Nucleon-pair Approximation Research Articles

Investigating the level structure of 206Rn: evolution from seniority regime to collective dynamics* *Supported partially by the National Natural Science Foundation of China(10975191, U1867210, 11375023, 11475014, 11575018, 12075169 , 12322506), and the National Key R&D program of China (2016YFA0400504)

Excited states of have been examined via the ( , 5n) fusion reaction at a beam energy of 78 MeV. A number of transitions and levels are identified by the γ-γ coincidence measurement, further enriching the level scheme of . The full configuration shell model and nucleon-pair approximation (NPA) were utilized to investigate the single-particle configurations and seniority structures in . The results of these two calculations suggest that and states exhibit only a 50% component of a seniority-two state associated with a broken neutron pair. The collectivity of these two states primarily arises from configuration mixing due to residual proton-neutron interactions. Furthermore, and states are predominantly characterized by a seniority-two state marked by a broken proton pair.

Toward optimizing the basis states in the nucleon-pair approximation: Variational factorization

Toward optimizing the basis states in the nucleon-pair approximation: Variational factorization

Recent progress in configuration–interaction shell model

Since Mayer and Jensen employed the single-particle shell model to interpret the magic numbers, various microscopic nuclear models have been developed to study the nuclear force and structure. The configuration–interaction shell model (CISM), performed in truncated model space with the inclusion of the residual interaction, is one widely-used nuclear structure model. In the last decade, CISM has progressed in investigating the cross-shell excitation in exotic light nuclei, the similarity and difference in mirror nuclei, and the isomerism and seniority conservation in medium and heavy nuclei. Additionally, researchers have attempted to construct effective Hamiltonians for nuclei near [Formula: see text]Sn and [Formula: see text]Pb through a unified way in the CISM framework. In parallel, related models, including the nucleon-pair approximation (NPA) approach, the Monte Carlo shell model (MCSM), the projected shell model (PSM), the Gamow shell model (GSM), etc., have also been extensively developed and validated in the last decade. This paper reviews the recent progress in CISM and some related models.

State-of-the-art nucleon-pair approximation to the nuclear shell model

State-of-the-art nucleon-pair approximation to the nuclear shell model

β -decay half-lives of the r -process waiting-point isotones of N=81 and 82 nuclei

In this paper we perform systematic calculations of $\ensuremath{\beta}$-decay half-lives for the $r$-process waiting-point isotones of $N=81$ and 82, in the framework of nucleon-pair approximation. A phenomenological form of the Hamiltonian, which includes single-particle energies, pairing plus quadrupole interactions, and Gamow-Teller interactions is assumed, and the theoretical $\ensuremath{\beta}$-decay energies are taken based on the Bayesian machine learning. A good agreement with experimental values of half-lives is achieved by using the nucleon-pair truncated model space in which the $S$-pair condensation dominates. Our calculated half-lives are found to be very sensitive to $\ensuremath{\beta}$-decay energies on the one hand, and are quite robust with respect to extension of the model space and variation of interaction parameters on the other hand.

Robustness of pair structures for nuclear yrast states* *Supported by the National Natural Science Foundation of China (11875134, 11875188, 12175071, 11975151, 11961141003), and the Shanghai Key Laboratory of Particle Physics and Cosmology (21DZ2271500-2)

In this study, we investigate the robustness of pair structures for nuclear yrast states, that is, whether the structures of relevant collective pairs as building blocks of different yrast states are the same. We focus on deformed and transitional nuclei and study the yrast states of Si, Cr, and Xe, whose experimental values are 2.60, 2.40, and 2.16, respectively, using the nucleon-pair approximation (NPA) and shell-model effective interactions. For each yrast state, we consider optimized pair structures to be those providing the energy minimum for this state. To find the minimum, many full NPA calculations are performed with varying pair structures, and the numerical optimization procedure of the conjugate gradient method is implemented. Our results suggest that optimized pair structures remain the same for all states within a rotational band of a deformed nucleus. Our results also suggest that after backbending, that is, changing of the intrinsic state, the structure of the S pair, which is essential to build the monopole pairing correlation, remains approximately unchanged, whereas the structures of the non-S pairs, which are essential to build the quadrupole correlation, change significantly.

