Shell-model Hamiltonian Research Articles

Double-step truncation procedure for large-scale shell-model calculations

We present a procedure that is helpful to reduce the computational complexity of large-scale shell-model calculations, by preserving as much as possible the role of the rejected degrees of freedom in an effective approach. Our truncation is driven first by the analysis of the effective single-particle energies of the original large-scale shell-model hamiltonian, so to locate the relevant degrees of freedom to describe a class of isotopes or isotones, namely the single-particle orbitals that will constitute a new truncated model space. The second step is to perform an unitary transformation of the original hamiltonian from its model space into the truncated one. This transformation generates a new shell-model hamiltonian, defined in a smaller model space, that retains effectively the role of the excluded single-particle orbitals. As an application of this procedure, we have chosen a realistic shell-model hamiltonian defined in a large model space, set up by seven and five proton and neutron single-particle orbitals outside 88Sr, respectively. We study the dependence of shell-model results upon different truncations of the original model space for the Zr, Mo, Ru, Pd, Cd, and Sn isotopic chains, showing the reliability of this truncation procedure.

Nuclear level density: Shell-model approach

The knowledge of the nuclear level density is necessary for understanding various reactions including those in the stellar environment. Usually the combinatorics of Fermi-gas plus pairing is used for finding the level density. Recently a practical algorithm avoiding diagonalization of huge matrices was developed for calculating the density of many-body nuclear energy levels with certain quantum numbers for a full shell-model Hamiltonian. The underlying physics is that of quantum chaos and intrinsic thermalization in a closed system of interacting particles. We briefly explain this algorithm and, when possible, demonstrate the agreement of the results with those derived from exact diagonalization. The resulting level density is much smoother than that coming from the conventional mean-field combinatorics. We study the role of various components of residual interactions in the process of thermalization, stressing the influence of incoherent collision-like processes. The shell-model results for the traditionally used parameters are also compared with standard phenomenological approaches.

Nuclear clustering using a modern shell model approach

Nuclear clustering, alpha decays, and multiparticle correlations are important components of nuclear dynamics. In this work we put forward the cluster-nucleon configuration interaction model. We use the modern configuration interaction approach with advanced realistic shell-model Hamiltonians in order to study clustering phenomena; the study is facilitated by the algebraic properties of many-nucleon configurations in the harmonic oscillator basis. Using a translationally invariant formalism we built cluster channels that satisfy the Pauli exclusion principle as well as orthogonality and normalization conditions. We formally justify the formalism and demonstrate that, within the new method, clustering strengths satisfy sum rules that are consistent with the statistical properties of nuclear reactions with composite particles. Using properly renormalized cluster form factors our approach appears to resolve long-standing problems related to absolute normalization of the alpha clustering spectroscopic factors and their behavior in $sd$-shell nuclei. Our methods are demonstrated in studies of $\ensuremath{\alpha}$ spectroscopic factors in $sd$-shell nuclei and in $^{16}\mathrm{O}$ treated in $p\ensuremath{-}sd$ shells. Comparison with experimental data supports the validity of the approach.

Spin Responses in Nuclei and Nuclear Weak Processes in Stars

New shell-model Hamiltonians which can successfuly describe spin-dependent transition rates in nuclei are applied to study nuclear weak processes in stars. New v-induced reaction cross sections in 12C and 56Fe evaluated by the new Hamiltonians are shown to reproduce well the experimental data. Nucleosynthesis of light elements in supernova explosions (SNe) as well as v oscillation effects are discussed with the new cross sections. Electron capture and β-decay rates in stellar environments are re-evaluated in fp- and sd-shell nuclei. Nucleosynthesis in Type-Ia SNe, rp-process and X-ray burst are discussed with the new reaction rates in Ni isotopes. Important roles of accurate e-capture and β-decay rates in sd-shell nuclei on the cooling of stars with 8-10 solar masses by nuclear URCA processes and the fate of the stars are demonstrated. β-decay half-lives of waiting-point nuclei at N =126 are evaluated by shell- model calculations, and r-process nucleosynthesis up to Th and U region in both core-collapse SNe and binary neutron star mergers are studied.

