Sn Isotopes Research Articles

Effect of pairing correlation on low-lying quadrupole states in Sn isotopes * *Supported by the National Natural Science Foundation of China (11975096, 11875027, 11775014) and the Fundamental Research Funds for the Central Universities (2020NTST06)

We examined the low-lying quadrupole states in Sn isotopes in the framework of fully self-consistent Hartree-Fock+BCS plus QRPA. We focus on the effect of the density-dependence of pairing interaction on the properties of the low-lying quadrupole state. The SLy5 Skyrme interaction with surface, mixed, and volume pairings is employed in the calculations, respectively. We find that the excitation energies and the corresponding reduced electric transition probabilities of the first 2+ state are different, given by the three pairing interactions. The properties of the quasiparticle state, two-quasiparticle excitation energy, reduced transition amplitude, and transition densities in 112Sn are analyzed in detail. Two different mechanisms, the static and dynamical effects, of the pairing correlation are also discussed. The results show that the surface, mixed, and volume pairings indeed affect the properties of the first 2+ state in the Sn isotopes.

Electric dipole polarizability in neutron-rich Sn isotopes as a probe of nuclear isovector properties

The determination of nuclear symmetry energy, and in particular, its density dependence, is a long-standing problem for nuclear physics community. Previous studies have found that the product of electric dipole polarizability $\alpha_D$ and symmetry energy at saturation density $J$ has a strong linear correlation with $L$, the slope parameter of symmetry energy. However, current uncertainty of $J$ hinders the precise constraint on $L$. We investigate the correlations between electric dipole polarizability $\alpha_D$ (or times symmetry energy at saturation density $J$) in Sn isotopes and the slope parameter of symmetry energy $L$ using the quasiparticle random-phase approximation based on Skyrme Hartree-Fock-Bogoliubov. A strong and model-independent linear correlation between $\alpha_D$ and $L$ is found in neutron-rich Sn isotopes where pygmy dipole resonance (PDR) gives a considerable contribution to $\alpha_D$, attributed to the pairing correlations playing important roles through PDR. This newly discovered linear correlation would help one to constrain $L$ and neutron-skin thickness $\Delta R_\textnormal{np}$ stiffly if $\alpha_D$ is measured with high resolution in neutron-rich nuclei. Besides, a linear correlation between $\alpha_D J$ in a nucleus around $\beta$-stability line and $\alpha_D$ in a neutron-rich nucleus can be used to assess $\alpha_D$ in neutron-rich nuclei.

The kinks in charge radii across N = 82 and 126 revisited

We revisit the studies of the isotopic shift in the charge radii of even–even isotopes of Sn and Pb nuclei at N = 82, and 126, respectively, within the relativistic mean-field (RMF) and relativistic-Hartree–Bogoliubov (RHB) approach. The shell model is also used to estimate isotopic shift in these nuclei, for the first time, to the best of our knowledge. The ground state single-particle energies (SPEs) are calculated for non-linear NL3 and NL3* and density-dependent DD-ME2 parameter sets compared with the experimental data, wherever available. We establish a correlation between the filling of single-particle levels and the isotopic shift in occupation probabilities. The obtained SPE from the RMF and RHB approaches are in line with those used in the shell model and experimental data for both the Sn and Pb isotopic chains. The shell model calculated isotopic shift agrees with RMF and RHB approaches that explain the experimental data quite well in case the of Pb nuclei beyond N=126.

