Single-particle Level Scheme Research Articles

Fully microscopic scission-point model to predict fission fragment observables

We present an upgraded version of the SPY model, called SPY2 for version 2 of the scission point yield, to estimate mainly the yields and the kinetic energy distributions of fission fragments: The theoretical framework is similar to our previous version, i.e., a statistical scission point model, but this version is based on fully microscopic nuclear ingredients describing the fragments properties at the scission point. These include the static properties of some 7000 nuclei at 120 axial quadrupole deformations, such as binding energies, proton densities, single-particle level schemes, and states densities, coherently calculated within the constrained Hartree-Fock-Bogoliubov model on the basis of the Skyrme BSk27 interaction. The use of microscopic ingredients has been extended to the proton density distribution and the nuclear states densities. Considering realistic proton densities of fragments allows us to improve the definition of the scission point as well as the prediction of the kinetic energy distribution and to link the kinetic energy to the diffuseness of the fragments' proton density. New microscopic nuclear states densities improve the general coherence of the model, in particular regarding the inclusion of the odd-even pairing effect. In this updated SPY2 version, the calculation of the fission yields and kinetic energy distributions is significantly improved and found to be in relatively good agreement with experiments, at least qualitatively. A detailed study is performed for three well known fissioning systems, namely, thermal neutron induced fission of $^{235}\mathrm{U}$ and $^{239}\mathrm{Pu}$ and spontaneous fission of $^{252}\mathrm{Cf}$. A systematic analysis of the fission mode as well as mean fragments deformation and total kinetic energies has been performed for some 2000 fissioning nuclei with $78\ensuremath{\le}Z\ensuremath{\le}110$ lying between the proton and neutron drip lines.

Inversion of the Odd-Even Effect in Cold Fission from the Time-Dependent Pairing Equations

A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones for excitations energies of the fragments smaller than 4 MeV. In this contribution, a microscopic model is proposed for the explanation of this odd-even effect in cold fission. This explanation is based on a mixing configuration mechanism that is produced during the fission process. This configuration mixing mechanism is obtained dynamically by solving a the generalized system of time-dependent pairing equations, which include a pair-breaking effect. The time dependent equations of motion for the pair breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two fission fragment. The single particle level scheme was calculated with the Woods-Saxon superasymmetric two center shell model, providing a continuous variation of the single particle energies and of the wave functions from one nucleus up to two separated fragments. A first rule can be extracted from this model. The even-even fission products cannot be obtained at zero excitation energies because of the existence of dynamical excitations produced in the avoided- level-crossing regions when the nuclear system deforms slowly.

β-decay rates of odd-mass neutron-rich isotopes in the deformed quasiparticle random-phase approximation with realistic interactions

The deformed quasiparticle random-phase approximation with realistic nucleon-nucleon interactions is extended for the ${\ensuremath{\beta}}^{\ensuremath{-}}$ decay of odd-mass neutron-rich Kr, Sr, Zr, and Mo isotopes, from their longest-lived isotopes to the experimentally unknown nuclei. The particle-particle and particle-hole channels of residual interactions are handled in large single-particle model spaces, based on the Br\"uckner $G$ matrix with charge-dependent Bonn nucleon-nucleon forces. Both allowed Gamow-Teller and first-forbidden transitions are considered and different treatments for odd-mass systems are emphasized. The sensitivity of the calculated results to the single-particle level scheme and the particle-particle strength is discussed. The calculated Gamow-Teller strengths are analyzed, together with the contributions from first-forbidden transitions. The calculated half-lives are found to agree well with the experimental data over the orders of magnitude from ${10}^{\ensuremath{-}2}$ to ${10}^{3}$ s.

Microscopic description of the odd-even effect in cold fission

The time-dependent equations of motion for the pair-breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two-fission fragment. The single-particle level scheme was calculated with the Woods--Saxon superasymmetric two-center shell model. This model provides a continuous variation of the energies from one nucleus up to two separated fragments. The dissipated energy resorts from the time-dependent pairing equations. A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones. This odd-even effect for cold fission was explained microscopically.

Dissipation for Z-channels in fission, alpha-decay and cluster emission

The dissipated energy in fission, cluster and alpha decay was estimated microscopically by using the time dependent pairing equations. The single particle level schemes were determined in the framework of the superasymmetric two center shell model. A strong dependence of the dissipated energy as function of the mass asymmetry is evidenced. The dissipated energy is much lower for alpha decay and cluster emission than for fission.

