Calculation Of Level Densities Research Articles

Shell-model-based investigation on level density of Xe and Ba isotopes

Nuclear level density is important for evaluating nuclear reaction processes, but its microscopic investigation is not frequently performed. The present work applies a recently developed shell-model-based method to estimate the level density of fission products $^{133--137}\mathrm{Xe}$ and $^{135--138}\mathrm{Ba}$. The monopole-based universal interaction, ${V}_{\text{MU}}$, and the M3Y type spin-orbit interaction are combined to construct the shell-model Hamiltonian. The model space is truncated based on the binding energy of each configuration, estimated from the monopole interaction. The calculated level densities of $^{133--137}\mathrm{Xe}$ and $^{135--138}\mathrm{Ba}$ are in good agreement with available experimental data. The effects of spin-orbit and tensor forces on the nuclear level density and the shell effects in the spin distribution are discussed.

Statistical Fluctuations of Energy Spectra in the Isobar A = 68 Nuclei

Statistical fluctuations of nuclear energy spectra for the isobar A = 68 were examined by means of the random matrix theory together with the nuclear shell model. The isobar A = 68 nuclei are suggested to consist of an inert core of 56Ni with 12 nucleons in f5p-space (2p3/2, 1f5/2 and 2p1/2 orbitals). The nuclear excitation energies, required by this work, were obtained through performing f5p-shell model calculations using the isospin formalism f5pvh interaction with realistic single particle energies. All calculations of the present study were conducted using the OXBASH code. The calculated level densities were found to have a Gaussian shape. The distributions of level spacing P(s) and statistic for the considered classes of states, obtained with full Hamiltonian of f5pvh (absence of the off-diagonal Hamiltonian) calculations, showed a chaotic (regular) behavior and coincided well with the distribution of Gaussian orthogonal ensemble (Poisson). Moreover, these distributions were independent of spin (J) and isospin (T)

Excitation functions of the Zn(p,xn)Ga reactions

The excitation functions for 64,66,67,68,70Zn(p,n)64,66,67,68,70Ga and 64,66,67,68,70Zn(p,2n)63,65,66,67,69Ga have been calculated for incident proton energies from threshold values to 30 MeV using the statistical model code TALYS-1.6. The (p,n) and (p,2n) cross sections have been calculated using both phenomenological Koning and Delaroche (KD) and semimicroscopic Jeukenne-Lejeune-Mahaux-Bruyeres (JLMB) optical model potentials. The phenomenological back-shifted Fermi gas model (BFM) and microscopic Hartree-Fock (HF) approaches for calculation of level densities have been compared in the present work. The pre-equilibrium process has been treated using exciton model. The sensitivity of cross section data to different models of optical model potential and level density have been investigated. It is found that the (p,n) and (p,2n) cross section calculations with semimicroscopic JLMB optical model potential and microscopic HF level density show agreement with the data. The (p,n) and (p,2n) cross section calculations have been compared with corresponding cross sections from nuclear data library, TENDL-2019. The (p,xn) cross sections obtained will be useful in several applications, particularly, for optimization of production routes for Ga isotopes.

Role of magic numbers in thermodynamic quantities of 206Pb and 138Ba using BCS and Lipkin-Nogami models

The level density is calculated using the Barden-Cooper-Schriefer (BCS) and Lipkin-Nogami (LN) models for 206Pb and 138Ba nuclei and compared with the experimental data. It has been shown that the calculated level densities using both models are highly matched with the experimental data. The spin cut off factor, moment of inertia, excitation energy, and entropy were calculated using the above mentioned models. The protons in the 206Pb nucleus occupy a closed shell and the 138Ba nucleus has a closed neutron shell. It was revealed that the contribution of the nucleons with magic number to the mentioned thermodynamic quantities is less than that of the nucleons from open shells at low temperature.

The nuclear level density parameter

The nuclear level density parameter in non Equi-Spacing Model (NON-ESM), Equi-Spacing Model (ESM) and the Backshifted Energy Dependent Fermi Gas model (BSEDFG) was determined for 106 nuclei; the results are tabulated and compared with the experimental works. It was found that there are no recognizable differences between our results and the experimental -values. The calculated level density parameters have been used in computing the state density as a function of the excitation energies for 58Fe and 246Cm nuclei. The results are in a good agreement with the experimental results from earlier published work.

