We link the structure of nuclei around $^{100}$Sn, the heaviest doubly magic nucleus with equal neutron and proton numbers ($N=Z=50$), to nucleon-nucleon ($NN$) and three-nucleon ($NNN$) forces constrained by data of few-nucleon systems. Our results indicate that $^{100}$Sn is doubly magic, and we predict its quadrupole collectivity. We present precise computations of $^{101}$Sn based on three-particle--two-hole excitations of $^{100}$Sn, and reproduce the small splitting between the lowest $J^π=7/2^+$ and $5/2^+$ states. Our results are consistent with the sparse available data.
By studying the 109Xe-->105Te-->101Sn superallowed alpha-decay chain, we observe low-lying states in 101Sn, the one-neutron system outside doubly magic 100Sn. We find that the spins of the ground state (J = 7=2) and first excited state (J = 5=2) in 101Sn are reversed with respect to the traditional level ordering postulated for 103Sn and the heavier tin isotopes. Through simple arguments and state-of-the-art shell model calculations we explain this unexpected switch in terms of a transition from the single-particle regime to the collective mode in which orbital-dependent pairing correlations, …
We present a systematic configuration-interaction shell model calculation on the structure of light tin isotopes with a new global optimized effective interaction. The starting point of the calculation is the realistic CD-Bonn nucleon-nucleon potential. The unknown single-particle energies of the $1d_{3/2}$, $2s_{1/2}$ and $0h_{11/2}$ orbitals and the T=1 monopole interactions are determined by fitting to the binding energies of 157 low-lying yrast states in $^{102-132}$Sn. We apply the Hamiltonian to analyze the origin of the spin inversion between $^{101}$Sn and $^{103}$Sn that was observed …
The dipole polarizability of stable even-mass tin isotopes 112,114,116,118,120,124 was extracted from inelastic proton scattering experiments at 295 MeV under very forward angles performed at RCNP. Predictions from energy density functionals cannot account for the present data and the polarizability of 208Pb simultaneously. The evolution of the polarizabilities in neighboring isotopes indicates a kink at 120Sn while all model results show a nearly linear increase with mass number after inclusion of pairing corrections.
We measure cross sections for residual nuclide formation in the mass range 6 < A < 97 caused by bombardment with protons and deuterons of 3.65 GeV/nucleon energy of enriched tin isotopes (112-Sn, 118-Sn, 120-Sn, 124-Sn). The experimental data are compared with calculations by the codes FLUKA, LAHET, CEM03, and LAQGSM03. Scaling behavior is observed for the whole mass region of residual nuclei, showing a possible multifragmentation mechanism for the formation of light products (6 < A < 31). Our analysis of the isoscaling dependence also shows a possible contribution of multifragmentation to the…
The occupancies and vacancies of the valence neutron orbitals across the stable tin isotopic chain from $112\leq A\leq 124$ have been determined. These were inferred from the cross sections of neutron-adding and -removing reactions. In each case, the reactions were chosen to have good angular-momentum matching for transfer to the low- and high-$\ell$ orbitals present in this valence space. These new data are compared to older systematic studies. The effective single-neutron energies are determined by combining information from energy centroids determined from the adding and removing reactions.…
We present results from large-scale shell-model calculations of even and odd tin isotopes from 134Sn to 142}Sn with a shell-model space defined by the 1f7/2,2p3/2,0h9/2,2p1/2,1f5/2,0i13/2 single-particle orbits. An effective two-body interaction based on modern nucleon-nucleon interactions is employed. The shell-model results are in turn analyzed for their pairing content using a generalized seniority approach. Our results indicate that a pairing-model picture captures a great deal of the structure and the correlations of the lowest lying states for even and odd isotopes.
