Odd-even Staggering Research Articles

Number of particles in fission fragments

In current simulations of fission, the number of protons and neutrons in a given fission fragment is almost always obtained by integrating the total density of particles in the sector of space that contains the fragment. Because of the antisymmetry of the many-body wave function of the whole nucleus, this procedure systematically gives noninteger numbers of particles in the fragments. We introduce a novel sampling method to estimate rigorously the probability of finding $Z$ protons and $N$ neutrons in a fission fragment without resorting to projectors, which can sometimes give unwieldy results. When applied on standard Hartree-Fock-Bogoliubov many-body states, we show that our approach reproduces indeed the results of full particle-number projection. We then estimate the charge and mass number dispersion of several scission configurations in $^{240}$Pu with and without pairing correlations included. We show that odd-even effects in the charge probability naturally occur within our approach, which could explain the well-known odd-even staggering of charge distributions. Our method is applicable either in static calculations of scission configurations such as, e.g., in the macroscopic-microscopic approach or energy density functional theory, but also in explicitly time-dependent density functional theory simulations of fission.

Isospin influence on the decay of compound nuclei formed in 78,82Kr + 40Ca and 78,86Kr + 40,48Ca reactions

Isospin influence on the decay of hot and rotating compound systems 118,122Ba⁎ and 118,134Ba⁎ formed in 78,82Kr + 40Ca and 78,86Kr + 40,48Ca reactions at incident energies 5.5 and 10 MeV/A, respectively, has been studied using the dynamical cluster-decay model. Binding energies used in the model have been calculated using semi-empirical mass formula of Davidson et al., where the bulk and asymmetry constants have been adjusted to reproduce the ground state mass excess for all the nuclei which have been added to AME2003 and mass table in Möller et al. 1995 to build AME2016 and FRDM2012 data. The theoretical mass excess is considered only for those cases where experimental mass excess was not available. The shell corrections have been calculated with shell closures at 2, 8, 14, 28, 50, 82, 126, 184 for both protons and neutrons and 258 for neutron only. The influence of isospin and incident energies have been seen on the following: mass and charge distribution of the fragments, de-excitation/evaporation, binary decay/fission processes of the compound system, odd-even staggering, isotopic composition, angular momentum distribution of preformation probabilities and decay cross-sections for evaporation and fission like events. The calculated cross-sections for evaporation and fission like events have been compared with experiments and the role of angular momentum on decay/fission processes have been discussed.

Consistency of two different approaches to determine the strength of a pairing residual interaction in the rare-earth region

Two fits of the pairing residual interaction in the rare-earth region are independently performed. One is made on the odd-even staggering of masses by comparing measured and explicitly calculated three-point binding-energy differences centered on odd-even nuclei. Another deals with the moments of inertia of the first 2+ states of well deformed even-even nuclei upon comparing experimental data with the results of Inglis-Belyaev moments (supplemented by a crude estimate of the so-called Thouless-Valatin corrections). The sample includes 24 even-even and 31 odd-mass nuclei selected according to two criteria: they should have good rotor properties and should not correspond to low pairing-correlation regimes in their ground states. Calculations are performed in the self-consistent Hartree-Fock plus BCS framework (implementing a self-consistent blocking in the case of odd-mass nuclei). The Skyrme SIII parametrization is used in the particle-hole channel and the fitted quantities are the strengths of |Tz|=1 proton and neutron seniority residual interactions. As a result, the two fits yield sets of strengths in excellent agreement: about 0.1% for the neutron parameters and 0.2% for protons. In contrast when one performs such a fit on odd-even staggering from quantities deduced from BCS gaps or minimal quasiparticle energies in even-even nuclei, as is traditional, one obtains results significantly different from those obtained in the same nuclei by a fit of moments of inertia. As a conclusion, beyond providing a phenomenological tool for microscopic calculations in this region, we have illustrated the proposition made in the seminal paper of Bohr, Mottelson and Pines that moments of inertia and odd-even staggering in selected nuclei were excellent measuring sticks of nuclear pairing correlations.