Energy partition in low energy fission

The intrinsic excitation energy of fission fragments is dynamically evaluated in terms of the time dependent pairing equations. These equations are corroborated with two conditions. One of them fixes the number of particles and the another separates the pairing active spaces associated to the two fragments in the vicinity of the scission configuration. The fission path is obtained in the frame of the macroscopic-microscopic model. The single particle level schemes are obtained within the two center Woods-Saxon shell model. It is shown that the available intrinsic dissipated energy is not shared proportionally to the masses of the two fission fragments. If the heavy fragment possesses nucleon numbers close to the magic ones, the accumulated intrinsic excitation energy is lower than that of the light fragment.

RIPL – Reference Input Parameter Library for Calculation of Nuclear Reactions and Nuclear Data Evaluations

We describe the physics and data included in the Reference Input Parameter Library, which is devoted to input parameters needed in calculations of nuclear reactions and nuclear data evaluations. Advanced modelling codes require substantial numerical input, therefore the International Atomic Energy Agency (IAEA) has worked extensively since 1993 on a library of validated nuclear-model input parameters, referred to as the Reference Input Parameter Library (RIPL). A final RIPL coordinated research project (RIPL-3) was brought to a successful conclusion in December 2008, after 15 years of challenging work carried out through three consecutive IAEA projects. The RIPL-3 library was released in January 2009, and is available on the Web through (http://www-nds.iaea.org/RIPL-3/). This work and the resulting database are extremely important to theoreticians involved in the development and use of nuclear reaction modelling (ALICE, EMPIRE, GNASH, UNF, TALYS) both for theoretical research and nuclear data evaluations. The numerical data and computer codes included in RIPL-3 are arranged in seven segments: MASSES contains ground-state properties of nuclei for about 9000 nuclei, including three theoretical predictions of masses and the evaluated experimental masses of Audi et al. (2003). DISCRETE LEVELS contains 117 datasets (one for each element) with all known level schemes, electromagnetic andmore » {gamma}-ray decay probabilities available from ENSDF in October 2007. NEUTRON RESONANCES contains average resonance parameters prepared on the basis of the evaluations performed by Ignatyuk and Mughabghab. OPTICAL MODEL contains 495 sets of phenomenological optical model parameters defined in a wide energy range. When there are insufficient experimental data, the evaluator has to resort to either global parameterizations or microscopic approaches. Radial density distributions to be used as input for microscopic calculations are stored in the MASSES segment. LEVEL DENSITIES contains phenomenological parameterizations based on the modified Fermi gas and superfluid models and microscopic calculations which are based on a realistic microscopic single-particle level scheme. Partial level densities formulae are also recommended. All tabulated total level densities are consistent with both the recommended average neutron resonance parameters and discrete levels. GAMMA contains parameters that quantify giant resonances, experimental gamma-ray strength functions and methods for calculating gamma emission in statistical model codes. The experimental GDR parameters are represented by Lorentzian fits to the photo-absorption cross sections for 102 nuclides ranging from {sup 51}V to {sup 239}Pu. FISSION includes global prescriptions for fission barriers and nuclear level densities at fission saddle points based on microscopic HFB calculations constrained by experimental fission cross sections.« less

EMPIRE: Nuclear Reaction Model Code System for Data Evaluation

EMPIRE is a modular system of nuclear reaction codes, comprising various nuclear models, and designed for calculations over a broad range of energies and incident particles. A projectile can be a neutron, proton, any ion (including heavy-ions) or a photon. The energy range extends from the beginning of the unresolved resonance region for neutron-induced reactions (∽ keV) and goes up to several hundred MeV for heavy-ion induced reactions. The code accounts for the major nuclear reaction mechanisms, including direct, pre-equilibrium and compound nucleus ones. Direct reactions are described by a generalized optical model (ECIS03) or by the simplified coupled-channels approach (CCFUS). The pre-equilibrium mechanism can be treated by a deformation dependent multi-step direct (ORION + TRISTAN) model, by a NVWY multi-step compound one or by either a pre-equilibrium exciton model with cluster emission (PCROSS) or by another with full angular momentum coupling (DEGAS). Finally, the compound nucleus decay is described by the full featured Hauser-Feshbach model with γ-cascade and width-fluctuations. Advanced treatment of the fission channel takes into account transmission through a multiple-humped fission barrier with absorption in the wells. The fission probability is derived in the WKB approximation within the optical model of fission. Several options for nuclear level densities include the EMPIRE-specific approach, which accounts for the effects of the dynamic deformation of a fast rotating nucleus, the classical Gilbert-Cameron approach and pre-calculated tables obtained with a microscopic model based on HFB single-particle level schemes with collective enhancement. A comprehensive library of input parameters covers nuclear masses, optical model parameters, ground state deformations, discrete levels and decay schemes, level densities, fission barriers, moments of inertia and γ-ray strength functions. The results can be converted into ENDF-6 formatted files using the accompanying code EMPEND and completed with neutron resonances extracted from the existing evaluations. The package contains the full EXFOR (CSISRS) library of experimental reaction data that are automatically retrieved during the calculations. Publication quality graphs can be obtained using the powerful and flexible plotting package ZVView. The graphic user interface, written in Tcl/Tk, provides for easy operation of the system. This paper describes the capabilities of the code, outlines physical models and indicates parameter libraries used by EMPIRE to predict reaction cross sections and spectra, mainly for nucleon-induced reactions. Selected applications of EMPIRE are discussed, the most important being an extensive use of the code in evaluations of neutron reactions for the new US library ENDF/B-VII.0. Future extensions of the system are outlined, including neutron resonance module as well as capabilities of generating covariances, using both KALMAN and Monte-Carlo methods, that are still being advanced and refined.