High-precision ab initio calculations of nuclear binding energies: Tin isotopes from dripline to dripline
Technische Universität Darmstadt（达姆施塔特工业大学）；ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH（GSI重离子研究中心极端物质研究所EMMI）；KU Leuven, Instituut voor Kern- en Stralingsfysica（荷语鲁汶大学核与辐射物理研究所）；Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles（布鲁塞尔自由大学天体物理学与天文学研究所）；Brussels Laboratory of the Universe – BLU-ULB（布鲁塞尔宇宙实验室）；IRFU, CEA, Université Paris-Saclay（巴黎萨克雷大学CEA IRFU）；Max-Planck-Institut für Kernphysik（马克斯·普朗克核物理研究所）
We calculate the low-lying spectra of heavy tin isotopes from A=120 to A=130 using the 2s1d0g_{7/2}0h_{11/2} shell to define the model space. An effective interaction has been derived using 132Sn as closed core employing perturbative many-body techniques. We start from a nucleon-nucleon potential derived from modern meson exchange models. This potential is in turn renormalized for the given medium, 132Sn, yielding the nuclear reaction matrix, which is then used in perturbation theory to obtain the shell model effective interaction.
We show in the Skyrme-Hartree-Fock approach that unambiguous correlations exist between observables of finite nuclei and nuclear matter properties. Using this correlation analysis to existing data on the neutron skin thickness of Sn isotopes, we find important constraints on the value E_{sym}(rho_0) and density slope L of the nuclear symmetry energy at saturation density. Combining these constraints with those from recent analyses of isospin diffusion and double neutron/proton ratio in heavy ion collisions leads to a value of L=58\pm 18 MeV approximately independent of E_{sym}(ρ_0).
The distribution of isoscalar monopole strength in the neutron-even 112-124Sn-isotopes has been computed using a relativistic random-phase-approximation approach. The accurately-calibrated model used here (``FSUGold'') has been successful in reproducing both ground-state observables as well as collective excitations - including the giant monopole resonance (GMR) in 90Zr, 144Sm, and 208Pb. Yet this same model significantly overestimates the GMR energies in the Sn isotopes. It is argued that the question of ``Why is Tin so soft?'' becomes an important challenge to the field and one that should b…
The quasiparticle time blocking approximation (QTBA) is applied to describe E1 excitations in the even-even tin isotopes. Within the model pairing correlations, two-quasiparticle (2q), and 2q*phonon configurations are included. Thus the QTBA is an extension of the quasiparticle random phase approximation to include quasiparticle-phonon coupling. Calculational formulas are presented in case of neutral excitations in the spherically symmetric system. The main equations are written in the coordinate representation that allows to take into account single-particle continuum completely. The E1 photo…
A realistic shell-model study is performed for neutron-deficient tin isotopes up to mass A=108. All shell-model ingredients, namely two-body matrix elements, single-particle energies, and effective charges for electric quadrupole transition operators, have been calculated by way of the many-body perturbation theory, starting from a low-momentum interaction derived from the high-precision CD-Bonn free nucleon-nucleon potential. The focus has been put on the enhanced quadrupole collectivity of these nuclei, which is testified by the observed large B(E2;0+ -> 2+)s. Our results evidence the crucia…
"Why are the tin isotopes soft?" has remained, for the past decade, an open problem in nuclear structure physics: models which reproduce the isoscalar giant monopole resonance (ISGMR) in the "doubly-closed shell" nuclei, $^{90}$Zr and $^{208}$Pb, overestimate the ISGMR energies of the open-shell tin and cadmium nuclei, by as much as 1 MeV. In an effort to shed some light onto this problem, we present results of detailed studies of the ISGMR in the molybdenum nuclei, with the goal of elucidating where--and how--the softness manifests itself between $^{90}$Zr and the cadmium and tin isotopes. Th…
We study the couplings between collective vibrations such as the isovector giant dipole and isoscalar giant quadrupole resonances in tin isotopes in the framework of the time-dependent Hartree-Fock theory with a Skyrme energy density functional. These couplings are a source of anharmonicity in the multiphonon spectrum. In particular, the residual interaction is known to couple the isovector giant dipole resonance with the isoscalar giant quadrupole resonance built on top of it, inducing a nonlinear evolution of the quadrupole moment after a dipole boost. This coupling also affects the dipole m…