Shell-model study of spin modes in nuclei and nuclear forces

Spin-dependent modes in nuclei are studied by shell-model method with the use of new shell-model Hamiltonians which properly take into account important roles of tensor interactions. New Hamiltonians can describe spin degrees of freedom in nuclei remarkably well. Nuclear weak processes at stellar environments are investigated based on these successes. New neutrino-nucleus reaction cross sections on 12C are applied to light-element synthesis in supernova explosions. The production rate for 11B/7Li is pointed out to be useful to determine v-oscillation parameters, in particular, v-mass hierarchy. New e-capture rates in Ni isotopes are obtained and implications for element synthesis are discussed. The monopole-based universal interaction is applied to study structure of p-sd shell nuclei and 40Ar as well as v-induced reactions on 40Ar. Repulsive corrections in the isospin T=1 monopoles are shown to be important for proper shell evolutions in neutron-rich carbon isotopes. The repulsive correction is pointed out to be due to three-body forces, in particular, the Fujita-Miyazawa force. Roles of the three-body forces on the shell evolution of neutron-rich calcium isotopes, the closed- shell nature of 48 Ca and M1 transition in 48 Ca are studied on top of the two-body G-matrix obtained by including core-polarization effects in larger spaces (≤24ℏω). Effects of the inclusion of g9/2-shell are also discussed.

Mirror energy difference and the structure of loosely bound proton-rich nuclei aroundA=20

The properties of loosely bound proton-rich nuclei around A = 20 are investigated within the framework of nuclear shell model. In these nuclei, the strength of the effective interactions involving the loosely bound proton s1=2 orbit are significantly reduced in comparison with those in their mirror nuclei. We evaluate the reduction of the effective interaction by calculating the monopole-baseduniversal interaction (VMU) in the Woods-Saxon basis. The shell-model Hamiltonian in the sd shell, such as USD, can thus be modified to reproduce the binding energies and energy levels of the weakly bound proton-rich nuclei around A = 20. The effect of the reduction of the effective interaction on the structure and decay properties of these nuclei is also discussed.

Precise comparison of the Gaussian expansion method and the Gamow shell model

We perform a detailed comparison of results of the Gamow Shell Model (GSM) and the Gaussian Expansion Method (GEM) supplemented by the complex scaling (CS) method for the same translationally-invariant cluster-orbital shell model (COSM) Hamiltonian. As a benchmark test, we calculate the ground state $0^{+}$ and the first excited state $2^{+}$ of mirror nuclei $^{6}$He and $^{6}$Be in the model space consisting of two valence nucleons in $p$-shell outside of a $^{4}$He core. We find a good overall agreement of results obtained in these two different approaches, also for many-body resonances.

Isospin Symmetry Violation insd-Shell Nuclei

We construct a new set of isospin-nonconserving (INC) shell-model Hamiltonians and apply it (i) to study the isobaric-mass multiplet equation (IMME) beyond a quadratic form in the A = 35, 3/2 + quartet, (ii) to compute the isospin mixing corrections to superallowed Fermi branch of 1 + → 1 + β decay of 32 Cl, and (iii) to obtain the isospin-forbidden branching ratios of β -delayed proton emission from 25 Si.

New neutrino-nucleus reaction cross sections at solar, reactor and supernova neutrino energies

Remarkable improvements in the evaluation of neutrino-nucleus reaction cross sections are obtained based on new shell-model Hamiltonians with proper tensor compo- nents. New �-induced reaction cross sections on 12 C, 13 C, 56 Fe, 56 Ni and 40 Ar are pre- sented, and predictions for nucleosynthesis in supernova explosions, �-oscillation effects and low-energy reactor and solar neutrino detection are discussed based on these new cross sections.

Split Isobaric Analog State inNi55: Case of Strong Isospin Mixing

Study of β+ decay of the exotic Tz=-3/2 nucleus 55Cu, via delayed γ rays, has revealed a strongly isospin mixed doublet (4599-4579 keV) in 55Ni, which represents the fragmented and previously unknown isobaric analog of the ground state of 55Cu. The observed small log ft values to both states in the doublet confirm the superallowed Fermi β decay. The near degeneracy of a pair of 3/2- levels in 55Ni results in the strong isospin mixing. The isospin mixing matrix element between the T=3/2 and T=1/2 levels is inferred from the experiment to be 9(1) keV, which agrees well with the matrix element of the charge symmetry breaking shell model Hamiltonian of Ormand and Brown. A precise value of the half-life of 55Cu at 57(3) ms was also obtained.