Inverse odd-even staggering in nuclear charge radii and possible octupole collectivity in At217,218,219 revealed by in-source laser spectroscopy

Hyperfine-structure parameters and isotope shifts for the 795-nm atomic transitions in $^{217,218,219}$At have been measured at CERN-ISOLDE, using the in-source resonance-ionization spectroscopy technique. Magnetic dipole and electric quadrupole moments, and changes in the nuclear mean-square charge radii, have been deduced. A large inverse odd-even staggering in radii, which may be associated with the presence of octupole collectivity, has been observed. Namely, the radius of the odd-odd isotope $^{218}$At has been found to be larger than the average of its even-$N$ neighbors, $^{217,219}$At. The discrepancy between the additivity-rule prediction and experimental data for the magnetic moment of $^{218}$At also supports the possible presence of octupole collectivity in the considered nuclei.

Sensitivity of BCS solutions to different odd–even staggering finite-difference formulas

The present work studies the effect of odd–even staggering (OES) value of nuclear masses on the solution of the BCS set of equations. Three finite-difference formulas for the OES are used. First, a comparison between the three different formulas applied to even–even Ti, Fe, Ge, Mo, Te and Sm isotopic chains is introduced. The effect of the three formulas on the outputs of the BCS equations is then investigated. The comparison between the three different formulas as well as their effect on some of the BCS outputs is clearly significant. One of the BCS output parameters, the valence shell occupancy, is compared with the corresponding available experimental values.

Two-neutron removal cross sections from C15,16 at around 240 MeV/nucleon

The cross sections for two-neutron removal from $^{15,16}\mathrm{C}$ on a carbon target have been measured at around 240 MeV/nucleon. The measured cross section for $^{15}\mathrm{C}$ is smaller than for $^{16}\mathrm{C}$. The trends of the cross sections for $^{15,16}\mathrm{C}$ are inconsistent with the previously reported experimental data, but in agreement with the theoretical predictions based on eikonal-model calculations. Based on the present results of $^{15,16}\mathrm{C}$ combined with the available experimental data on $^{17--20}\mathrm{C}$, an odd-even staggering of the two-neutron removal cross sections along the neutron-rich carbon isotopic chain from $^{15}\mathrm{C}$ to $^{20}\mathrm{C}$ is observed, and this feature is well reproduced by the theoretical calculations.

Number of spin- J states and odd-even staggering for identical particles in a single- j shell

In this work, new recursion relations for the number of spin-J states for identical particles in a single-j shell are presented. Such relations are obtained using the generating-function technique, which enables one to exhibit an odd-even staggering in the spin distribution of an even number of fermions in a single-j shell: the number of states with an even value of J is larger than the number of states with an odd value of J. An analytical expression of the excess of states with an even value of J is provided, and its asymptotic behavior for large values of j is discussed.

Influence of the nuclear level density on the odd-even staggering in Fe56+p spallation at energies from 300 to 1500 MeV/nucleon

Background: Special attention has been paid to study the shell effect and odd-even staggering (OES) in the nuclear spallation.Purpose: In this paper, we investigate the influence of the nuclear level density on the OES in the $^{56}\mathrm{Fe}+p$ spallations at energies from 300 to 1500 MeV/nucleon.Method: The isospin-dependent quantum molecular dynamics (IQMD) model is applied to produce the highly excited and equilibrium remnants, which is then de-excited using the statistical model gemini. The excitation energy of the heaviest hot fragments is applied to match the IQMD model with the gemini model. In the gemini model, the statistical description of the evaporation are based on the Hauser-Feshbach formalism, in which level density prescriptions are applied.Results: By investigating the OES of the excited pre-fragments, it is found that the OES originates at the end of the decay process when the excitation energy is close to the nucleon-emission threshold energy, i.e., the smaller value of the neutron separation energy and proton separation energy. The strong influence of level density on the OES is noticed. Two types of the nuclear level densities, the discrepancy of which is only about 7% near the nucleon emission threshold energy, are used in the model. However, the calculated values of the OES differ by the factor of 3 for the relevant nuclei.Conclusions: It is suggested that, although the particle-separation energies play a key role in determining the OES, the level density at excitation energy lower than the particle-separation energies should be taken into consideration

Universal odd-even staggering in isotopic fragmentation and spallation cross sections of neutron-rich fragments

An evident odd-even staggering (OES) in fragment cross sections has been experimentally observed in many fragmentation and spallation reactions. However, quantitative comparisons of this OES effect in different reaction systems are still scarce for neutron-rich nuclei near the neutron drip line. By employing a third-order difference formula, the magnitudes of this OES in extensive experimental cross sections are systematically investigated for many neutron-rich nuclei with ($N\ensuremath{-}Z$) from 1 to 23 over a broad range of atomic numbers ($Z\ensuremath{\approx}3--50$). A comparison of these magnitude values extracted from fragment cross sections measured in different fragmentation and spallation reactions with a large variety of projectile-target combinations over a wide energy range reveals that the OES magnitude is almost independent of the projectile-target combinations and the projectile energy. The weighted average of these OES magnitudes derived from cross sections accurately measured in different reaction systems is adopted as the evaluation value of the OES magnitude. These evaluated OES magnitudes are recommended to be used in fragmentation and spallation models to improve their predictions for fragment cross sections.