Low-energy E1 strength in neutron-rich $^{132-164}$Sn isotopes is analyzed in the framework of the Skyrme random phase approximation (RPA) with different Skyrme forces. A double folding procedure is applied to take into account the energy-dependent width effects beyond RPA. All the considered Skyrme forces indicate a soft prolate deformation in the open shell isotopes $^{142-164}$Sn. The integrated E1 strength in the energy region of the pygmy resonance grows with the neutron number. The influence of deformation on the integrated strength near the particle emission thresholds (which is of a ke…
The ``breathing mode'' of neutron-rich nuclei is our window into the incompressibility of neutron-rich matter. After much confusion on the interpretation of the experimental data, consistency was finally reached between different models that predicted both the distribution of isoscalar monopole strength in finite nuclei and the compression modulus of infinite matter. However, a very recent experiment on the Tin isotopes at the Research Center for Nuclear Physics(RCNP) in Japan has again muddled the waters. Self-consistent models that were successful in reproducing the energy of the giant monop…
Proton elastic scattering off Sn isotopes at $E_p=295$ MeV in the mass number range of $A=$116--124 was investigated using calculation employing relativistic impulse approximation (RIA) with theoretical densities obtained for the Sn isotopes from relativistic Hartree-Bogoliubov (RHB) and nonrelativistic Skyrme Hartree-Fock-Bogoliubov (SHFB) calculations of spherical nuclei. In the RIA calculations, a modified version of the Murdock and Horowitz model that includes a density dependence in the effective nucleon-nucleon ($NN$) interaction was used. A calculation using the theoretical density obta…
The isotopic systematics of Sn isotopes in the range A=116--124 were investigated by combining the nuclear structure and reaction calculations for the analysis of Sn(p,p) reactions at 295 MeV. The Sn(p,p) reactions were calculated employing relativistic impulse approximation (RIA) with theoretical densities obtained for the Sn isotopes from relativistic Hartree-Bogoliubov (RHB) calculations with the DD-ME2 interaction and nonrelativistic Skyrme Hartree-Fock-Bogoliubov (SHFB) calculations with the SKM* interaction. Calculation using the DD-ME2 density reproduced the experimental data for $^{122…
We present the first ab initio calculations for open-shell nuclei past the tin isotopic line, focusing on Xe isotopes as well as doubly-magic Sn isotopes. We show that, even for moderately hard interactions, it is possible to obtain meaningful predictions and that the NNLOsat chiral interaction predicts radii and charge density distributions close to the experiment. We then make a new prediction for ${}^{100}$Sn. This paves the way for ab initio studies of exotic charge density distributions at the limit of the present ab initio mass domain, where experimental data is becoming available. The p…
For the first time, the shell structure of open-shell nuclei is described in a fully self-consistent extension of the covariant energy density functional theory. The approach implies quasiparticle-vibration coupling for superfluid systems. One-body Dyson equation formulated in the doubled quasiparticle space of Dirac spinors is solved for nucleonic propagators in tin isotopes which represent the reference case: the obtained energies of the single-quasiparticle levels and their spectroscopic amplitudes are in agreement with data. The model is applied to describe the shell evolution in a chain o…
The E1 strength function for 15 stable and unstable Sn even-even isotopes from A=100 till A=176 are calculated using the self-consistent microscopic theory which, in addition to the standard (Q)RPA approach, takes into account the single-particle continuum and the phonon coupling. Our analysis shows two distinct regions for which the integral characteristics of both the giant and pygmy resonances behave rather differently. For neutron-rich nuclei, starting from $^{132}$Sn, we obtain a giant E1 resonance which significantly deviates from the widely-used systematics extrapolated from experimenta…
The Relativistic Continuum Hartree Bogoliubov theory (RCHB), which is the extension of the Relativistic Mean Field and the Bogoliubov transformation in the coordinate representation, has been used to study tin isotopes. The pairing correlation is taken into account by a density-dependent force of zero range. RCHB is used to describe the even-even tin isotopes all the way from the proton drip line to the neutron drip line. The contribution of the continuum which is important for nuclei near the drip-line has been taken into account. The theoretical $S_{2n}$ as well as the neutron, proton, and m…
