. The calculation starts with a given effective Hamiltonian in the original space without having to introduce any free parameters due to truncation. Tests against full shell-model calculations in the ds shell are performed for energy levels of 28Si and for electromagnetic transitions in the (ds) 6 T = 0 space.