Valence Proton Numbers Research Articles

Evolution of collectivity and neutron-proton interactions

In this paper we study the correlation between the evolution of nuclear collective motion and neutron-proton interactions. We construct empirical neutron-proton interactions (denoted by ${V}_{\mathrm{NP}}$) for nuclei in the $(28<Z<50,$ $50<N<82)$, $(50<Z<82,$ $50<N<82$), and $(50<Z<82,$ $82<N<126)$ major shells. The ratios between ${V}_{\mathrm{NP}}$ and ${N}_{\mathrm{n}}{N}_{\mathrm{p}}$ (the product of valence proton number and valence neutron number) are found to be different for particle-particle, particle-hole, and hole-hole regions of the same major shell, suggesting different average strengths of neutron-proton interactions for these regions. Very interestingly, collective observables of even-even nuclei, e.g., ${2}^{+}$ and ${4}^{+}$ state energies, which follow different trajectories in the ${N}_{\mathrm{n}}{N}_{\mathrm{p}}$ scheme for particle-particle, particle-hole, and hole-hole regions, are found to form one unified and compact trajectory if those collective observables are plotted in terms of total neutron-proton interactions, ${V}_{\mathrm{NP}}$.

Systematic study of unfavored α -decay half-lives of closed-shell nuclei related to ground and isomeric states

In present work, the unfavored $\mathcal{\alpha}$ decay half-lives and $\mathcal{\alpha}$ preformation probabilities of closed shell nuclei related to ground and isomeric states around $Z=82$, $N=82$ and 126 shell closures are investigated by adopting two-potential approach from the perspective of valence nucleon (hole) and isospin asymmetry of the parent nucleus. The results indicate that $\mathcal{\alpha}$ preformation probability is linear dependence on $N_p N_n$ or $N_p N_n I$, the same as the case of favored $\mathcal{\alpha}$ decay in our previous work [X.-D. Sun \textit{et al.}, Phys. Rev. C 94, 024338 (2016)]. $N_p$, $N_n$, and $I$ represent the number of valence protons (holes), the number of valence neutrons (holes), and the isospin of the parent nucleus, respectively. Fitting the $\mathrm{\alpha}$ preformation probabilities data extracted from the differences between experimental data and calculated half-lives without a shell correction, we give two linear formulas of the $\mathcal{\alpha}$ preformation probabilities and the values of corresponding parameters. Based on the formulas and corresponding parameters, we calculate the $\mathrm{\alpha}$ decay half-lives for those nuclei. The calculated results can well reproduce the experimental data.

Effectively-truncated large-scale shell-model calculations and nuclei around 100Sn

This paper presents a short overview of a procedure we have recently introduced, dubbed the double-step truncation method, which is aimed to reduce the computational complexity of large-scale shell-model calculations. Within this procedure, one starts with a realistic shell-model Hamiltonian defined in a large model space, and then, by analyzing the effective single particle energies of this Hamiltonian as a function of the number of valence protons and/or neutrons, reduced model spaces are identified containing only the single-particle orbitals relevant to the description of the spectroscopic properties of a certain class of nuclei. As a final step, new effective shell-model Hamiltonians defined within the reduced model spaces are derived by way of a unitary transformation of the original large-scale Hamiltonian. A detailed account of this transformation is given and the merit of the double-step truncation method is illustrated by discussing few selected results for 96Mo, described as four protons and four neutrons outside 88Sr. Some new preliminary results for light odd-tin isotopes from A = 101 to 107 are also reported.

Systematic study of favoredα-decay half-lives of closed shell odd-Aand doubly-odd nuclei related to ground and isomeric states

In this work, we systematically investigate the favored $\ensuremath{\alpha}$-decay half-lives and $\ensuremath{\alpha}$ preformation probabilities of both odd-$A$ and doubly-odd nuclei related to ground and isomeric states around the doubly magic cores at $Z=82, N=82$ and at $Z=82, N=126$, respectively, within a two-potential approach from the view of the valence nucleon (or hole). The results show that the $\ensuremath{\alpha}$ preformation probability is linearly related to ${N}_{\text{p}}{N}_{\text{n}}$ or ${N}_{\text{p}}{N}_{\text{n}}I$, where ${N}_{\text{p}}, {N}_{\text{n}}$, and $I$ are the number of valence protons (or holes), the number of valence neutrons (or holes), and the isospin of the parent nucleus, respectively. Fitting the $\ensuremath{\alpha}$ preformation probabilities data extracted from the differences between experimental data and calculated half-lives without a shell correction, we give two analytic formulas of the $\ensuremath{\alpha}$ preformation probabilities and the values of corresponding parameters. Using those formulas and the parameters, we calculate the $\ensuremath{\alpha}$-decay half-lives for those nuclei. The calculated results can well reproduce the experimental data.

