Neutron Resonance Spacings Research Articles

Indispensability of cross-shell contributions in neutron resonance spacing

Spin- and parity-dependent nuclear level densities (NLDs) are obtained for a configuration interaction shell model using a numerically efficient spectral distribution method. The calculations are performed for 24Na and 25,26,27Mg nuclei using full sd-pf model space that incorporates the cross-shell excitations from the sd to the pf-shell. The obtained NLDs are then employed to determine the s-wave neutron resonance spacing (D0), which is one of the crucial inputs for the predictions of astrophysical reaction rates. Although the considered nuclei are not neutron-rich, the contributions from cross-shell excitations to the pf-shell are indispensable for explaining the experimental data for D0, which otherwise are significantly overestimated.

Nuclear level density and γ -ray strength function of Ni63

The nuclear level density (NLD) and $\gamma$-ray strength function ($\gamma$SF) of $^{63}\mathrm{Ni}$ have been investigated using the Oslo method. The extracted NLD is compared with previous measurements using particle evaporation and those found from neutron resonance spacing. The $\gamma$SF was found to feature a strong low energy enhancement that could be explained as M1 strength based on large scale shell model calculations. Comparison of $\gamma$SFs measured with the Oslo method for various $\mathrm{Ni}$ isotopes reveals systematic changes to the strength below $5$ MeV with increasing mass.

Astrophysical reaction rates with realistic nuclear level densities

Realistic nuclear level densities (NLDs) obtained within the spectral distribution method (SDM) are employed to study nuclear processes of astrophysical interest. The merit of SDM lies in the fact that the NLDs corresponding to many body shell model Hamiltonian consisting of residual interaction can be obtained for the full configurational space without recourse to the exact diagnolization of huge matrices. We calculate NLDs and s-wave neutron resonance spacings which agree reasonably well with the available experimental data. By employing these NLDs, we compute reaction cross-sections and astrophysical reaction rates for radiative neutron capture in few Fe-group nuclei, and compare them with experimental data as well as with those obtained with NLDs from phenomenological and microscopic mean-field models. The results obtained for the NLDs from SDM are able to explain the experimental data quite well. These results are of particular importance since the configuration mixing through the residual interaction naturally accounts for the collective excitations. In the mean-field models, the collective effects are included through the vibrational and rotational enhancement factors and their NLDs are further normalized at low energies with neutron resonance data.

Independent normalization for γ -ray strength functions: The shape method

The Shape method, a novel approach to obtain the functional form of the $\gamma$-ray strength function ($\gamma$SF) in the absence of neutron resonance spacing data, is introduced. When used in connection with the Oslo method the slope of the Nuclear Level Density (NLD) is obtained simultaneously. The foundation of the Shape method lies in the primary $\gamma$-ray transitions which preserve information on the functional form of the $\gamma$SF. The Shape method has been applied to $^{56}$Fe, $^{92}$Zr, $^{164}$Dy, and $^{240}$Pu, which are representative cases for the variety of situations encountered in typical NLD and $\gamma$SF studies. The comparisons of results from the Shape method to those from the Oslo method demonstrate that the functional form of the $\gamma$SF is retained regardless of nuclear structure details or $J^\pi$ values of the states fed by the primary transitions.

Microscopie Calculations of Nuclear Level Densities and Astrophysical Applications

A new level density formula based on a microscopic calculation of the thermodynamic quantities using the deformed Hartree-Fock-BCS method is proposed. In the microscopic approach shell, pairing and deformation effects on the thermodynamic quantities are treated consistently. The final level density formula is shown to be in close agreement with experimental neutron resonance spacings and low energy states. The impact of the newly-determined level densities on nuclear reaction cross sections and rates of relevance in astrophysics applications is presented.

Phenomenological level density model with hybrid parameterization of deformed and spherical state densities

ABSTRACT A phenomenological level density model that has different level density parameter sets for the state densities of the deformed and the spherical states and the optimization of the parameters using experimental data of the average s-wave neutron resonance spacing are presented. The transition to the spherical state from the deformed one is described using the parameters derived from a microscopic nuclear structure calculation. The nuclear reaction calculation has been performed by the statistical model using the present level density. Resulting cross sections for various reactions with the spherical, deformed, and transitional target nuclei show a good agreement with the experimental data, which indicates the effectiveness of the present model. The role of the rotational collective enhancement in the calculations of those cross sections is also discussed.

Level Density Inputs in Nuclear Reaction Codes and the Role of the Spin Cutoff Parameter

The proton spectrum from the 57Fe(α, p) reaction has been measured and analyzed with the Hauser-Feshbach model of nuclear reactions. Different input level density models have been tested. It was found that the best description is achieved with either Fermi-gas or constant temperature model functions obtained by fitting them to neutron resonance spacings and to discrete levels and using the spin cutoff parameter with much a weaker excitation energy dependence than predicted by the Fermi-gas model.