Evolution of magnetic dipole strength in Sn100−140 isotope chain and the quenching of nucleon g factors

The evolution of electromagnetic transitions along isotope chains is of particular importance for the nuclear structure and dynamics, as well as for the r-process nucleosynthesis. Recent measurement of inelastic proton scattering on even-even $^{112-124}$Sn isotopes provides a novel insight into the isotopic dependence of E1 and M1 strength distributions. We investigate M1 transitions in even-even $^{100-140}$Sn isotopes from a theoretical perspective, based on relativistic nuclear energy density functional. The M1 transition strength distribution is characterized by an interplay between single and double-peak structures, that can be understood from the evolution of single-particle states, their occupations governed by the pairing correlations, and two-quasiparticle transitions involved. It is shown that discrepancy between model calculations and experiment on B(M1) transition strength is considerably reduced than previously known, and the quenching of the g-factors for the free nucleons needed to reproduce the experimental data on M1 transition strength amounts $g_{eff}/g_{free}$=0.80-0.93. Since some of the B(M1) strength above the neutron threshold may be missing in the inelastic proton scattering measurement, further experimental studies are required to confirm if only small modifications of the bare g-factors are actually needed when applied in finite nuclei.

$${\varvec{\alpha }}$$ clustering and neutron-skin thickness of carbon isotopes

The interplay between the formation of neutron skin and alpha cluster at the dilute surface of neutron-rich nuclei is one of the interesting subjects in the study of neutron-rich nuclei and nuclear clustering. A theoretical model has predicted that the growth of neutron skin will prevent the alpha clustering at the nuclear surface. Quite recently, this theoretical perspective; the suppression of alpha clustering by the neutron-skin formation was first confirmed experimentally in Sn isotopes as the reduction of the (p, p alpha) reaction cross-section. Motivated by the novel discovery, in this work, we have investigated the relationship between the neutron-skin thickness and alpha clustering in C isotopes. Based on the analysis by the antisymmetrized molecular dynamics, we show that the alpha spectroscopic factor at nuclear exterior decreases in neutron-rich C isotopes, and the clustering suppression looks correlated with the growth of the neutron-skin thickness.

Relativistic Dirac Analyses of Polarized Proton Scatterings from Sn isotopes

Relativistic Dirac Analyses of Polarized Proton Scatterings from Sn isotopes

Bayesian evaluation of charge yields of fission fragments of U239

Recent experiments [Phys. Rev. Lett. 123, 092503 (2019); Phys. Rev. Lett. 118, 222501 (2017)] have made remarkable progress in measurements of the isotopic fission-fragment yields of the compound nucleus $^{239}\mathrm{U}$, which is of great interests for fast-neutron reactors and for benchmarks of fission models. We apply the Bayesian neural network (BNN) approach to learn existing evaluated charge yields and infer the incomplete charge yields of $^{239}\mathrm{U}$. We found that the two-layer BNN is improved compared to the single-layer BNN for overall performance. Our results support the normal charge yields of $^{239}\mathrm{U}$ around Sn and Mo isotopes. The role of odd-even effects in charge yields has also been studied.

Quadrupole dominance in the light Sn and in the Cd isotopes

[Background] The BE2 rates of the Sn isotopes for $N\le 64$ exhibit enhancements hitherto unexplained. The same is true for the Cd isotopes. [Purpose] Describe the electromagnetic properties of the Sn and Cd isotopes [Method] Shell model calculations with a minimally renormalized realistic interaction, supplemented by Quasi and Pseudo-SU3 symmetries and Nilsson-SU3 selfconsistent calculations. [Results for $N\le 64$] Shell model calculations with the neutron effective charge as single free parameter describe well the BE2(2>0) and BE2(4>2) rates for $N\le 64$ in the Cd and Sn isotopes. The former exhibit weak permanent deformation corroborating the prediction of a Pseudo-SU3 symmetry, which remains of heuristic value in the latter, where the pairing force erodes the quadrupole dominance. Calculations in $10^7$ and $10^{10}$-dimensional spaces exhibit almost identical patterns: A vindication of the shell model. [Results for $N\ge 64$] Nilsson-SU3 calculations describe BE2 patterns in [112-120]Cd and [116-118]Sn isotopes having sizable quadrupole moment of non-rotational origin denoted as q-vibrations. No calculations are proposed for the heavier species, for which the conventional seniority dscription is assumed for Sn, while in Cd the quadrupole moments change sign. [Conclusion] A radical reexamination of traditional interpretations in the region has been shown to be necessary, in which quadrupole dominance plays a major role. What emerges is a bumpy but coherent view.