Nuclear masses of even-even nuclei with the proton number $8\leq Z\leq 50$ (O to Sn isotopes) from proton drip line to neutron drip line are investigated using the triaxial relativistic Hartree-Bogoliubov (RHB) theory with the relativistic density functional PC-PK1, and the dynamical correlation energies (DCEs) associated with the rotational motion and the quadrupole shape vibrational motion are taken into account by the five-dimensional collective Hamiltonian (5DCH) method. The root-mean-square deviation with respect to the experimental masses reduces from 2.50 MeV to 1.59 MeV after the consi…
Finite nuclei such as those found in the chain of even tin isotopes from ^{102}Sn to ^{130}Sn, exhibit a near constancy of the 2^+_1-0^+_1 excitation energy, a constancy which can be related to strong pairing correlations and the near degeneracy in energy of the relevant single particle orbits. Large shell-model calculations for these isotopes reveal that the major contribution to pairing correlations in the tin isotopes stems from the ^1S_0 partial wave in the nucleon-nucleon interaction. Omitting this partial wave and the ^3P_2 wave in the construction of an effective interaction, results in…
The region around the doubly magic nuclide $^{100}Sn$ is very interesting for nuclear physics studies in terms of structure, reaction and nuclear astrophysics. The main ingredients in nuclear structure studies using the shell model are the single-particle energies and the two-body matrix elements. To obtain the former, experimental data of $^{101}Sn$ isotope spectrum are necessary. Since there is not enough experimental data, different approaches are used in the literature to obtain single-particle energies. In sn100pn interaction, the hole excitation spectrum was used in $^{131}Sn$ to determi…
The lightest Xenon isotopes are studied in the framework of the Interacting Shell Model (ISM). The valence space comprises all the orbits lying between the magic closures N=Z=50 and N=Z=82. The calculations produce collective deformed structures of triaxial nature that encompass nicely the known experimental data. Predictions are made for the (still unknown) N=Z nucleus 108-Xe. The results are interpreted in terms of the competition between the quadrupole correlations enhanced by the pseudo-SU(3) structure of the positive parity orbits and the pairing correlations brought in by the 0h11/2 orbi…
One of the main ingredients in nuclear structure studies using shell model are the single-particle energy (spe). In order to obtain these values accurately, experimental data is needed. The region around the doubly magic nuclide 100Sn is very interesting for nuclear studies in terms of structure, reaction and nuclear astrophysics. Experimental spectrum data for the 101Sn isotope is required for nuclear shell model studies to be carried out in this region. Since there is not enough experimental data in the literature, different approaches are used for the obtaining spes for the region such as u…
We have computed dipole strength distributions for nickel and tin isotopes within the Relativistic Quasiparticle Time Blocking approximation (RQTBA). These calculations provide a good description of data, including the neutron-rich tin isotopes $^{130,132}$Sn. The resulting dipole strengths have been implemented in Hauser-Feshbach calculations of astrophysical neutron capture rates relevant for r-process nucleosynthesis studies. The RQTBA calculations show the presence of enhanced dipole strength at energies around the neutron threshold for neutron rich nuclei. The computed neutron capture rat…
The self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. with the set DF3-a of parameters fixed previously is used to calculate three kinds of quadrupole moments. At first, we examined systematically quadrupole moments of odd neighbors of semi-magic lead and tin isotopes and $N=50,N=82$ isotones. Second, we found quadrupole moments of the first $2^+$ states in the same two chains of isotopes. Finally, we evaluated quadrupole moments of odd-odd nuclei neighboring to double magic ones. Reasonable agreement with available experimental data has bee…
Root-mean-square charge radii are discussed in terms of spherical Energy Density Functional (EDF) models corrected for quadrupole deformations. Comparisons between experiment and theory are made for the absolute radii of all even-even nuclei, for the isotonic shift between cadmium and tin isotopes, the isotopic shifts of the calcium isotopes and the isotonic shift for nuclei with $ N=28 $. We conclude that the data are well described in this approach, except for the sharp rise just after the neutron magic numbers.