Pairing-energy coefficients of neutron-rich fragments in spallation reactions**Supported by National Natural Science Foundation of China (U1732135), Natural Science Foundation of Henan Province (162300410179) and Henan Normal University for the Excellent Youth (154100510007)

The ratio of pairing-energy coefficient to temperature (ap/T) of neutron-rich fragments produced in spallation reactions has been investigated by adopting an isobaric yield ratio method deduced in the framework of a modified Fisher model. A series of spallation reactions, 0.5A and 1A GeV 208Pb + p, 1A GeV 238U + p, 0.5A GeV 136Xe + d, 0.2A, 0.5A and 1A GeV 136Xe + p, and 56Fe + p with incident energy ranging from 0.3A to 1.5A GeV, has been analysed. An obvious odd-even staggering is shown in the fragments with small neutron excess (I ≡ N-Z), and in the relatively small-A fragments which have large I. The values of ap/T for the fragments, with I from 0 to 36, have been found to be in a range from −4 to 4, and most values of ap/T fall in the range from −1 to 1. It is suggested that a small pairing-energy coefficient should be considered in predicting the cross sections of fragments in spallation reactions. It is also concluded that the method proposed in this article is not good for fragments with A/As > 85% (where As is the mass number of the spallation system).

Candidate chiral doublet bands in Pm138

High-spin states of $^{138}\mathrm{Pm}$ have been populated using the $^{124}\mathrm{Te}(^{19}\mathrm{F},\phantom{\rule{0.28em}{0ex}}5\mathrm{n})^{138}\mathrm{Pm}$ reaction at a beam energy of 105 MeV. A new positive-parity side band with the same $\ensuremath{\pi}{h}_{11/2}\ensuremath{\bigotimes}\ensuremath{\nu}{h}_{11/2}$ configuration as that of the yrast band is observed in $^{138}\mathrm{Pm}$. The properties of the two positive-parity bands show general agreement with the fingerprints of chiral rotation and thus these two bands are suggested to be candidates for near degenerate chiral doublet bands. Besides, odd-even spin staggering of the $\ensuremath{\pi}{h}_{11/2}\ensuremath{\bigotimes}\ensuremath{\nu}{h}_{11/2}$ bands is studied systematically in odd-odd Cs, La, Pr, and Pm isotopes. As a result of this study we suggest that the spin value of lowest observed state of the yrast band be reassigned as 9 in $^{138}\mathrm{Pm}$. This new spin assignment is also supported by the argument of alignment additivity.

Role of deformation in odd-even staggering in reaction cross sections for Ne30,31,32 and Mg36,37,38 isotopes

We discuss the role of pairing anti-halo effect in the observed odd-even staggering in reaction cross sections for $^{30,31,32}$Ne and $^{36,37,38}$Mg isotopes by taking into account the ground state deformation of these nuclei. To this end, we construct the ground state density for the $^{30,31}$Ne and $^{36,37}$Mg nuclei based on a deformed Woods-Saxon potential, while for the $^{32}$Ne and $^{38}$Mg nuclei we also take into account the pairing correlation using the Hartree-Fock-Bogoliubov method. We demonstrate that, when the one-neutron separation energy is small for the odd-mass nuclei, a significant odd-even staggering still appears even with finite deformation, although the degree of staggering is somewhat reduced compared to the spherical case. This implies that the pairing anti-halo effect in general plays an important role in generating the odd-even staggering in reaction cross sections for weakly bound nuclei.