The shell effects in s-wave neutron resonance level densities ρ according to combinatorial calculations and on the basis of the semi-classical approach

The results of calculations of level densities ρ, in the vicinity of the neutron binding energy Sn, are presented. These results were obtained using the Böhning combinatorial method for the calculation of particle-hole state densities dependent on the number of decompositions of the nucleus excitation energy to energies of independent fermions. The calculation was based on the semi-classical model description in the computation of particle-hole state densities and then of the level densities ρ, and takes into account the existence of energy gaps Δ, located near the Fermi level, in a single particle level scheme. This procedure considerably improved and extended the Böhning calculation method. The results, which were obtained in this way for ρ, for 220 nuclei, reproduce the regularities observed in the experimental values of ρ, which are dependent on the neutron number N, and they agree with the experimental data within two orders of magnitude. In addition, the neutron resonance densities ρ were calculated on the basis of the particle-hole state densities obtained using the analytical formula from Böhning's paper. To make the calculations possible, the values of ‘complexity’ k, as given in the semi-classical model, and the spin factors R(J), according to the paper by Ryckbosch, were used.

Dissipation effects in low energy fission

The dissipation energy is computed using the microscopic time-dependent equations of motion along minimal action trajectories for three different values of the mass-asymmetries in the 236U fission. The single-particle level schemes are given by the two-center shell model. A qualitative agreement with the experimental mass distribution was obtained when the dissipation energy was taken into consideration in computing the penetrabilities.

Uncertainty in the0νββdecay nuclear matrix elements

The nuclear matrix elements $M^{0\nu}$ of the neutrinoless double beta decay ($0\nu\beta\beta$) are evaluated for $^{76}$Ge,$^{100}$Mo, $^{130}$Te, and $^{136}$Xe within the Renormalized Quasiparticle Random Phase Approximation (RQRPA) and the simple QRPA. Three sets of single particle level schemes are used, ranging in size from 9 to 23 orbits. When the strength of the particle-particle interaction is adjusted so that the $2\nu\beta\beta$ decay rate is correctly reproduced, the resulting $M^{0\nu}$ values become essentially independent on the size of the basis, and on the form of different realistic nucleon-nucleon potentials. Thus, one of the main reasons for variability of the calculated $M^{0\nu}$ within these methods is eliminated.

Landau–Zener Effect in Superfluid Nuclear Systems

The Landau–Zener effect is generalized for many-body systems with pairing residual interactions. The microscopic equations of motion are obtained and the 14C decay of 223Ra spectroscopic factors are deduced. An asymmetric nuclear shape parametrization given by two intersected spheres is used. The single particle level scheme is determined in the frame of the superasymmetric two-center shell model. The deformation energy is computed in the microscopic–macroscopic approximation. The penetrabilities are obtained within the WKB approximation. The fine structure of the cluster decay analyzed in the frame of this formalism gives a very good agreement with the experimental ratio of partial half-lives for transition to the first excited state and to the ground state.

Nuclear state density calculations: An exact recursive approach

Using single-particle level scheme as input, we have developed a code to calculate total and particle-hole state densities both for even and odd nuclei. A mathematically exact recursive method is used for the solution of the problem within Fermi gas model. No residual interaction is considered.

Combinatorial nuclear level densities based on the Gogny nucleon-nucleon effective interaction

A combinatorial method to calculate total level densities from an arbitrary single-particle level scheme is presented. Parity, angular momentum, pairing correlations as well as collective enhancements are explicitly treated. This method is employed using single-particle level schemes obtained from Hartree-Fock-Bogoliubov calculations based on the Gogny effective interaction. Sixty five even-even nuclei with masses 26 ?A? 250 are considered. Rather good agreements are obtained when comparing our predictions with experimental data for energies of the order of the neutron binding energies and for low excitation energies where discrete levels are experimentally observed.