Isospin nonconservation insd-shell nuclei

The question of isospin-symmetry breaking in nuclei of the $sd$ shell is addressed. We propose a new global parametrization of the isospin-nonconserving shell-model Hamiltonian which accurately describes experimentally known isobaric mass splittings. The isospin-symmetry violating part of the Hamiltonian consists of the Coulomb interaction and effective charge-dependent forces of nuclear origin. Particular attention has been paid to the effect of the short-range correlations. The behavior of $b$ and $c$ coefficients of the isobaric-mass-multiplet equation (IMME) is explored in detail. In particular, a high-precision numerical description of the staggering effect is proposed and contribution of the charge-dependent forces to the nuclear pairing is discussed. The Hamiltonian is applied to the study of the IMME beyond a quadratic form in the $A=32$ quintet, as well as to calculation of nuclear structure corrections to superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}$ Fermi $\ensuremath{\beta}$ decay and to amplitudes of Fermi transitions to nonanalog states in $sd$-shell nuclei.

Two-proton removal from44S and the structure of42Si

Newly-published $^{42}$Si gamma-ray spectra and a final-state-inclusive $^{42}$Si production cross section value, obtained in a higher-statistics intermediate-energy two-proton removal experiment from $^{44}$S, are considered in terms of the final-state-exclusive cross sections computed using proposed shell-model effective interactions for nuclei near N=28. Specifically, we give cross section predictions when using the two nucleon amplitudes of the two-proton overlaps $<^{42}$Si$ (J^\pi)\,|\,^{44}$S$>$ computed using the newly-proposed {\sc sdpf-mu} shell-model Hamiltonian. We show that these partial cross sections or their longitudinal momentum distributions should enable a less-tentative interpretation of the measured gamma-ray spectra and provide a more quantitative assessment of proposed shell-model Hamiltonians in this interesting and challenging region of the chart of nuclides.

Exotic nuclei and nuclear forces

I overview new aspects of the structure of exotic nuclei as compared to stable nuclei, focusing on several characteristic effects of nuclear forces. The shell structure of nuclei has been proposed by Mayer and Jensen, and has been considered to be kept valid basically for all nuclei, with well-known magic numbers, 2, 8, 20, 28, 50, …. Nuclear forces were shown, very recently, to change this paradigm. It will be presented that the evolution of shell structure occurs in various ways as more neutrons and/or protons are added, and I will present basic points of this shell evolution in terms of the monopole interaction of nuclear forces. I will discuss three types of nuclear forces. The first one is the tensor force. The tensor force is one of the most fundamental nuclear forces, but its first-order effect on the shell structure has been clarified only recently in studies on exotic nuclei. The tensor force can change the spin–orbit splitting depending on the occupation of specific orbits. This results in changes of the shell structure in many nuclei, and consequently some of Mayer–Jensen's magic numbers are lost and new ones emerge, in certain nuclei. This mechanism can be understood in an intuitive way, meaning that the effect is general and robust. The second type of nuclear forces is central force. I will show a general but unknown property of the central force in the shell-model Hamiltonian that can describe nuclear properties in a good agreement with experiment. I will then demonstrate how it can be incorporated into a simple model of the central force, and will discuss how this force works in the shell evolution. Actually, by combining this central force with the tensor force, one can understand and foresee how the same proton–neutron interaction drives the shell evolution, for examples such as Sn/Sb isotopes, N = 20 nuclei and Ni/Cu isotopes. The distribution of single-particle strength is discussed also in comparison to (e,e′p) experiment on 48Ca. The shell evolution affects shapes of nuclei through Jahn–Teller-type mechanism, and a very interesting example with exotic Si isotopes is discussed. The third type of nuclear force is a three-body force, which originates in the Δ particle excitation as proposed by Fujita and Miyazawa many years ago. This force is shown to produce a repulsive interaction between valence neutrons after averaging effects from the third nucleon in the core. The same three-body force is responsible for neutron stars. By including such effects of the three-body force, one can predict the correct drip line of oxygen isotopes, for instance. Thus, the landscape of atomic nuclei varies in going from stable to exotic nuclei due to particular nuclear forces, leading to a paradigm shift. This paper overviews some basic ideas and selected examples.

Renormalizing the double-beta operator for the shell model

We review work, much of it by the author and collaborators, to take omitted levels into account in shell-model calculations of neutrinoless double-beta decay. Previous work on this type of problem deals mainly with corrections to the shell-model Hamiltonian. Here we focus on corrections to the two-body transition operator that governs the rate of neutrinoless decay. Corrections in the form of two-body effective operators appear to be significant but have yet to be fully pinned down. Three-body effective operators may be significant as well.