Nuclear level density with proton resonance using Gaussian orthogonal ensemble theory

The Gaussian orthogonal ensemble (GOE) version of the random matrix theory (RMT) has been used to study the level density following up the proton interaction with 44Ca, 48Ti and 56Fe.
 A promising analysis method has been implemented based on the available data of the resonance spacing, where widths are associated with Porter Thomas distribution. The calculated level density for the compound nuclei 45Sc,49Vand 57Co shows a parity and spin dependence, where for Sc a discrepancy in level density distinguished from this analysis probably due to the spin misassignment .The present results show an acceptable agreement with the combinatorial method of level density.

Evaluated the level density for proton induced nuclear resonances in (P+48Ti) reaction using different models

The experimental proton resonance data for the reaction P+48Ti have been used to calculate and evaluate the level density by employed the Gaussian Orthogonal Ensemble, GOE version of RMT, Constant Temperature, CT and Back Shifted Fermi Gas, BSFG models at certain spin-parity and at different proton energies. The results of GOE model are found in agreement with other, while the level density calculated using the BSFG Model showed less values with spin dependence more than parity, due the limitation in the parameters (level density parameter, a, Energy shift parameter, E1and spin cut off parameter, σc). Also, in the CT Model the level density results depend mainly on two parameters (T and ground state back shift energy, E0), which are approximately constant in their behavior with the proton energy compared with GOE model. The RMT estimation used to calculate the corrections of the incompleteness proton resonance measurement data by using two methods; the conventional analysis method, which depends on the resonance widths and the updated, developed, tested and applied a new analysis method, which depends mainly on the resonance spacings. The spacing analysis method is found much less sensitive to non-statistical phenomena than is the width analysis method. Where the analysis of a given data set via these two independent analysis methods indicated the increasing in the reliability of the determination of the missing fraction of levels, the observed fraction f between 0.87+0.13−0.11 – 0.68+0.12−0.12 for different spin-parity of this reaction and then the distinguishability in the level density calculations can be achieved. The modified Porter Thomas distribution along with the maximum likelihood function have been used to get the missing levels corrections for 5 proton resonance sequences in the present reaction. To estimate the present long-range correlations for pure sequence of levels the mean square of the deviation of the cumulative number of levels from a fitted straight line represented by the Dyson and Mehta Δ3 statistic has been employed for spin parity 12+, and calculated the <Δ3> against the cumulative number of proton levels.

Investigation of gamma strength functions and level density models effects on photon induced reaction cross–section calculations for the fusion structural materials 46,50Ti, 51V, 58Ni and 63Cu

Scientists have been focused on fusion reactor studies to overcome the increasing energy demand. The materials, which have the potential to be used in fusion reactors must be resistant to the harmful effects of radiation in the manner of material itself. Selection of the appropriate materials to be used in nuclear reactors has a crucial importance to achieve the maximum efficiency and security. Ti, V, Ni and Cu are known as some of the constructional fusion materials. Existence of many knowledge about those materials provides countless advantages to the researchers and one of them is the cross–section, which basically means the probability of a nuclear reaction's occurrence. In addition to the cross–section, there exist some other parameters, which could be pointed as gamma strength function and level density models that affect the theoretical calculations. In this study, photon induced reaction cross–sections of 46,50Ti, 51V, 58Ni and 63Cu target isotopes have been calculated by using TALYS 1.8 code with different gamma strength functions in the giant dipole resonance region. For gamma strength functions Kopecky-Uhl generalised Lorentzian Model, Brink-Axel Lorentzian Model, Hartree-Fock BCS tables, Hartree-Fock-Bogolyubov tables and Goriely's Hybrid Model have been employed. To appoint the best gamma strength function model, the relative variance calculations have been performed. Also, reaction cross–sections have been recalculated by using the best gamma strength function models through the different level density options. Constant Temperature Fermi Gas Model, Back Shifted Fermi Gas Model and Generalised Super Fluid Model have been employed for level density calculations. Experimental data for the investigated reactions have been taken from EXFOR library and used for comparisons of the obtained calculation results.