Nuclear Level Density within Extended Superfluid Model with Collective State Enhancement

For nuclear level densities, an enhanced generalized superfluid model with different vibrational enhancement factors is studied. The ready-to-use tables for the asymptotic value of a-parameter of level density as well as for additional shift to excitation energy for 291 atomic nuclei are prepared using the chi-square fit of the theoretical values of neutron resonance spacing and cumulative number of low-energy levels to the experimental values. The systematics of these parameters as a function of mass number and neutron excess are obtained. The vibrational enhancement descriptions are additionally tested for a wide range of excitation energies. Simple methods of the vibrational enhancements lead to increasing the level density due to vibrational states by factor ∼ 2 to 3 for the isotopes A ∼ 100 with the excitation energies near the neutron binding energies.

Statistical nuclear properties (level densities, spin distributions)

A general overview is given on the phenomenological methods used to describe the level densities in nuclei. Two well-known two-parameter formulas of level densities, the Back-Shifted Fermi Gas (BSFG) model and the Constant Temperature (CT) model, were used. A common ingredient of both is the spin distribution function, which contains in Ericsons's parametrization the spin-cutoff parameter σ. A realistic description of the parameters of both spin distribution function and the two level density models has been obtained by fitting the experimental data of 310 nuclei between 18F and 251Cf, consisting of the complete level schemes at low excitation energies and the s-wave neutron resonance spacings at the neutron binding energy. We determine a simple formula for the spin-cutoff parameter as a function of mass number and excitation energy. Also, an even-odd spin staggering in the spin distribution of the even-even nuclei was observed, and described with a simple formula. Using this newly defined spin distribution function, an empirical set of parameters of the BSFG and CT models was determined by fitting both the low-energy levels and the neutron resonance spacings. For these parameters, simple formulas were proposed that involve only quantities available from the mass tables, and allow reasonable estimations of the level density parameters for nuclei far from stability. Both the BSFG and CT models describe equally well the level densities at energies up to at least the neutron binding energy. Finally, we discuss recent experimental evidence that the CT model is the more correct description of the nuclei in the low-excitation energy (pairing) regime.

Nuclear Level Densities with Microscopic Statistical Method Using a Consistent Residual Interaction

Nuclear level density (NLD) evaluated under the framework of the microscopic statistical method is examined. We calculate single-particle levels by the Skyrme-Hartree-Fock (SHF) + Bardeen-Cooper-Schrieffer (BCS) method, and the grand partition function is calculated with the same two-body interaction as used in the SHF+BCS method. The results are compared with preceding theoretical calculations using different residual interactions. Cumulative numbers and s-wave neutron resonance spacings are reproduced fairly well. The effect of a renormalization term neglected in conventional microscopic statistical methods is also investigated. In order to test a practical application, we calculate neutron capture cross section and compare it with experimental results.

Combinatorial level densities for practical applications

We review our calculated energy-, spin- and parity-dependent nuclear level densities based on the microscopic combinatorial model described in ref. [1]. We show that this model predicts the experimental sand p-wave neutron resonance spacings with a degree of accuracy comparable to that of the best global models available and also provides reasonable description of low energies cumulative number of levels as well as of the experimental data obtained by the Oslo group [2]. We also provide a renormalization recipe which enables to play with the tabulated results for practical applications. Finally, we study the impact of temperature dependent calculation on s-wave neutron resonance spacings.

Improved microscopic nuclear level densities within the Hartree-Fock-Bogoliubov plus combinatorial method

New developments have been brought to our energy-, spin-, and parity-dependent nuclear level densities based on the microscopic combinatorial model. As in our previous study, a detailed calculation of the intrinsic state density and of the rotational enhancement factor is included, but this time the vibrational contributions explicitly take the phonon excitations into account. The present model predicts the experimental $s$- and $p$-wave neutron resonance spacings with a degree of accuracy comparable to that of the best global models available. It is also shown that the model gives a reliable extrapolation at low energies where experimental data on the cumulative number of levels can be extracted. The predictions are also in good agreement with the experimental data extracted from the analysis of particle-$\ensuremath{\gamma}$ coincidence in the ($^{3}\mathrm{He}$, $\ensuremath{\alpha}\ensuremath{\gamma}$) and ($^{3}\mathrm{He}$, $^{3}\mathrm{He}$${}^{'}\ensuremath{\gamma}$) reactions. Total as well as partial level densities for more than 8500 nuclei are made available in a table format for practical applications. For the nuclei for which experimental $s$-wave spacings and enough low-lying states exist, renormalization factors are provided to reproduce simultaneously both observables. The same combinatorial method is used to estimate the nuclear level densities at the fission saddle points of actinides and at the shape isomer deformation. Finally, the new nuclear level densities are applied to the calculation of radiative neutron capture cross sections.

Test of nuclear level density inputs for Hauser-Feshbach model calculations

The energy spectra of neutrons, protons, and \ensuremath{\alpha}-particles have been measured from the $d+$$^{59}\mathrm{Co}$ and $^{3}\mathrm{He}+^{58}\mathrm{Fe}$ reactions leading to the same compound nucleus, $^{61}\mathrm{Ni}$. The experimental cross sections have been compared to Hauser-Feshbach model calculations using different input level density models. None of them have been found to agree with experiment. It manifests the serious problem with available level density parametrizations especially those based on neutron resonance spacings and density of discrete levels. New level densities and corresponding Fermi-gas parameters have been obtained for reaction product nuclei such as $^{60}\mathrm{Ni}$, $^{60}\mathrm{Co}$, and $^{57}\mathrm{Fe}$.