We report here an empirical formalism for predicting the cross-sections of neutron-rich nuclei produced in fragmentation of relativistic \b{eta}-unstable neutron-rich projectiles in the mid-mass region. The formalism is based on the abrasion-ablation picture of the projectile fragmentation reaction. It has been shown that the formalism can accurately reproduce the experimental cross-section data for nuclei produced in the fragmentation of 132Sn. The formalism, it is shown, can also reproduce the experimental cross-sections of neutron-rich nuclei produced in the fragmentation of 129Xe fairly ac…
Recent results of the description of quadrupole moments of odd semi-magic nuclei are briefly reviewed. They are based on the self-consistent theory of finite Fermi systems with account for the phonon-particle coupling (PC) effects. The self-consistent model for describing the PC effects was developed previously for magnetic moments. Account for the non-pole diagrams is an important ingredient of this model. In addition to previously reported results for the odd In and Sb isotopes, which are the proton-odd neighbors of even tin nuclei, we present new results for odd Bi isotopes, the odd neighbo…
The properties of $γ$ instability in rapidly rotating even-even $^{132-138}$Nd isotopes have been investigated using the pairing-deformation self-consistent total-Routhian-surface calculations in a deformation space of ($β_2, γ, β_4$). It is found that even-even $^{134-138}$Nd nuclei exhibit the triaxiality in both ground and excited states, even up to high-spin ones. The lightest isotope possesses a well-deformed prolate shape without $γ$ deformation component. The current numerical results are compared with previous calculations and available observables, showing basically a general agreemen…
Dipole magnetic moments of several long isotopic chains are analyzed within the self-consistent Finite Fermi System theory based on the Generalized Energy Density Functional method with exact account for the pairing and quasi-particle continuum. New data for nuclei far from the beta-stability valley are included in the analysis. For a number of semi-magic isotopes of the tin and lead chains a good description of the data is obtained, with accuracy of 0.1 - 0.2 mu_N. A chain of non-magic isotopes of copper is also analyzed in detail. It is found that the systematic analysis of magnetic moments …
Background: Following the 2007 precise measurements of monopole strengths in tin isotopes, there has been a continuous theoretical effort to obtain a precise description of the experimental results. Up to now, there is no satisfactory explanation of why the tin nuclei appear to be significantly softer than 208Pb. Purpose: We determine the influence of finite-range and separable pairing interactions on monopole strength functions in semi-magic nuclei. Methods: We employ self-consistently the Quasiparticle Random Phase Approximation on top of spherical Hartree-Fock-Bogolyubov solutions. We use t…
The role of three-nucleon forces (3NF) in the description of nuclear structure properties is nowadays a main topic in the field of microscopic many-nucleon calculations. We investigate the relative weight between effective two- and three-nucleon forces in neutron-rich nuclei beyond the doubly-closed 132Sn core within the realistic shell-model framework, studying the evolution of the spectroscopic properties of N=82 isotopes and heavy tin isotopes. This problem is tackled indirectly without explicitly taking into account effective 3NF through the comparison of the results of shell-model calcula…
Excitation energies and transition probabilities of the first $2^+$ excitations in even tin and lead isotopes as well as the quadrupole moments of odd neighbors of these isotopes are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. The effect of the density dependence of the effective pairing interaction is analyzed in detail by comparing results obtained with volume and surface pairing. The effect is found to be noticeable. For example, the $2^+$-energies are systematically higher at 200-300 keV for the volume paring …
The quadrupole moments of odd neighbors of semi-magic lead and tin isotopes and $N=50,N=82$ isotones are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. Two sets of parameters, DF3 and DF3-a, fixed previously are used. They differ by the spin-orbit and effective tensor force parameters, the latter being significantly bigger in the DF3-a functional. Results for the two functionals turned out to be rather different. The functional DF3-a leads to quadrupole moments in reasonable agreement with the experimental ones for m…
We study deep inelastic scattering on isospin asymmetric nuclei. In particular, the difference of the nuclear structure functions and the Gottfried sum rule for the lightest mirror nuclei, 3He and 3H, are investigated. It is found that such systems can provide significant information on charge symmetry breaking and flavor asymmetry in the nuclear medium. Furthermore, we propose a new method to extract the neutron structure function from radioactive isotopes far from the line of stability. We also discuss the flavor asymmetry in the Drell-Yan process with isospin asymmetric nuclei.