Sn(II) chloride speciation and equilibrium Sn isotope fractionation under hydrothermal conditions: A first principles study

Knowledge of Sn(II) speciation in aqueous and gaseous phases and the corresponding isotope effects are critical for understanding the transport and deposition of Sn in various geological and cosmochemical processes. In this study, we use first principles method to investigate the speciation of stannous (Sn(II)) chloride in fluids under hydrothermal and supercritical conditions. The results show that SnCl3−, SnCl2(H2O) and SnCl(H2O)2+ are stable in hydrothermal solutions at temperatures of up to 300 °C, with SnCl3− being the dominant species, whereas SnCl2 and SnCl2(H2O) are the stable species in vapor phases. Notably, SnCl2 is found to be stable under supercritical conditions. The reduced partition function ratios (β factors) for the stable Sn(II) species and three major Sn minerals (cassiterite, megawite, and romarchite) are also calculated by first principles methods. The calculation results show that under equilibrium, heavy Sn isotopes are preferentially partitioned into stannic (Sn(IV)) species, and gaseous species enrich heavy Sn isotopes relative to aqueous species. Based on the equilibrium Sn isotope fractionation factors derived in this study, we use a transport-precipitation model to evaluate the Sn isotope response to cassiterite precipitation in hydrothermal fluids. The modeling results show that significant Sn isotope variability could be produced during cassiterite precipitation, with temperature and Sn speciation being the primary controlling factors. Furthermore, by comparing the Sn isotope variability in natural cassiterite and those derived from the model, we argue that Sn should occur predominantly as Sn(IV) species in hydrothermal fluids during cassiterite precipitation in tin mineralizing systems.

Quasi-SU(3) coupling of ( 1h11/2,2f7/2 ) across the N=82 shell gap: Enhanced E2 collectivity and shape evolution in Nd isotopes

The essence of the quasi-SU(3) coupling scheme suggested by Zuker et al. is that the high-$j$ intruder orbit alone is insufficient to induce required large collectivity, and it is necessary for the high-$j$ orbit to correlate through the quadrupole interaction with another higher-lying orbit with $\mathrm{\ensuremath{\Delta}}j=2$. To extend this idea to the medium-heavy mass region, we investigate the systematics of energy levels and $B(E2)$ values for Sn, Te, Xe, Ba, Ce, Nd, and Sm isotopes by applying the recently proposed realistic PMMU shell model. The calculations are performed by using the projected Hartree-Fock-Bogolyubov plus generator coordinate method in the model space of ($1{g}_{9/2},1{g}_{7/2},2{d}_{5/2},2{d}_{3/2},3{s}_{1/2},1{h}_{11/2}$). The calculations describe well the experimental data over a wide range of nuclei. However, it is found that, in some nuclei close to the neutron midshell, $^{124,126}\mathrm{Ce}$, $^{130,132}\mathrm{Nd}$, and $^{134}\mathrm{Sm}$, the calculated $B(E2;{2}_{1}^{+}\ensuremath{\rightarrow}{0}_{1}^{+})$ values underestimate the data considerably. This problem can be resolved by inclusion of the $2{f}_{7/2}$ orbit into the model space, which, with $1{h}_{11/2}$, forms a quasi-SU(3) coupling scheme. It is shown that the $QQ$ interaction between the SU(3)-partner $1{h}_{11/2}\text{\ensuremath{-}}2{f}_{7/2}$ is a driving force for enhanced $E2$ collectivity in the above nuclei and is responsible for the shape evolution in Nd isotopes. In addition, the participation of the $2{f}_{7/2}$ orbit in the model space favors prolate shape in the ground state of $N\ensuremath{\le}76$ isotopes, thus subverting the oblate result from the same calculation but without $2{f}_{7/2}$.