Description for N = 126 Isotones 210Po and 212Rn with Particle-Hole Excited Nucleon-Pair Approximation and Realistic Effective Interaction

In this paper, we study yrast states of two N=126 isotones 210Po and 212Rn using the nucleon-pair approximation with particle–hole excitations and using a low-momentum interaction Vlow–k renormalized from the free CD-Bonn NN potential. An overall good agreement with experimental level structures, B(E2)s, and B(E3)s, is achieved. We also calculate the probabilities of neutron particle–hole excitations in these yrast states, with a focus on negative-parity states, which reflect the roles played by the neutron negative-parity configurations of one-particle-one-hole excitations across the N=126 shell gap and the negative-parity configurations of valence proton particles involving the 0i13/2 orbit. The N=126 shell gap is discussed in terms of energies of neutron one-particle-one-hole excitations.

Quadrupole phonon excitations and transition probabilities for low-lying states of neutron-rich Cd isotopes

In this paper we study the low-lying states of neutron-rich $^{122,124,126,128}\mathrm{Cd}$ and $^{130,132,134,136,138}\mathrm{Cd}$ within the nucleon-pair approximation of the shell model. We adopt the phenomenological Hamiltonian for $^{122,124,126,128}\mathrm{Cd}$, and the shell-model effective interaction $jj46$ for $^{130,132,134,136,138}\mathrm{Cd}$. The available experimental excitation energies and quadrupole transition probabilities are well reproduced by our calculation, and we also make predictions for very neutron-rich Cd nuclei. Based on our calculation, the $B(E2;{0}_{\mathrm{g}.\mathrm{s}.}^{+}\ensuremath{\rightarrow}{2}_{1}^{+})$ values exhibit an asymmetric feature with respect to the $N=82$ shell closure, which mainly comes from the contributions of the proton transition matrix elements. We also investigate for the eight open-shell Cd nuclei, whether two low-lying yrast states with spins differing by 2 can be connected by a quadrupole-phonon excitation, and here we take the proton and neutron quadrupole operators multiplied by $1/{r}_{0}^{2}$ as our quadrupole phonon operators. We calculate explicit overlaps between the low-lying yrast states and the constructed quadrupole-phonon states based on the low-lying yrast states with lower spins, and the results indicate that, for all eight open-shell Cd nuclei, $|{2}_{1}^{+}\ensuremath{\rangle}$ and $|{4}_{1}^{+}\ensuremath{\rangle}$ can be well described to be the states constructed by coupling the proton or neutron phonon to $|{0}_{\mathrm{g}.\mathrm{s}.}^{+}\ensuremath{\rangle}$ and $|{2}_{1}^{+}\ensuremath{\rangle}$, respectively. Very interestingly, for $^{126,124,122}\mathrm{Cd}$ and $^{136,138}\mathrm{Cd}$, the ${2}_{1}^{+}$ state can be well described to be both the proton phonon state and the neutron phonon state, which indicates a nonorthogonal feature of these two phonon states. We further present an analytic relation for the overlap between these two phonon states, which implies that the proton and neutron phonon states constructed using the quadrupole operators and the ${0}_{\mathrm{g}.\mathrm{s}.}^{+}$ state in an open-shell nucleus are almost impossibly orthogonal.

Nucleon-pair approximation for nuclei from spherical to deformed regions

In this paper we model low-lying states of atomic nuclei in the nucleon-pair approximation of the shell model, using three approaches to select collective nucleon pairs: the generalized seniority scheme, the conjugate gradient method, and the Hartree-Fock approach. We find the collective pairs obtained from the generalized seniority scheme provides a good description for nearly spherical nuclei, and those from the conjugate gradient method or the Hartree-Fock approach work well for transitional and deformed nuclei. Our NPA calculations using collective pairs with angular momenta 0, 2, and 4 (denoted by $SDG$ pairs) reproduce the nuclear shape evolution in the $N=26$ isotones, $^{46}$Ca, $^{48}$Ti, $^{50}$Cr, and $^{52}$Fe, and yield good agreement with full configuration-interaction calculations of low-lying states in medium-heavy transitional and deformed nuclei: $^{44-48}$Ti, $^{48}$Cr, $^{50}$Cr, $^{52}$Fe, $^{60-64}$Zn, $^{64,66}$Ge, $^{84}$Mo, and $^{108-112}$Xe. Finally, using the $SDGI$-pair approximation we describe low-lying states of $^{112,114}$Ba, cases difficult to reach by conventional configuration-interaction methods.