We perform Hartree-Fock-Bogoliubov (HFB) calculations for semi-magic Calcium, Nickel, Tin and Lead isotopes and $N$=20, 28, 50 and 82 isotones using density-dependent pairing interactions recently derived from a microscopic nucleon-nucleon interaction. These interactions have an isovector component so that the pairing gaps in symmetric and neutron matter are reproduced. Our calculations well account for the experimental data for the neutron number dependence of binding energy, two neutrons separation energy, and odd-even mass staggering of these isotopes. This result suggests that by introduci…
Excitation energies and transition probabilities of the first 2+ excitations in even lead, tin and nickel isotopes are calculated within the self-consistent Theory of Finite Fermi Systems based on the Energy Density Functional by Fayans et al. A reasonable agreement with available experimental data is obtained. The effect of the density dependence of the effective pairing interaction is analyzed in detail by comparing results obtained with volume and surface pairing. The effect is found to be noticeable, especially for the 2+ energies which are systematically higher at 200-300 keV for the volu…
New experimental data on 2+ energies of 136,138Sn confirms the trend of lower 2+ excitation energies of even-even tin isotopes with N > 82 compared to those with N< 82. However, none of the theoretical predictions using both realistic and empirical interactions can reproduce experimental data on excitation energies as well as the transition probabilities (B(E2; 6+ -> 4+)) of these nuclei, simultaneously, apart from one whose matrix elements have been changed empirically to produce mixed seniority states by weakening pairing. We have shown that the experimental result also shows good agreement …
We study the use of truncated normal-ordered three-nucleon interactions in ab initio nuclear structure calculations starting from chiral two- plus three-nucleon Hamiltonians evolved consistently with the similarity renormalization group (SRG). We present three key steps: (i) a rigorous benchmark of the normal-ordering approximation in the importance-truncated no-core shell model (IT-NCSM) for 4-He, 16-O, and 40-Ca; (ii) a direct comparison of the IT-NCSM results with coupled-cluster calculations at the singles and doubles level (CCSD) for 16-O; and (iii) first applications of SRG-evolved chira…
We construct an effective shell-model interaction for the valence space spanned by single-particle neutron and single-hole proton states in $^{100}$Sn. Starting from chiral nucleon-nucleon and three-nucleon forces and single-reference coupled-cluster theory for $^{100}$Sn we apply a second similarity transformation that decouples the valence space. The particle-particle components of the resulting effective interaction can be used in shell model calculations for neutron deficient tin isotopes. The hole-hole interaction can be used to calculate the $N = 50$ isotones south of $^{100}$Sn, and the…
We have developed a new Hartree-Fock-Bogoliubov (HFB) code which has been specifically designed to study ground state properties of nuclei near the neutron and proton drip lines. The unique feature of our code is that it takes into account the strong coupling to high-energy continuum states, up to an equivalent single-particle energy of 60 MeV. We solve the HFB equations for deformed, axially symmetric even-even nuclei in coordinate space on a 2-D lattice with Basis-Spline methods. For the p-h channel, the Skyrme (SLy4) effective N-N interaction is utilized, and for the p-p and h-h channel we …
We review the ab initio symmetry-adapted (SA) framework for determining the structure of stable and unstable nuclei, along with related electroweak, decay and reaction processes. This framework utilizes the dominant symmetry of nuclear dynamics, the shape-related symplectic Sp(3,R) symmetry, which has been shown to emerge from first principles and to expose dominant degrees of freedom that are collective in nature, even in the lightest species or seemingly spherical states. This feature is illustrated for a broad scope of nuclei ranging from helium to titanium isotopes, enabled by recent devel…