Isospin-density-dependent pairing from infinite nuclear matter to finite nuclei

The effective isospin-density dependent pairing interaction (P1) [S. S. Zhang, U. Lombardo and E. G. Zhao, Sci. Chin. Phys. Mech. Astro. {\bf 54}, 236 (2011)] extracted from neutron pairing gaps for $^1$S$_0$ in asymmetric nuclear matter calculations [S. S. Zhang, L. G. Cao, U. Lombardo, et al. Phys. Rev. C {\bf 81}, 044313 (2010)] is employed to study the bulk properties of Ca, Ni, Zr and Sn isotopes. The odd-even mass (OEM) staggering is calculated by the Skyrme Hartree-Fock plus BCS method (SHF + BCS) with the SkP interaction. For comparison, we study two other types of isovector effective pairing interactions. One is also extracted from pairing gaps of infinite nuclear matter by the Brueckner-Hartree-Fock (BHF) method but for free spectrum (P2). The other is obtained by fitting the empirical OEM (P3). An isoscalar effective pairing interaction (P4) is also adopted which is determined by fitting the empirical OEM. We find that interaction P1 can better describe the OEM staggering of Ni, Zr and Sn isotopes by $14.3 \%, 41 \%, 30.4 \%$ compared with interaction P2, in terms of root mean square deviations to the empirical OEM, respectively. On the other hand, the performance of P1 and P2 is comparable for Ca isotopes. For Ca and Ni isotopes, P1 behaves similarly as P3, but for Zr isotopes P1 is better than P3 by $\sim 34 \%$. Among the four pairing interactions studied, P4 performs the worst. Therefore, one may conclude that for neutron pairings in finite nuclei, the isovector pairings are preferred than the isoscalar one. It is quite interesting to note that the pairing interaction P1 extracted from nuclear matter calculations can describe pairing gaps of finite nuclei as well as or even better than the interaction P3 directly fitted to finite nuclei.

Evolution of collectivity in Xe118

A recoil-distance Doppler shift experiment has been performed using the $^{102}\mathrm{Pd}(^{19}\mathrm{F},p2n$) reaction at a beam energy of 73 MeV to measure the lifetime of excited states in $^{118}\mathrm{Xe}$. The differential decay-curve method using $\ensuremath{\gamma}\ensuremath{\gamma}$ coincidences and a gating procedure that allows to extract the lifetime without feeding assumptions has been employed. The lifetimes obtained for the yrast states up to spin-parity ${8}^{+}$ are compared with interacting boson model calculations and $^{118}\mathrm{Xe}$ can be classified as a transitional nucleus between the spherical and a deformed shape. Systematics of the $B(E2)$ values for the ${2}^{+}\ensuremath{\rightarrow}{0}^{+}$ and ${4}^{+}\ensuremath{\rightarrow}{2}^{+}$ transitions in the isotopic chains of tin, tellurium and xenon are presented. It is proposed that a ``critical point'' exists at which the $B(E2;{4}^{+}\ensuremath{\rightarrow}{2}^{+})/B(E2;{2}^{+}\ensuremath{\rightarrow}{0}^{+})$ ratio drops to unity for lower neutron numbers within the isotopic chain. The position of the ``critical point'' varies with proton number, i.e., it is presumed to be located at the same mass number $A=114$ in the Sn, Te, and Xe isotopes.

Development of an accurate DWIA model of coherent π0 —photoproduction to study neutron skins in medium heavy nuclei

Despite decades of studies which have seen the nuclear charge distribution being measured with increasing precision, the neutron distribution remains elusive. The difference between the neutron and proton distributions is often expressed as the difference of their root mean square radii: the neutron skin thickness. Recently, the A2 collaboration at MaMi has measured the skin thickness in lead through coherent pion photoproduction [1] with a very high precision. However, they do not include theoretical uncertainties, which can be significant for this process.A new reaction code in the distorted wave impulse approximation (DWIA) is developed to help the (ongoing) analysis of the recent measurement by the A2 collaboration at MaMi of the coherent pion photoproduction cross section on 116,120,124Sn isotopes [2] and to properly quantify the theoretical uncertainties.