Microscopic analysis of octupole shape transitions in neutron-rich actinides with relativistic energy density functional**Supported by National Natural Science Foundation of China (11475140, 11575148)

Quadrupole and octupole deformation energy surfaces, low-energy excitation spectra, and electric transition rates in eight neutron-rich isotopic chains − Ra, Th, U, Pu, Cm, Cf, Fm, and No − are systematically analyzed using a quadrupole-octupole collective Hamiltonian model, with parameters determined by constrained reflection-asymmetric and axially-symmetric relativistic mean-field calculations based on the PC-PK1 energy density functional. The theoretical results of low-lying negative-parity bands, odd-even staggering, average octupole deformations ⟨β3⟩, and show evidence of a shape transition from nearly spherical to stable octupole-deformed, and finally octupole-soft equilibrium shapes in the neutron-rich actinides. A microscopic mechanism for the onset of stable octupole deformation is also discussed in terms of the evolution of single-nucleon orbitals with deformation.

Probing the Ga31 ground-state properties in the region near Z=28 with high-resolution laser spectroscopy

Magnetic-dipole and electric-quadrupole moments for $^{65,67,69,75,79\text{--}82}\mathrm{Ga}$ are reported using the Collinear Resonance Ionization Spectroscopy (CRIS) technique at the ISOLDE facility, CERN. The moments of $^{65}\mathrm{Ga}$ have been measured for the first time: $\ensuremath{\mu}=+1.775(3)\phantom{\rule{4pt}{0ex}}{\ensuremath{\mu}}_{\mathrm{N}}$ and ${Q}_{s}=+21.0(15)\phantom{\rule{4pt}{0ex}}e\phantom{\rule{0.16em}{0ex}}{\mathrm{fm}}^{2}$. These results are compared to shell-model calculations using the GXPF1 and JUN45 interactions and the trends of the moments approaching the region of the doubly magic $^{56}\mathrm{Ni}$ are discussed. Additionally, new values for the change in mean-square charge radii for $^{65}\mathrm{Ga}$ and $^{67}\mathrm{Ga}$ allow investigation into the odd-even staggering in the region below $N=40$.

Monopole pairing correlations with random interactions

In this paper we study the monopole pairing correlation in ${0}^{+}$ states of semimagic nuclei under random interactions. We calculate overlaps between $S$-pair wave functions and exact shell-model wave functions. No dominance of the $S$-pair components is shown: In a single-$j$ shell ${0}^{+}$ states have random overlaps with the $S$-pair wave function; in a multi-$j$ space the ${0}_{1}^{+}$ state has a small overlap with the $S$-pair wave function, and this is particularly pronounced if the single-particle splitting is small and/or the spins $j$ of single-particle orbits are large. For the ${0}_{1}^{+}$ state in a multi-$j$ space, the generalized seniority scheme and the seniority truncation of the shell model provide similar results. We study the odd-even staggering of one-neutron separation energies under random interactions. The odd-even staggering phenomenon survives in random samples where all the even members have spin-zero ground states but disappears in random samples where all the even members have spin-nonzero ground states.

Weizsäcker-Skyrme-type nuclear mass formula incorporating two combinatorial radial basis function prescriptions and their application

Within the improved Weizs\acker-Skyrme (WS)-type nuclear mass formulas, we systematically calculated one-nucleon and two-nucleon separated energy, \ensuremath{\alpha}-decay and \ensuremath{\beta}-decay energies, and the odd-even staggering (OES) of nuclear binding energies. As a result, the root-mean-square (rms) deviations of 2267 nuclei within the new improved WS-type mass formula are dropped from 493 to 167 keV, where 2267 nuclei are extracted from the atomic mass evaluation of 2012. Simultaneously, all the rms deviations of one-nucleon and two-nucleon separation energies and decay energies ${Q}_{\ensuremath{\alpha}},{Q}_{{\ensuremath{\beta}}^{\ensuremath{-}}},{Q}_{{\ensuremath{\beta}}^{+}}$, and ${Q}_{\mathrm{EC}}$ for more than 3000 nuclei are cut down by about 100--400 keV. Further, some basic physical observations of 988 boundary nuclei are predicted for providing reference to experiments. Finally, the overall neutron OESs and proton OESs have been systemically investigated and the residual error satisfies a normal distribution. The pairing gaps ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{n}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{p}}$ of the isotopes of O, Ca, Ni, Zr, Sn, Gd, Qs, Pb, Pa, Ds and the isotonic magic chains of $N=28,50,82,126$ and even-even nuclei are also studied with dramatic improvements obtained. Especially, the rms of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{n}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{p}}$ in these nuclei have been reduced by about 200 keV. The above physical quantities show important information for nuclear charts and the features of nuclear structure.