Symmetric and asymmetric structural evolutions of Te isotopes across the N = 82 shell closure * *Supported by National Natural Science Foundation of China (11875188, 11905130 and 12075169), Sichuan Science and Technology Program (2019JDRC0017), the Doctoral Program of Southwest University of Science and Technology (18zx7147), and Shanghai Sailing Program (19YF1434200)

Systematic calculations of low-lying energy levels, transitions, and g factors of even-even tellurium isotopes with mass numbers from 128 to 140 are performed via the nucleon-pair approximation (NPA) of the shell model with phenomenological multipole-multipole interactions. An optimal agreement is obtained between the calculated results and experimental data. The yrast band structures of nuclei below and above the shell closure are compared and discussed. In particular, the evolutionary differences of values and factors, with respect to the symmetry of , are attributed to the dominant contribution differences in neutron and proton excitations, respectively.

Nucleon-pair approximation with uncoupled representation * *Support by the Sichuan Science and Technology Program (2019JDRC0017), Doctoral Program of Southwest University of Science and Technology (18zx7147), National Natural Science Foundation of China (11705100), Youth Innovations and Talents Project of Shandong Provincial Colleges, and Universities (201909118), Higher Educational Youth Innovation Science and Technology Program Shandong Province (2020KJJ004), National Natural Science Foundation of

In this paper, we propose an approach to nucleon-pair approximation (NPA) with m-scheme bases, in which the collective nucleon pairs are represented in terms of antisymmetric matrices, and commutations between nucleon pairs are given using a matrix multiplication that avoids angular-momentum couplings and recouplings. Therefore the present approach significantly simplifies the NPA computation. Furthermore, it is formulated on the same footing with and without isospin.

Exact sum rules with approximate ground states

Electromagnetic and weak transitions tell us a great deal about the structure of atomic nuclei. Yet modeling transitions can be difficult: it is often easier to compute the ground state, if only as an approximation, than excited states. One alternative is through transition sum rules, in particular the non-energy-weighted and energy-weighted sum rules, which can be computed as expectation values of operators. We investigate by computing sum rules for a variety of nuclei, comparing the numerically exact full configuration-interaction shell model, as a reference, to Hartree–Fock, projected Hartree–Fock, and the nucleon pair approximation. These approximations yield reasonable agreement, which we explain by prior work on the systematics of transition moments.

From deformed Hartree-Fock to the nucleon-pair approximation

The nucleon-pair approximation (NPA) can be a compact alternative to full configuration-interaction (FCI) diagonalization in nuclear shell-model spaces, but selecting good pairs is a long-standing problem. While seniority-based pairs work well for near-spherical nuclides, they do not work well for deformed nuclides with strong rotational bands. We propose an alternate approach. We show how one can write any Slater determinant for an even number of particles as a general pair condensate, from which one can project out pairs of good angular momentum. We implement this by generating unconstrained Hartree-Fock states in a shell model basis and extracting S, D, and G pairs. The subsequent NPA calculations yield good agreement with FCI results using the same effective interactions.

Variational approach for pair optimization in the nucleon pair approximation

We propose a pair-condensate variational approach (PCV) to determine a set of the most important collective pairs in the description of low-lying states in atomic nuclei. Having available the precise details on these key collective pairs -- their spin, parity, and structure -- can be particularly useful in calculations based on the nucleon-pair approximation (NPA), helping to reduce their uncertainties. In trial calculations for the transitional Ba isotopes, our variational approach describes the evolution of quadrupole-deformation properties similar to Hartree-Fock treatments, while at the same time highlighting the $\gamma$ softness of $^{132}$Ba. Our approach can conclusively determine which collective pairs are critical for obtaining the lowest possible yrast, quasi-beta, quasi-gamma bands, producing both the level structure of these bands and related B(E2) values in reasonable consistency with experiment. These trial calculations suggest that with our PCV approach the NPA can be meaningfully applied to transitional nuclei with a wide spectrum of shapes. We also show that while neutron negative-parity pairs could in principle have an important impact on backbending in $^{132}$Ba, they are not favored for this nucleus.