Calculation of nuclear charge radii with a trained feed-forward neural network

A feed-forward neural network model is trained to calculate the nuclear charge radii. The model trained with input data set of proton and neutron number $Z,N$, the electric quadrupole transition strength $B(E2)$ from the first excited 2$^+$ state to the ground state, together with the symmetry energy. The model reproduces well not only the isotope dependence of charge radii, but also the kinks of charge radii at the neutron magic numbers $N=82$ for Sn and Sm isotopes, and also $N=126$ for Pb isotopes. The important role of $B(E2)$ value is pointed out to reproduce the kink of the isotope dependence of charge radii in these nuclei. Moreover, with the inclusion of the symmetry energy term in the inputs, the charge radii of Ca isotopes are well reproduced. This result suggests a new correlation between the symmetry energy and charge radii of Ca isotopes. The Skyrme HFB calculation is performed to confirm the existence of this correlation in a microscopic model.

AbInitio Computation of Charge Densities for Sn and Xe Isotopes.

We present the first abinitio 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 NNLO_{sat} 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 abinitio studies of exotic charge density distributions at the limit of the present abinitio mass domain, where experimental data is becoming available. The present study closes the gap between the largest isotopes reachable by abinitio methods and the smallest exotic nuclei accessible to electron scattering experiments.

Bayesian inference of nuclear symmetry energy from measured and imagined neutron skin thickness in Sn116,118,120,122,124,130,132, Pb208 , and Ca48

The neutron skin thickness $\Delta r_{np}$ in heavy nuclei has been known as one of the most sensitive terrestrial probes of the nuclear symmetry energy around $\frac{2}{3}$ of the saturation density $\rho_0$ of nuclear matter. Existing neutron skin data mostly from hadronic observables suffer from large uncertainties and their extraction from experiments are often strongly model dependent. While waiting eagerly for the promised model-independent and high-precision neutron skin data for $^{208}$Pb and $^{48}$Ca from the parity-violating electron scattering experiments (PREX-II and CREX at JLab as well as MREX at MESA), within the Bayesian statistical framework using the Skyrme-Hartree-Fock model we infer the posterior probability distribution functions (PDFs) of the slope parameter $L$ of the nuclear symmetry energy at $\rho_0$ from imagined $\Delta r_{np}(^{208}$Pb$)=0.15$, 0.20, and 0.30 fm as well as $\Delta r_{np}(^{48}$Ca$)=0.12$, 0.15, and 0.25 fm, with different $1\sigma$ error bars. The results are compared with the PDFs of $L$ inferred using the same approach from the available $\Delta r_{np}$ data for Sn isotopes from hadronic probes. They are also compared with results from a recent Bayesian analysis of the radius and tidal deformability data of canonical neutron stars from GW170817 and NICER. The neutron skin data for Sn isotopes gives $L=45.5^{+26.5}_{-21.6}$ MeV surrounding its mean value or $L=53.4^{+18.6}_{-29.5}$ MeV surrounding its maximum {\it a posteriori} value, respectively, with the latter smaller than but consistent with the $L=66_{-20}^{+12}$ MeV from the neutron star data within their 68\% confidence intervals. In order to provide additionally useful information on $L$ extracted from the $\Delta r_{np}$ of Sn isotopes, the experimental error bar of $\Delta r_{np}(^{208}$Pb) should be at least smaller than 0.06 fm aimed by some current experiments.

A unified view of the first-excited 2+ and 3− states of Cd, Sn and Te isotopes

Symmetries are known to play an important role in the low lying level structure of Sn isotopes, mostly in terms of the seniority and generalized seniority schemes. In this paper, we revisit the multi-j generalized seniority approach for the first excited $2^+$ and $3^-$ states in the Cd ($Z=48$), Sn ($Z=50$) and Te ($Z=52$) isotopes, where the Cd and Te isotopes represent two-proton hole and two-proton particle nuclei, thus involving both kind of particles (protons and neutrons) in contrast to Sn isotopes. Interestingly, the approach based on neutron valence space alone is able to explain the B(E2) and B(E3) trends respectively for the $2^+$ and $3^-$ states in all the three Cd, Sn and Te isotopes. The new results on the inverted parabolic behavior of B(E3) values in Cd and Te isotopes are understood in a manner identical to that of Sn isotopes by using the generalized seniority scheme. No shell quenching is supported by these calculations; hence, the neutron magic numbers, $N=50$ and $N=82$, remain robust in these isotopic chains. It is quite surprising that the generalized seniority continues to be reasonably successful away from the semi-magic region, thus providing a unifying view of the $2^+$ and $3^-$ states.