Fission dynamics with microscopic level densities

Working within the Langevin framework of nuclear shape dynamics, we study the dependence of the evolution on the degree of excitation. As the excitation energy of the fissioning system is increased, the pairing correlations and the shell effects diminish and the effective potential-energy surface becomes ever more liquid-drop like. This feature can be included in the treatment in a formally well-founded manner by using the local level densities as a basis for the shape evolution. This is particularly easy to understand and implement in the Metropolis treatment where the evolution is simulated by means of a random walk on the five-dimensional lattice of shapes for which the potential energy has been tabulated. Because the individual steps between two neighboring lattice sites are decided on the basis of the ratio of the statistical weights, what is needed is the ratio of the local level densities for those shapes, evaluated at the associated local excitation energies. For this purpose, we adapt a recently developed combinatorial method for calculating level densities which employs the same single-particle levels as those that were used for the calculation of the pairing and shell contributions to the macroscopic-microscopic deformation-energy surface. For each nucleus under consideration, the level density (for a fixed total angular momentum) is calculated microscopically for each of the over five million shapes given in the three-quadratic-surface parametrization. This novel treatment, which introduces no new parameters, is illustrated for the fission fragment mass distributions for selected uranium and plutonium cases.

Particle entropy and depairing in hot nuclei

The nuclear level densities and single particle entropies are predicted for nuclei in the mass region [Formula: see text] within a framework of statistical theory of hot nuclei method. In this method, particle-number and energy conservation as well as nuclear pairing correlations are included in the partition function of grand canonical ensemble. The suppression of pairing correlations is distinctly noticed in temperature dependence of entropies between the critical temperatures [Formula: see text] MeV and [Formula: see text] MeV for [Formula: see text], [Formula: see text] and [Formula: see text] isotopes of the elements. These structural thermodynamic entropies are interpreted as a remarkable signature of the superfluid to normal phase transition connected to the vanishing of pairing gap. The calculated level densities are compared with recent experimental values. In addition, the single particle entropy of intermediate-mass nuclei is depicted as half of the entropy of mid-shell nuclei in the rare-earth region. As a consequence, the [Formula: see text] shell closure of [Formula: see text]V carries low entropy at low excitation energy presents an interesting analogy to the [Formula: see text] shell closure of [Formula: see text]Ni. Merely, in the case of odd–even [Formula: see text] has higher entropy than the even–even [Formula: see text] nucleus.

Deexcitation Modes in Spallation Nuclear Reactions

Spallation nuclear reactions in the range of 0.2 to 1.2 GeV are studied using the CRISP code. A new approach for the deexcitation stage of the compound nucleus was introduced. For the calculations of the level densities, this approach is based on the Back-shifted Fermi gas model (BSFG), which takes into account pairing effects and shell corrections, whereas the calculation of the fission barriers were performed by means of the Extended Thomas-Fermi plus Strutinsky Integral (ETFSI) method, which is a high-speed approximation to the Hartree-Fock method with pairing correlations treated as in the usual BCS plus blocking approach. This procedure is more appropriate to calculate level densities for exotic nuclei. Satisfactory results were obtained and compared with experimental data obtained in the GSI experiments. As another important result, we highlight some directions for the development of a qualitatively superior version of the CRISP code with the implementation of more realistic and suitable physical models to be applied in stable and exotic nuclei that participate in the process. This new version of the code includes several substantial changes in the decay of the hot compound nucleus which allow satisfactory agreement with the experimental data and a reduction of the adjustment parameters.

Level densities of heavy nuclei in the shell model Monte Carlo approach

Nuclear level densities are necessary input to the Hauser-Feshbach theory of compound nuclear reactions. However, the microscopic calculation of level densities in the presence of correlations is a challenging many-body problem. The configurationinteraction shell model provides a suitable framework for the inclusion of correlations and shell effects, but the large dimensionality of the many-particle model space has limited its application in heavy nuclei. The shell model Monte Carlo method enables calculations in spaces that are many orders of magnitude larger than spaces that can be treated by conventional diagonalization methods and has proven to be a powerful tool in the microscopic calculation of level densities. We discuss recent applications of the method in heavy nuclei.