Generalized particle-hole state densities within the equidistant spacing model

We present an improved mathematical method to compute particle-hole state densities, with high numerical accuracy, for an arbitrary single particle level scheme. For the simplest case, the equidistant spacing model (ESM), we compare our exact particle-hole state density calculations with several existing and new analytical formulae. The latter are derived using the method of Berger and Martinot and lead to a generalization of the Obložinský formula for the two-component particle-hole state density. Our new formulae, which include correction terms up to any order, are discussed for various particle-hole configurations over a broad energy range.

Realistic microscopic level densities for spherical nuclei.

Nuclear level densities play an important role in nuclear reactions such as the formation of the compound nucleus. We develop a microscopic calculation of the level density based on a combinatorial evaluation from a realistic single-particle level scheme. This calculation makes use of a fast Monte Carlo algorithm allowing us to consider large shell model spaces which could not be treated previously in combinatorial approaches. Since our model relies on a microscopic basis, it can be applied to exotic nuclei with more confidence than the commonly used semiphenomenological formuals. An exhaustive comparison of our predicted neutron s-wave resonance spacings with experimental data for a wide range of nuclei is presented.

Yrast-yrare interaction strength and bandcrossing frequency.

Accurate particle-number-conserving calculation shows that in a single-j model the yrast-yrare interaction V is always strong and no periodic oscillation of V with the degree of shell filling is found, in contrast to the results obtained by the Hartree-Fock-Bogoliubov approximation. To understand the behavior of V and bandcrossing frequency ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, the spin-alignment, seniority structure, configuration structure, and ``quasiparticle structure'' of the yrast and yrare bands are analyzed in detail. Calculation in a two-j model (high-j intruder orbits plus normal orbits of opposite parity) was also carried out to illustrate schematically that a weak yrast-yrare interaction may occur for certain single-particle level scheme. The coexistence of normal low-j orbits with high-j intruder orbits is indispensible for a sharp backbending observed in some realistic nuclei.

A re-examination of the band structure of 223Ra and reflection asymmetry in this nucleus

A thorough examination of the level structure of 223Ra provides evidence that leads to an extensive re-evaluation of spin assignments and hence band structure. Altogether six bands corresponding to three parity-coupled pairs are identified and the interpretation of the data suggests that this N=135 nucleus may possess an octupole deformation. The band structure is interpreted within the framework of calculations using the Strutinski shell-correction method with a single-particle level scheme obtained with a Woods-Saxon mean field. The minimisation procedure for the total energy is achieved in a multideformation space including higher-order multipolarity deformations up to lambda =7. The importance of higher-order multipolarity deformation parameters is underlined.

Comparison of average s-wave resonance spacings from proton and neutron resonances.

Average s-wave resonance spacings were derived from high-resolution proton resonance data for 11 odd-Z nuclei in the mass range A=41--67 and compared with the average s-wave spacings obtained from neutron resonances for their odd-N isobars. After careful correction for missing levels the neutron and proton results are consistent; proton resonance data can be used to increase the data base for nuclear level densities. After correction to a common excitation energy, the spacings show a correlation with the shell correction energy ${E}_{\mathrm{shell}}$, especially for A=41, 55, and 59, where data exist for three isobars with quite different shell correction energies. The dependence of the average level spacing D\ifmmode\bar\else\textasciimacron\fi{} on ${E}_{\mathrm{shell}}$ is approximately exponential, in good agreement with the theoretical models of Ignatyuk et al. and Kataria et al. The data also have been used to calculate asymptotic a parameters in these two models. The mass dependence of these parameters can be described by the simple relation a=0.137A with an rms deviation of 4.3%. Attempts to describe the data within the Fermi gas model resulted in much larger fluctuations of the parameters with A and Z. Finally, the data were compared to the predictions of the microscopic Fermi gas model, using the single particle level scheme of Seeger and Perisho. At present, the quality of this description is still somewhat worse than with the phenomenological models of Ignatyuk et al. and Kataria et al.

Generators of collective vibrations in heavy nuclei

Solving harmonic equations of motion for a realistic single-particle potential, the generators of various isoscalar collective vibrations up to a multipolarity of 4 are extracted. Transition strengths, transition currents, and energy systematics are studied in comparison with full RPA calculations, fluid dynamical calculations, and experimental data. It is shown that all these states can be well described by simple generators and that the influence of the detailed structure of the residual two-body force is small compared with that of the single-particle level scheme and of geometric constraints.