Evolution of proton single-particle states in neutron-rich Sb isotopes beyond N=82

The β decay of the semimagic Sn isotopes Sn136,137,138 has been studied at the Radioactive Isotope Beam Factory at the RIKEN Nishina Center. The first experimental information on excited states was obtained for Sb137 while, in the case of Sb136, the established excitation scheme could be extended by ten previously unidentified levels. In the decay of the most-neutron-rich isotope Sn138, two γ rays were observed for the first time. The new experimental results, in combination with state-of-the-art shell-model calculations, provide the first information with respect to the evolution of the 0g7/2 and 1d5/2 proton single-particle states with increasing neutron number beyond N=84.

Provenance of tin in the Late Bronze Age Balkans based on probabilistic and spatial analysis of Sn isotopes

We report the largest published dataset to date of Sn-isotopic compositions of Bronze Age artifacts (338) along with 150 cassiterite samples (75 new) from six potential tin ore sources from which the tin in these artifacts were thought to have likely originated. The artifacts are from a broad area, Central Europe through the Central Balkans, and the six tin sources are Cornwall, three sites in the Erzgebirge, and two sites in Serbia. A clustering analysis on mean site-level isotopic values of δ124Sn identifies regional variation that can be attributed to the use of different tin ore sources in different regions. Therefore, geographically meaningful regions were identified to group the Bronze Age artifact assemblages and a probabilistic, Bayesian analysis was performed to determine the proportional contribution of each tin source to each regional assemblage. Artifacts enriched in heavy isotopes (δ124Sn > 0.7‰) that cluster in west-central Serbia are likely associated with the ores from Mt. Cer in west Serbia. Mixed artifact assemblages (high and low δ124Sn) in this region are attributed to the use of cassiterite from the two Serbian sites (Mt. Cer and Mt. Bukulja). Moderate composition artifacts that occur north of the Middle Danube in Vojvodina, Transylvania, and Central Europe are likely associated primarily with ores from the West Pluton of the Erzgebirge. Compositionally light bronzes (δ124Sn < 0.2‰) in southern Serbia and the lower Danube river valley cannot be linked to a documented ore source. There is no indication of the use of ores from Cornwall or the East Pluton of the Erzgebirge in Central Europe and the Balkans during the Late Bronze Age.

Core swelling in spherical nuclei: An indication of the saturation of nuclear density

Background: Nuclear radius is one of the most important and basic properties of atomic nuclei and its evolution is closely related to the saturation of the nuclear density in the internal region but the systematics of the nuclear radii for the neutron-rich unstable nuclei is not well known. Purpose: Motivated by the recent interaction cross section measurement which indicates the 48Ca core swelling in the neutron-rich Ca isotopes, we explore the mechanism of the enhancement of the neutron and proton radii for spherical nuclei. Methods: Microscopic Hartree-Fock calculations with three sets of Skyrme-type effective interactions are performed for the neutron-rich Ca, Ni and Sn isotopes. The total reaction cross sections for the Ca isotopes are evaluated with the Glauber model to compare them with the recent cross section data. Results: We obtain good agreement with the measured cross sections and charge radii. The neutron and proton radii of the various "core" configurations are extracted from the full Hartree-Fock calculation and discuss the core swelling mechanism. Conclusions: The core swelling phenomena occur depending on the properties of the occupying valence single-neutron states to minimize the energy loss that comes from the saturation of the densities in the internal region, which appears to be prominent in light nuclei such as Ca isotopes.