Moments Method for Shell-Model Level Density

The modern form of the Moments Method applied to the calculation of the nuclear shell-model level density is explained and examples of the method at work are given. The calculated level density practically exactly coincides with the result of full diagonalization when the latter is feasible. The method provides the pure level density for given spin and parity with spurious center-of-mass excitations subtracted. The presence and interplay of all correlations leads to the results different from those obtained by the mean-field combinatorics.

Effects of level density parameter on the superheavy production in cold fusion

By using semiclassical method and considering Woods–Saxon and Coulomb potentials, the level density parameter a was calculated for three superheavy nuclei 270110, 278112 and 290116. Obtained results showed that the value of level density parameter of these nuclei is near to the simple relation a ≈ A/10. In framework of the dinuclear system model, the effects of level density parameter on the probability of the formation of a compound nucleus, the ratio of neutron emission width and fission width, and evaporation residue cross-section of three cold fusion reactions 62 Ni +208 Pb , 70 Zn +208 Pb and 82 Se +208 Pb , leading to superheavy elements were investigated. The findings indicate that the level density parameter play a significant role in calculations of heavy-ion fusion–fission reactions. The obtained results in the case of a = A/12 have larger values in comparison with calculated level density parameter with Woods–Saxon potential (a WS ) and a = A/10. The theoretical results of the evaporation residue cross-section are very sensitive to the choice of level density parameter. The calculated values with a WS are in good agreement with experimental values.

Calculating Level Densities of Heavy Nuclei by the Shell Model Monte Carlo Method

The microscopic calculation of nuclear level densities in the presence of correlations is a difficult many-body problem. The shell model Monte Carlo method provides a powerful technique to carry out such calculations using the framework of the configuration-interaction shell model in spaces that are many orders of magnitude larger than spaces that can be treated by conventional methods. We present recent applications of the method to the calculation of level densities and their collective enhancement factors in heavy nuclei. The calculated level densities are in close agreement with experimental data.

Recent Advances in the Microscopic Calculations of Level Densities by the Shell Model Monte Carlo Method

The shell model Monte Carlo (SMMC) method enables calculations in model spaces that are many orders of magnitude larger than those that can be treated by conventional methods, and is particularly suitable for the calculation of level densities in the presence of correlations. We review recent advances and applications of SMMC for the microscopic calculation of level densities. Recent developments include (i) a method to calculate accurately the ground-state energy of an odd-mass nucleus, circumventing a sign problem that originates in the projection on an odd number of particles, and (ii) a method to calculate directly level densities, which, unlike state densities, do not include the spin degeneracy of the levels. We calculated the level densities of a family of nickel isotopes $^{59-64}$Ni and of a heavy deformed rare-earth nucleus $^{162}$Dy and found them to be in close agreement with various experimental data sets.

Temperature-dependent combinatorial level densities with the D1M Gogny force

The combinatorial model of nuclear level densities has now reached a level of accuracy comparable to that of the best global analytical expressions without suffering from the limits imposed by the statistical hypothesis on which the latter expressions rely. In particular, it provides naturally, non-Gaussian spin distribution as well as nonequipartition of parities which are known to have a significant impact on cross section predictions at low energies. Our previous global model [S. Goriely, S. Hilaire, and A. J. Koning, Phys. Rev. C 78, 064307 (2008)] suffered from deficiencies, in particular in the way the collective effects---both vibrational and rotational---were treated. We have recently improved the level density calculations using simultaneously the single-particle levels and collective properties predicted by a newly derived Gogny interaction [S. Goriely, S. Hilaire, M. Girod, and S. P\'eru, Phys. Rev. Lett. 102, 242501 (2009)], therefore enabling a microscopic description of energy-dependent shell, pairing, and deformation effects. In addition, for deformed nuclei, the transition to sphericity is coherently taken into account on the basis of a temperature-dependent Hartree-Fock calculation which provides at each temperature the structure properties needed to build the level densities. This new method is described and shown to give reasonable results with respect to available experimental data.