Low-lying states of even-even N=80 isotones within the nucleon-pair approximation

In this paper we study low-lying states of even-even $N=80$ isotones including $^{130}\mathrm{Sn}, ^{132}\mathrm{Te}, ^{134}\mathrm{Xe}, ^{136}\mathrm{Ba}$, and $^{138}\mathrm{Ce}$, within the nucleon-pair approximation of the shell model. We calculate low-lying energy levels of these nuclei with both positive and negative parities. The wave functions of yrast ${2}_{1}^{+}\ensuremath{-}{10}_{1}^{+}$ states and ${1}_{1}^{\ensuremath{-}}\ensuremath{-}{11}_{1}^{\ensuremath{-}}$ states of these nuclei are analyzed in detail. Our calculations show that most of these states have a very simple structure in nucleon-pair basis: the ${2}_{1}^{+}$ states of $^{132}\mathrm{Te}, ^{134}\mathrm{Xe}, ^{136}\mathrm{Ba}$, and $^{138}\mathrm{Ce}$ are dominated by one ${D}^{+}$ neutron pair and spin $J=2$ proton excitation; the ${4}_{1}^{+}$ and ${6}_{1}^{+}$ states of $^{132}\mathrm{Te}$ and $^{134}\mathrm{Xe}$ are dominated by the proton excitation, while the ${8}_{1}^{+}$ and ${10}_{1}^{+}$ states are dominated by ${S}^{+}$ pairs of protons and one spin-eight and spin-ten pair of valence neutron holes (i.e., seniority-two excitations); for negative-parity states, the lowest states with spin four to seven of $^{132}\mathrm{Te}, ^{134}\mathrm{Xe}, ^{136}\mathrm{Ba}$, and $^{138}\mathrm{Ce}$ are essentially given by ${S}^{+}$ pairs of protons and one broken pair with spin four to seven consists of ${h}_{11/2}{d}_{3/2}$ valence neutron holes. The necessity of $J=10$ pairing interaction in the phenomenological monopole plus quadrupole shell-model Hamiltonian for $N=80$ isotones is demonstrated with analytical formulas for simple nucleon-pair configurations, and the pattern of $g$ factors is discussed based on simple arguments. Our calculated $B(E2)$ transition rates and $g$ factors of the low-lying states are well consistent with the experimental data.

Electromagnetic properties of neutron-rich Ge isotopes * * Supported by National Natural Science Foundation of China (11875188), Open Project Fund of Shanghai Key Laboratory of Particle Physics and Cosmology (18DZ2271500-2), Sichuan Science and Technology Program (2019JDRC0017), and the Doctoral Program of Southwest University of Science and Technology (18zx7147)

The electric quadrupole moment and the magnetic moment (or the factor) of low-lying states in even-even nuclei 72-80Ge and odd-mass nuclei 75-79Ge are studied in the framework of the nucleon pair approximation (NPA) of the shell model, assuming the monopole and quadrupole pairing plus quadrupole-quadrupole interaction. Our calculations reproduce well the experimental values of and for 72,74,76Ge, as well as the yrast energy levels of these isotopes. The structure of the states and the contributions of the proton and neutron components in and are discussed in the -pair truncated shell-model subspace. The overall trend of and as a function of the mass number , as well as their signs, are found to originate essentially from the proton contribution. The negative value of in 72,74Ge is suggested to be due to the enhanced quadrupole-quadrupole correlation and configuration mixing.

Nucleon-pair approximation with particle-hole excitations

In this paper, the formalism for the nucleon-pair approximation (NPA) with particle-hole excitations is developed. By using this approach, one is able to perform multiple-major-shell calculations. In this formulation, we use a particle-hole mixed representation, i.e., the particle-hole conjugate transformation is used for operators of the lower major shells, while operators of the upper major shells remain unchanged. We consider collective valence-particle pairs in the upper major shells, collective valence-hole pairs in the lower major shells, and collective pairs consisting of one valence particle in the upper shells and one valence hole in the lower shells to construct our model space. Matrix elements for the shell-model Hamiltonian, both for effective interactions and for phenomenological pairing plus multipole-multipole interactions, are derived in the nucleon-pair basis. As a special case, analytical formulas for doubly magic nuclei with excitations up to two-particle--two-hole are presented. To exemplify this approach, we calculate $^{100}\mathrm{Sn}$ considering both proton and neutron up to four-particle--four-hole excitations, where valence particles are in the 50-82 major shell and valence holes in the 28-50 major shell, with the low-momentum nucleon-nucleon interaction derived from the CD-Bonn potential.