Toward a global description of nuclear charge radii: Exploring the Fayans energy density functional

In this work, we study the odd-even effect in binding energies and charge radii, and systematic behavior of differential radii, to identify the underlying components of the effective nuclear interaction. We apply nuclear density functional theory using a family of Fayans and Skyrme energy density functionals fitted to similar datasets but using different optimization protocols. We inspect various correlations between differential charge radii, odd-even staggering in energies and radii, and nuclear matter properties. Detailed analysis is carried out for medium-mass and heavy semi-magic nuclei with a particular focus on the Ca chain. By making the surface and pairing terms dependent on density gradients, the Fayans functional offers the superb simultaneous description of odd-even staggering effects in energies and charge radii. Conversely, when the data on differential radii are added to the pool of fit-observables, the coupling constants determining the strengths of the gradient terms of Fayans functional are increased by orders of magnitude. The Skyrme functional optimized in this work with the generalized Fayans pairing term offers results of similar quality. We quantify these findings by performing correlation analysis based on the statistical linear regression technique. The nuclear matter parameters characterizing Fayans and Skyrme functionals optimized to similar datasets are fairly close.

Is a Trineutron Resonance Lower in Energy than a Tetraneutron Resonance?

We present quantum MonteCarlo calculations of few-neutron systems confined in external potentials based on local chiral interactions at next-to-next-to-leading order in chiral effective field theory. The energy and radial densities for these systems are calculated in different external Woods-Saxon potentials. We assume that their extrapolation to zero external-potential depth provides a quantitative estimate of three- and four-neutron resonances. The validity of this assumption is demonstrated by benchmarking with an exact diagonalization in the two-body case. We find that the extrapolated trineutron resonance, as well as the energy for shallow well depths, is lower than the tetraneutron resonance energy. This suggests that a three-neutron resonance exists below a four-neutron resonance in nature and is potentially measurable. To confirm that the relative ordering of three- and four-neutron resonances is not an artifact of the external confinement, we test that the odd-even staggering in the helium isotopic chain is reproduced within this approach. Finally, we discuss similarities between our results and ultracold Fermi gases.

Observation of new neutron-rich Mn, Fe, Co, Ni, and Cu isotopes in the vicinity of Ni78

Neutron-rich nuclei in the vicinity of $^{78}$Ni were produced using a $^{238}$U beam at the RIKEN Radioactive Isotope Beam Factory. The particle-identification plot for the in-flight fission fragments highlights the first observation of eight new isotopes: $^{73}$Mn, $^{76}$Fe, $^{77,78}$Co, $^{80,81,82}$Ni, and $^{83}$Cu. Although the β-decay half-lives of $^{77}$Co and $^{80}$Ni were recently reported by Xu [Phys. Rev. Lett. 113, 032505 (2014)]PRLTAO0031-900710.1103/PhysRevLett.113.032505 using data from the same experiment, the current work provides the first direct, quantitative evidence for the existence of these isotopes. The experimental production cross sections are reproduced in a satisfactory manner by theoretical predictions. An odd-even staggering of the cross sections was observed, and the effect appears to become more pronounced for the most exotic nuclei that were investigated. The staggering effect was interpreted as an increase of the neutron-evaporation probability for odd-N isotopes, owing to the decrease of the neutron-separation energy, Sn. The predicted cross section for $^{80}$Ni is significantly overestimated, which may be related to a weak binding of the neutron pair above the N=50 shell closure.

Penning-trap mass spectrometry and mean-field study of nuclear shape coexistence in the neutron-deficient lead region

We present a study of nuclear shape coexistence in the region of neutron-deficient lead isotopes. The midshell gold isotopes Au180,185,188,190 (Z=79), the two long-lived nuclear states in At197 (Z=85), and the neutron-rich nuclide At219 were produced by the ISOLDE facility at CERN and their masses were determined with the high-precision Penning-trap mass spectrometer ISOLTRAP. The studied gold isotopes address the trend of binding energies in a region of the nuclear chart where the nuclear charge radii show pronounced discontinuities. Significant deviations from the atomic-mass evaluation were found for Au188,190. The new trend of two-neutron separation energies is smoother, although it does reveal the onset of deformation. The origin of this effect is interpreted in connection to the odd-even staggering of binding energies, as well as theoretically by Hartree-Fock-Bogoliubov calculations including quasiparticle blocking. The role of blocking for reproducing the large odd-even staggering of charge radii in the mercury isotopic chain is illustrated.