Nuclear Weak Processes and Astrophysical Applications

Nuclear weak processes are investigated based on new shell model Hamiltonians, which give successful description of spin responses in nuclei, and applied to astrophysical problems. Neutrino-induced reactions on 12C and synthesis of light elements by supernova neutrinos, and effects of contamination of 13C, whose natural isotopic abundance is 1.1%, on inclusive ν-12C reactions are discussed. Spin-dipole transitions and ν-induced reactions on 16O are studied by using a new Hamiltonian with proper tensor components, and compared with conventional calculations and previous CRPA results. Gamow-Teller transition strength in 40Ar and ν-induced reactions on 40Ar by solar neutrinos are studied based on monopole-based-universal interaction (VMU). We finally discuss electron capture reactions on Ni isotopes in stellar environments.

Scale Invariance, Self Similarity and Critical Behavior in Classical and Quantum Systems

Symmetry and self-affinity or scale invariance are related concepts. We explore the fractal properties of fluctuations in dynamical systems, using some of the available tools in the context of time series analysis. We carry out a power spectrum study in the Fourier domain, the method of detrended fluctuation analysis and the investigation of autocorrelation function behavior. Our study focuses on two particular examples, the logistic module-1 map, which displays properties of classical dynamical systems, and the excitation spectrum of a schematic shell-model Hamiltonian, which is a simple system exhibiting quantum chaos.

Validity of the generalized density matrix method for the microscopic calculation of a collective/bosonic Hamiltonian

Recently a procedure by generalized density matrix (GDM) is proposed for calculating a collective/bosonic Hamiltonian microscopically from the shell-model Hamiltonian. In this work we examine the validity of the method by comparing the GDM results with that of the exact shell-model diagonalization in a number of models. It is shown that the GDM method reproduces the low-lying collective states quite well, both for energies and transition rates, across the whole region going from vibrational to gamma-unstable and deformed nuclei.

Effective shell-model hamiltonians from realistic nucleon–nucleon potentials within a perturbative approach

This paper discusses the derivation of an effective shell-model hamiltonian starting from a realistic nucleon–nucleon potential by way of perturbation theory. More precisely, we present the state of the art of this approach when the starting point is the perturbative expansion of the Qˆ-box vertex function. Questions arising from diagrammatics, intermediate-states and order-by-order convergences, and their dependence on the chosen nucleon–nucleon potential, are discussed in detail, and the results of numerical applications for the p-shell model space starting from chiral next-to-next-to-next-to-leading order potentials are shown. Moreover, an alternative graphical method to derive the effective hamiltonian, based on the Zˆ-box vertex function recently introduced by Suzuki et al., is applied to the case of a non-degenerate (0+2)ħω model space. Finally, our shell-model results are compared with the exact ones obtained from no-core shell-model calculations.

Validity of pair approximations for nuclei in open shells

We study the validity of pair approximations of the nuclear shell model (SM) for a few realistic nuclei (${}^{130\ensuremath{-}131}$Te and ${}^{132}$I here) in open shells (i.e., nuclei with both valence protons and valence neutrons). We take a phenomenological shell model Hamiltonian, which includes the single-particle energy term, monopole and quadrupole pairing interactions for like particles, and quadrupole-quadrupole interactions between all valence nucleons. We make comparisons of energy levels and $B(E2)$ values calculated by pair approximations with those in the SM calculations. The overlaps between wave functions of the nucleon pair approximation (NPA) and those of the SM are presented explicitly for the low-lying states. Our calculated results demonstrate that the NPA is a remarkable approximation of the SM for these nuclei. The important role played by pairs with negative parity in some low-lying states is discussed.

Gamow-Teller and First-Forbidden Transition Strengths in Astrophysical Processes

Gamow-Teller (GT) and first-forbidden (FF) transitions in nuclei and their possible consequences on astrophysical processes in stars are discussed. GT transition strengths in fp-shell nuclei are studied by shell model calculations with the use of new shell model Hamiltonians, GXPF1. Neutrino-nucleus reaction cross sections are re-evaluated and compared with previous investigations. Proton emission cross sections on 56Ni are pointed out to be enhanced and lead to the enhancement of the production yields of 55Mn and 59Co in population III supernovae. Next, electron capture reactions on Ni and Fe isotopes in steller environments are studied. The capture rates depend sensitively on the distributions of the GT strengths. Capture rates on 58Ni and 60Ni obtained by GXPF1 are found to reproduce rather well those obtained from experimental GT strengths. Finally, beta decays of the N = 126 isotones are studied by shell model calculations taking into account both the GT and FF transitions. The FF transitions are found to be important to reduce the half-lives by twice to several times of those by the GT contributions only. Possible implications on the short half-lives of the waiting point nuclei on the r-process nucleosynthesis are discussed for certain astrophysical conditions.