Shell model study ofT=0states forCd96by the nucleon-pair approximation

In this paper we study the nucleon-pair approximation for $T=0$ states of $^{96}\mathrm{Cd}$ in the $1{p}_{1/2}1{p}_{3/2}0{f}_{5/2}0{g}_{9/2}$ shell with the JUN45 interaction. The lowest seniority scheme and the isoscalar spin-one pair approximation are not enough to describe the states. The isoscalar spin-aligned pair approximation is reasonably good for the yrast ${2}^{+}, {4}^{+}, {6}^{+}, {12}^{+}, {14}^{+}$, and ${16}^{+}$ states as pointed out previously. Not only the yrast positive-parity states but also nonyrast states and negative-parity states are well described by both the isovector pair approximation and the isoscalar pair approximation. We calculate overlaps between nucleon-pair basis states and shell-model wave functions. The largest overlaps and the corresponding nucleon-pair basis states are presented. We find that isovector spin-zero pairs, isovector spin-two pairs, and isoscalar spin-aligned pairs are the dominant building blocks in these states.

Nucleon-pair states of even-even Sn isotopes based on realistic effective interactions

In this paper we study yrast states of $^{128,126,124}\mathrm{Sn}$ and $^{104,106,108}\mathrm{Sn}$ by using the monopole-optimized realistic interactions in terms of both the shell model (SM) and the nucleon-pair approximation (NPA). For yrast states of $^{128,126}\mathrm{Sn}$ and $^{104,106}\mathrm{Sn}$, we calculate the overlaps between the wave functions obtained in the full SM space and those obtained in the truncated NPA space, and find that most of these overlaps are very close to 1. Very interestingly, for most of these states with positive parity and even spin or with negative parity and odd spin, the SM wave function is found to be well represented by one nucleon-pair basis state, viz., a simple picture of ``nucleon-pair states'' (nucleon-pair configuration without mixings) emerges. In $^{128,126}\mathrm{Sn}$, the positive-parity (or negative-parity) yrast states with spin $J>10$ (or $J>7$) are found to be well described by breaking one or two $S$ pairs in the ${10}_{1}^{+}$ (or ${7}_{1}^{\ensuremath{-}}$) state, i.e., the yrast state of seniority-two, spin-maximum, and positive-parity (or negative-parity), into non-$S$ pair(s). Similar regularity is also pointed out for $^{104,106}\mathrm{Sn}$. The evolution of $E2$ transition rates between low-lying states in $^{128,126,124}\mathrm{Sn}$ is discussed in terms of the seniority scheme.

Nucleon-pair states of even-evenN=82isotones

In this paper we study low-lying states of five $N=82$ isotones, $^{134}\mathrm{Te}, ^{136}\mathrm{Xe}, ^{138}\mathrm{Ba}, ^{140}\mathrm{Ce}$ and $^{142}\mathrm{Nd}$, within the framework of the nucleon-pair approximation (NPA). For the low-lying yrast states of $^{136}\mathrm{Xe}$ and $^{138}\mathrm{Ba}$, we calculate the overlaps between the wave functions obtained in the full shell-model (SM) space and those obtained in the truncated NPA space, and find that most of these overlaps are very close to 1. Very interestingly and surprisingly, for most of these yrast states, the SM wave functions are found to be well approximated by one-dimensional, optimized pair basis states, which indicates a simple picture of ``nucleon-pair states''. The positive-parity yrast states with spin $Jg6$ in these nuclei, as well as the ${8}_{2}^{+}$ state, are found to be well described by breaking one or two $S$ pair(s) of the ${6}_{1}^{+}$ or ${6}_{2}^{+}$ state (low-lying, seniority-two, spin-maximum, and positive-parity); similarly, negative-parity yrast states with spin $Jg9$ are well represented by breaking one or two $S$ pair(s) of the ${9}_{1}^{\ensuremath{-}}$ state (low-lying, seniority-two, spin-maximum, and negative-parity). It is shown that the low-lying negative-parity yrast states of $^{136}\mathrm{Xe}$ and $^{138}\mathrm{Ba}$ are reasonably described to be one-octupole-phonon excited states. The evolution of the ${6}_{1}^{+}$ and ${6}_{2}^{+}$ states for the five isotones are also systematically investigated.