Global microscopic nuclear level densities within the HFB plus combinatorial method for practical applications

New energy-, spin- and parity-dependent nuclear level densities based on the microscopic combinatorial model are proposed for practical applications. The combinatorial model includes a detailed microscopic calculation of the intrinsic state density and of the rotational enhancement factor, but a phenomenological treatment of the vibrational effect. The calculations make a coherent use of nuclear structure properties determined within the deformed Skyrme–Hartree–Fock–Bogolyubov framework. The present model predicts the experimental s-wave and p-wave neutron resonance spacings with a degree of accuracy comparable to that of the best global models available. It is also shown that the model gives reliable extrapolations to low energies where experimental data on the cumulative number of levels can be extracted. Level densities for more than 8500 nuclei are made available in a table format for practical applications.

Unified model of nuclear mass and level density formulas

Nuclear ground and excited state properties are described by using parameter systematics on mass and level density formulas. The formulas are based on an analytical expression of the single-particle state density by introducing the shell-pairing correlation in a new way. Main features are the shell, pairing, and deformation effects on the droplet model near the ground state, which are washed out at higher excitation energies. The main aim of the paper is to provide in the analytical framework the improved energy dependent shell, pairing, and deformation corrections generalized to the collective enhancement factors, which offer a systematic prescription over a great number of nuclear reactions. The new formulas are shown to be in close agreement with not only the empirical nuclear mass data but also the measured slow neutron resonance spacings and experimental systematics observed in the excitation energy dependent properties.

Systematics of nuclear level density parameters

The level density parameters for backshifted Fermi gas (both without and with energy-dependent level density parameter) and constant-temperature models have been determined for 310 nuclei between $^{18}\mathrm{F}$ and $^{251}\mathrm{Cf}$ by fitting of the complete level schemes at low excitation energies and s-wave neutron resonance spacings at the neutron binding energies. Simple formulas are proposed for the description of the two parameters of each of these models, which involve only quantities available from the mass tables. These formulas may constitute a reliable tool for extrapolating to nuclei far from stability, where nuclear level densities cannot be measured.

Systematics of nuclear level density parameters

The level density parameters for the back-shifted Fermi gas (both without and with energy-dependent level density parameter) and the constant temperature models have been determined for 310 nuclei between 18F and 251Cf by fitting the complete level schemes at low excitation energies and the s-wave neutron resonance spacings at the neutron binding energies. Simple formulae are proposed for the description of the two parameters of each of these models, which involve only quantities available from the mass tables. These formulae may constitute a reliable tool for extrapolating to nuclei far from stability, where nuclear level densities cannot be measured.

Calculation of Neutron Resonance Spacing with Microscopic Theory

Nuclear level spacings calculated with a microscopic theory are compared with spacings determined from neutron resonance experiment. The gross features of the experimental data due to nuclear shells are reproduced with the microscopic theory. The experimental data for nuclei with statistically deformed nuclei have also been tested with level density formula including low energy rotational levels. The experimental data for the actinide nuclei and the lanthanide nuclei are found to be consistent with the theory which includes collective rotational levels.

Combinatorial level densities from a relativistic structure model

A new model for calculating nuclear level densities is investigated. The single-nucleon spectra are calculated in a relativistic mean-field model with energy-dependent effective mass, which yields a realistic density of single-particle states at the Fermi energy. These microscopic single-nucleon states are used in a fast combinatorial algorithm for calculating the non-collective excitations of nuclei. The method, when applied to magic and semi-magic nuclei, such as 60Ni, 114Sn and 208Pb, reproduces the cumulative number of experimental states at low excitation energy, as well as the s-wave neutron resonance spacing at the neutron binding energy. Experimental level densities above 10 MeV are reproduced by multiplying the non-collective level densities by a simple vibrational enhancement factor. Problems to be solved in the extension to open-shell nuclei are discussed.

On Consistent Description of Nuclear Level Density

The back–shifted Fermi gas (BSFG) model has been used for description of the nuclear level density at excitation energies up to the nucleon binding energy by fitting the latest experimental low–lying discrete levels and average s–wave nucleon resonance spacings D0. The analysis of the ratio of the proton and neutron resonance spacings corresponding to the same compound nucleus has led to the ratio IIr=(0.75±0.06) of the effective moment of inertia of the nucleus to the rigid-body value Ir for the nucleus 51V. The energy-dependent level density parameter and the method proposed by Koning and Chadwick for choosing the appropriate shell correction energy have been adopted for energies above the nucleon binding energy. Finally, a consistent description of all experimental data related to the nuclear level density, i.e. the low–lying discrete levels, the nucleon–resonance data, and the level density data above the nucleon binding energy, has been obtained for nuclei within A–ranges 24–41, 55–70, and 104–114.