Evolution of quadrupole collectivity inN=80isotones toward theZ=64subshell gap: TheB(E2;21+→01+)value ofSm142

It was shown that the evolution of the $B(E2;{2}_{1}^{+}\ensuremath{\rightarrow}{0}_{1}^{+})$ values in $N$ = 80 isotones from Te to Nd is affected by the underlying subshell structure. This manifests itself in the observation of the local suppression of the $B(E2)$ value at $Z=58$ with respect to the neighboring nuclei $^{136}\mathrm{Ba}$ and $^{140}\mathrm{Nd}$. To investigate this shell sensitivity toward the $Z=64$ subshell gap, the $B(E2;{2}_{1}^{+}\ensuremath{\rightarrow}{0}_{1}^{+})$ value of the unstable nucleus $^{142}\mathrm{Sm}$ was measured utilizing the projectile Coulomb excitation technique. The radioactive ion beam (RIB) experiment was performed at the REX-ISOLDE facility at CERN. The $B(E2)$ value of $32\phantom{\rule{0.16em}{0ex}}(4)\phantom{\rule{0.16em}{0ex}}\mathrm{W}.\mathrm{u}.$ reflects the impact of the $\ensuremath{\pi}(1{g}_{7/2}\phantom{\rule{0.16em}{0ex}}2{d}_{5/2})$ subshell closure at $Z=64$ with respect to a linear scaling of collectivity with valence proton number.

Nucleon separation energies in the valence correlation scheme

In this paper we study the one- and two-nucleon separation energies (${S}_{n}$, ${S}_{p}$, ${S}_{2n}$, and ${S}_{2p}$) by using the Atomic-Mass-Evaluation-2011 Preview. We show the linear dependence of separation energies, previously investigated for ${S}_{n}$ and ${S}_{p}$ of even-even nuclei, in terms of $\ensuremath{\alpha}{N}_{p}+\ensuremath{\beta}{N}_{n}$ (${N}_{p}$ and ${N}_{n}$ is the valence proton number and valence neutron number, respectively, with respect to the nearest magic closure), hold in a broader sense. It is applicable equally well to odd-mass and odd-odd nuclei. New odd-even staggerings are found for ${S}_{n}$ and ${S}_{p}$, and are discussed by using the pairing interaction and the symmetry energy. Predictive power of these simple relations is discussed.

Effective valence proton numbers for nuclei withZ∼64

The subshell effect for nuclei with proton number $Z\ensuremath{\sim}$ 64 has been known for many years. The most economic way to consider this effect is to use the effective valence proton number. In this Brief Report we extract effective valence proton numbers for nuclei in this region by using the systematics of the first ${2}^{+}$ energies (${E}_{{2}_{1}^{+}}$) of even-even nuclei, the ratios of the first ${4}^{+}$ and ${6}^{+}$ state energies with respect to ${E}_{{2}_{1}^{+}}$ (${R}_{4}$ and ${R}_{6}$), the $B(E2)$ values, the quadrupole deformation parameters ${e}_{2}$, and anomalous $g$ factors of the ${2}_{1}^{+}$ state for even-even nuclei. It is noticed that these physical quantities saturate when ${N}_{\mathrm{p}}{N}_{\mathrm{n}}$, the product of the valence proton number and the valence neutron number, is large enough; on the other hand, they go to saturation at different ``speeds.'' We show that the subshell effect is more evident for ${E}_{{2}_{1}^{+}}$ and yrast state energy ratios (${R}_{4}$ and ${R}_{6}$), and relatively less for other quantities.