Nuclear level densities in the constant-spacing model

A new method to calculate level densities for non-interacting Fermions within the constant-spacing model with a finite number of states is developed. We show that asymptotically (for large numbers of particles or holes) the densities have Gaussian form. We improve on the Gaussian distribution by using analytical expressions for moments higher than the second. Comparison with numerical results shows that the resulting sixth-moment approximation is excellent except near the boundaries of the spectra and works globally for all particle/hole numbers and all excitation energies.

Population of rotational bands in superheavy nuclei

Using the statistical approach, we study the population of ground-state rota- tional bands of superheavy nuclei produced in the fusion-evaporation reactions 208 Pb( 48 Ca, 2n) 254 No, 206 Pb( 48 Ca, 2n) 252 No, and 204 Hg( 48 Ca, 2n) 250 Fm. We calculate relative intensi- ties of E2-transitions between the rotational states and entry spin distributions of the resid- ual nuclei, evaporation residue cross sections, and excitation functions for these reactions. Fermi-gas model is used for the calculation of level density, and damping of shell effects both with excitation energy and angular momentum is taking into account. The results are in a good agreement with the experiment data. theoretical approaches which predict the next doubly magic shell closure beyond 208 Pb. Reaching the region of superheavy nuclei in the vicinity of this closure, so called island of stability, is a challenge for the modern nuclear physics. The macroscopic-microscopic approaches predict the magic nucleus with Z=114 (1). The closed shell Z = 114 disappears in the framework of self-consistent mean-field models (2-6) which result the strong shell effects at Z=120-126. Although all these models provide close binding energies for nuclei with Z �114, their predictions for the spins of the ground states, quasiparticle spectra, and for the moments of inertia are rather different. Therefore, one can test the theoretical models by comparing their structure predictions with the spectroscopical data. Experimental study of rotational bands of superheavies is especially hopeful, as the limits of sta- bility of these nuclei in spin and excitation energy are governed by the fission barrier, which is mostly defined by shell component. The investigation of high-spin spectra also provides information about angular momentum dependence of the shell effects. Unfortunately, there are still a little experimental data for the heaviest evaporation residues, as their production cross sections are too small for the spec- troscopical analysis. However, using the in-beam studies, the structure of high-spin excited states up to � 20~ has been recently identified for the nuclei 254 No, 252 No, and 250 Fm, produced with the cross sections 0.2-3 µb (7-10). The nuclei studied are produced in fusion-evaporation reactions, which can be successfully de- scribed using the statistical approach. In Refs. (11-14) we have already calculated evaporation residue cross sections and excitation functions for superheavy isotopes using their properties, predicted by different theoretical models (15-18). The results are in a good agreement with the experimental data. In the present paper, we extend the statistical approach to describe the population of yrast rotational states in the residual superheavy nuclei and the production of evaporation residues in the complete fusion reactions 206,208 Pb( 48 Ca, 2n) 252,254 No and 204 Hg( 48 Ca, 2n) 250 Fm, taking into consideration the survival of compound nuclei against fission at di fferent angular momenta. The relative intensities of E2-transitions and entry spin distributions of the residual nuclei can be derived from the calculated population cross sections of the rotational states and compared with the experimental data (7-10).

Monte Carlo Simulation of Prompt Fission Gamma Emission

Prompt fission gamma spectra and multiplicities are investigated through the Monte Carlo code FIFRELIN which has been developed recently [1]. Up to now, this code was mainly used to study the characteristics (spectrum and multiplicity) of the prompt neutrons emitted from excited Fission Fragments (FF). When prompt neutron emission is over, the remaining available excitation energy of the FF is assumed to be dissipated as prompt gamma emission. An algorithm, similar to the one used by F. Becvar [2] to treat gamma cascades, has been implemented into the code. It is based on the known discrete levels of the excited fragments as well as on the theoretical models needed to calculate level densities and strength functions. All nuclear structure data are provided by the RIPL3.0 library [3]. Preliminary results relative to gamma deexcitation spectrum and multiplicity for single nuclei are presented. A total gamma spectrum for 252Cf spontaneous fission is also emphasized. Influence of the several methods, models and assumptions on the calculated fission gamma spectra is discussed as well as the prospective for such a simulation.