Enhanced Quadrupole Collectivity atN=40: The Case of Neutron-Rich Fe Isotopes

The transition rates for the 2(1)+ states in (62,64,66)Fe were studied using the recoil distance Doppler-shift technique applied to projectile Coulomb excitation reactions. The deduced E2 strengths illustrate the enhanced collectivity of the neutron-rich Fe isotopes up to N = 40. The results are interpreted using the generalized concept of valence proton symmetry which describes the evolution of nuclear structure around N = 40 as governed by the number of valence protons with respect to Z ≈ 30. The trend of collectivity suggested by the experimental data is described by state-of-the-art shell-model calculations with a new effective interaction developed for the fpgd valence space.

Residual proton-neutron interactions and theNpNnscheme

We investigate the correlation between integrated proton-neutron interactions obtained by using the up-to-date experimental data of binding energies and the product ${N}_{p}{N}_{n}$ of valence proton number and valence neutron number with respect to the nearest doubly closed nucleus. We make corrections on a previously suggested formula for the integrated proton-neutron interaction. Our results demonstrate a nice, nearly linear, correlation between the integrated $p$-$n$ interaction and ${N}_{p}{N}_{n}$, which provides us with a firm foundation of the applicability of the ${N}_{p}{N}_{n}$ scheme to nuclei far from the stability line.

Simple parametrization of an α-decay spectroscopic factor in the 150 ⩽ A ⩽ 200 region

Half-life values for α-decay in the mass region A = 150–200 have been calculated in the microscopic super asymmetric fission model. The interaction between the α-particle and the daughter nucleus has been obtained in the double folding model with the microscopic interaction DDM3Y1. Theoretical densities for the nuclei involved have been calculated in the relativistic mean-field approach with the Lagrangian density FSU Gold. Spectroscopic factors for α-decay, calculated as the ratios of the theoretically calculated and experimentally measured half-life values, show a simple dependence on the mass number and the product of the number of valence protons and neutrons. The implications of the parametrization have been discussed. Finally, based on this simple relation, α-decay half-life values in a number of nuclei have been predicted.

A Simple Systematical Law of E21+ Values for HeavyNuclei in the NpNn Scheme and the Evolution of the Z = 64 Shell Gap

A systematics of excitation energy of the first 2+ state E21+in even–even heavy nuclei (A⩾120) is studied in the NpNnscheme. It is found that a simple exponential function describes thedependence of E21+ values on NpNn values very well. Inaddition, the Z = 64 shell gap is reexamined by investigating thesystematics of the 52⩽Z⩽66 region. It is found that theZ = 64 shell gap is largest at N = 82 and becomes smaller with eitherincreasing or decreasing of neutron numbers. The effects of this shellgap become negligible for counting the valence proton numbers whenN⩾92 or N⩽72.

Empirical Formula for the Excitation Energies of the First 2+ and 3- States in Even-Even Nuclei

We report empirical findings that a simple formula in terms of the mass number $A$, the valence proton number $N_p$, and the valence neutron number $N_n$ can describe the essential trends of excitation energies $E_x$ of the first $2^+$ and $3^-$ states in even-even nuclei throughout the periodic table. The formula reads as $E_x = \alpha A^{-\beta} + \exp (- \lambda N_p) + \exp (- \lambda N_n)$. The parameter $\beta$ in the first term is determined by the mass number $A$ dependence of the bottom contour line of the excitation energy systematics. The other two parameters $\alpha$ and $\lambda$ are fitted by minimizing the $\chi^2$ value between the measured and calculated excitation energies. Our results suggest that the single large-$j$ shell simulation can be applied to the excitation energies of the first $2^+$ and $3^-$ states in even-even nuclei.

Generalization of the N(p)N(n)N(p)N(n) scheme and the structure of the valence space

The N(p)N(n) scheme, which has been extensively applied to even-even nuclei, is found to be a very good benchmark for odd-even, even-odd, and doubly-odd nuclei as well. There are no apparent shifts in the correlations for these four classes of nuclei. The compact correlations highlight the deviant behavior of the Z = 78 nuclei and are used to deduce effective valence proton numbers near Z = 64 as well as to study the evolution of the Z = 64 subshell gap.

Measured magnetic moments and shape coexistence in the neutron-deficient nuclei 184,186,188Pt.

A novel implantation-decay technique has been employed to measure the magnetic moments of the ${2}_{1}^{+}$ states in the neutron-deficient nuclei ${}^{184}\mathrm{Pt}$, ${}^{186}\mathrm{Pt}$, and ${}^{188}\mathrm{Pt}$. The magnetic moment systematics for the even Pt isotopes now extend from ${}_{78}^{184}{\mathrm{Pt}}_{106}$ to ${}_{78}^{198}{\mathrm{Pt}}_{120}$, spanning the upper half of the valence neutron shell. Despite the prolate-to-oblate shape transition near $A\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}190$, they remain remarkably constant. The $g({2}_{1}^{+})$ values for ${}^{184,186,188}\mathrm{Pt}$ are consistent with shape-coexistence models in which the deformed configuration has a larger effective number of valence protons than the less-deformed configuration.

Multiparticle-octupole coupling and magnetic moments of [formula omitted] isomers in N = 126 isotones

The influence of particle-vibration coupling on the g-factors of the |(h 9 2 ) n〉 isomers in the N = 126 isotones is assessed using the multiparticle-octupole-coupling model. According to the model, admixtures of the configuration |(h 9 2 ) n−1 f 7 2 〉 in the yrast 8 + and 21 2 − states, nominally associated with the configuration |(h 9 2 ) n〉 , increase with n. On its own, the octupole mixing mechanism therefore predicts g-factors for these states that increase with the number of valence protons. This trend is the opposite of that predicted by core-polarization blocking. Combining multiparticle-octupole coupling and first-order core-polarization blocking significantly reduces the discrepancy between the experimental and theoretical g-factors of these states. We conclude that the observed breakdown in additivity for the g-factors of the |(h 9 2 n )〉 isomers in the N = 126 isotones arises primarily from first-order core-polarization blocking and the combination of configuration mixing due to multiparticle-octupole coupling and shell-model residual interactions.

Effect of valence nucleons on RMS charge radii and surface thickness

The nucleonic promiscuity factor P=NpNn/(Np+Nn), where Np(Nn) is the number of valence protons (neutrons) or holes, is shown to be a useful and physically meaningful parameter in the description of RMS charge radii. The introduction of P in the radius formula R approximately=Rc+0.07P/Rc (Rc; droplet model 'closed-shell radius') reduces the chi 2/n' value by a factor of three. The empirically found mass number dependence of the P-dependent contribution to radii is in agreement with the qualitative theoretical expectation for the average strength of the isoscalar p-n interaction. A significant correlation between surface diffusity and promiscuity as well as between diffusity and radius is pointed out.

Magnetic moments of the 21+ states in 146,148Ce.

The g factors of the 2/sub 1//sup +/ states in /sup 146,148/Ce have been measured using the time integral perturbed angular correlation technique. The levels were populated by the radioactive decay of mass-separated /sup 146,148/La which was produced by thermal neutron fission of /sup 235/U. The measured g(2/sub 1//sup +/) values are 0.24 +- 0.05 and 0.37 +- 0.06 for /sup 146/Ce and /sup 148/Ce, respectively. The results are interpreted as a probe of the number of valence protons participating in the collective motion. Data for Ba, Ce, Nd, and Sm nuclei in the region of N = 90 show a distinct change in the value of g(2/sub 1//sup +/) as the number of neutrons increases from 86 to 90. This change is attributed to the effects of the strong interaction between the ..pi..h/sub 11/2/ and ..nu..h/sub 9/2/ orbits which causes a gradual dissipation of the Z = 64 subshell gap.

Empirically determined limits on absolute boson numbers for well deformed nuclei

In the context of the consistent Q formalism of the interacting boson approximation, which contains only one essential parameter, denoted chi, it is possible to investigate the limits of allowable boson numbers N for well deformed nuclei. This is done by examining the agreement with experiment of certain E2 branching ratios as a function of N. The results show that, in general, the best agreement is obtained for Nroughly-equalN/sub IBA/, where N/sub IBA/ is the usual interacting boson approximation value defined as the sum of half the number of valence protons and neutrons; moreover, significant discrepancies with experiment are obtained if N<((3/4))N/sub IBA/. The effect of additional terms in the interacting boson approximation Hamiltonian is also investigated.