Back-shifted Fermi Gas Research Articles

Properties of112Cdfrom the(n,n′γ)reaction: Levels and level densities

Levels in ${}^{112}\mathrm{Cd}$ have been studied through the ${(n,n}^{\ensuremath{'}}\ensuremath{\gamma})$ reaction with monoenergetic neutrons. An extended set of experiments that included excitation functions, $\ensuremath{\gamma}$-ray angular distributions, and $\ensuremath{\gamma}\ensuremath{\gamma}$ coincidence measurements was performed. A total of 375 $\ensuremath{\gamma}$ rays were placed in a level scheme comprising 200 levels (of which 238 $\ensuremath{\gamma}$-ray assignments and 58 levels are newly established) up to 4 MeV in excitation. No evidence to support the existence of 47 levels as suggested in previous studies was found, and these have been removed from the level scheme. From the results, a comparison of the level density is made with the constant temperature and back-shifted Fermi gas models. The back-shifted Fermi gas model with the Gilbert-Cameron spin cutoff parameter provided the best overall fit. Without using the neutron resonance information and only fitting the cumulative number of low-lying levels, the level density parameters extracted are a sensitive function of the maximum energy used in the fit.

Estimations of level densities and radiative strength functions from experimental data and probable change in the properties of nuclei at E exc ⋍ 3–4 MeV

Investigations of two-step γ cascades following thermal-neutron capture by heavy nuclei reveal that the density of excited levels is significantly smaller than that which is predicted by an exponential extrapolation like that of the back-shifted Fermi gas model. Data on two-step cascades allow one to determine in a model-independent way the most probable energy dependences of the level density and radiative strength functions, virtually over the whole excitation-energy interval below the neutron binding energy B n . Data for more than 30 heavy nuclei cannot be understood without assuming a considerable change in the nuclear properties at E exc ⋍ 3–4 MeV.

Hyperdeformation and Clusterization in the Actinide Region

The 233U(d,pf)234U, and the 235U(d,pf)236U reactions have been studied with high energy resolution. The observed fission resonances were described as members of rotational bands with rotational parameters characteristic of hyperdeformed nuclear shapes. Information on the K values of the bands and for the J values of the band members has been obtained from fission fragment angular distribution measurements. The level density of the most strongly excited states has been compared to the prediction of the back-shifted Fermi-gas formula and the energy of the ground state in the third minimum has been estimated for 234U. The fission fragment mass distribution of the hyperdeformed states in 236U has also been measured. The width of the mass distribution, coincident with the hyperdeformed bands, is significantly smaller than the ones obtained in coincidence with background regions below and above the resonances, which suggests a pear-shaped di-nuclear configuration of 236U in the third well of the potential barrier.

Effects of Differing Energy Dependences in Three Level-Density Models on Calculated Cross Sections

Three level-density formalisms commonly used for cross-section calculations are examined. Residual nuclides in neutron interaction with 58Ni are chosen to quantify the well-known differences in the energy dependences of the three formalisms. Level-density parameters for the Gilbert and Cameron model are determined from experimental information. Parameters for the back-shifted Fermi-gas and generalized superfluid models are obtained by fitting their level densities at two selected energies for each nuclide to those of the Gilbert and Cameron model, forcing the level densities of the three models to be as close as physically allowed. The remaining differences are in their energy dependences that, it is shown, can change the calculated cross sections and particle emission spectra significantly, in some cases or energy ranges by a factor of 2.

Level-density parameters in the back-shifted Fermi gas model

The parameters a and δeff appearing in the back-shifted Fermi gas model are determined for about 3000 nuclei on the basis of modern estimated experimental data and the proposed systematics. For 272 of these nuclei, the parameters are deduced from experimental data on the cumulative numbers of low-lying levels and on mean spacings between S-wave neutron resonances at the neutron binding energy in the nuclei. For 952 nuclei, the parameter δeff is calculated by using the cumulative numbers of low-lying levels and values of the parameter a that were obtained via an interpolation from the points corresponding to the aforementioned 272 nuclei. For the remaining nuclei, the parameters a and δeff are obtained on the basis of the proposed systematics. An expression is constructed for taking into account the damping of shell effects with increasing excitation energy of nuclei. The results are compared with those from other studies.

On the excitation energy of the ground state in the third minimum of [formula omitted

The 233 U(d,pf) 234 U reaction has been studied with high energy resolution. The observed fission resonances in the excitation energy range of 4.75 ≤E ∗≤ 5.40 MeV were described as members of rotational bands with rotational parameters characteristic to the hyperdeformed nuclear shape (ℏ 2/2 θ=2.1±0.2 keV). Information on the K values of the bands has been obtained from fission fragment angular distribution measurements. The level density of the most strongly excited J=3 states has been compared to the prediction of the back-shifted Fermi-gas formula and the energy of the ground state in the third minimum has been estimated to be E III = 3.1 ± 0.4 MeV.

β-delayed proton decays of theTz=1/2series nuclei,81Zrand85Mo

The decays of ${T}_{z}=1/2$ series nuclei ${}^{81}\mathrm{Zr}$ and ${}^{85}\mathrm{Mo}$ have been restudied via their $\ensuremath{\beta}$-delayed proton decay by using p-$\ensuremath{\gamma}$ coincidence measurements. The $\ensuremath{\beta}$-delayed proton spectra populating the first excited states of the final nuclei have been obtained. By using statistical model calculations with level densities calculated with a backshifted Fermi gas model, the spin and parity for the ground state of ${}^{81}\mathrm{Zr}$ have been tentatively assigned to be ${3/2}^{\ensuremath{-}}$ and those for ${}^{85}\mathrm{Mo}$ to be ${1/2}^{\ensuremath{-}}.$ Their ${Q}_{\mathrm{e}.\mathrm{c}.}\ensuremath{-}{B}_{p}$ values are $4.7\ifmmode\pm\else\textpm\fi{}0.2$ and $5.1\ifmmode\pm\else\textpm\fi{}0.2 \mathrm{MeV},$ which correspond to mass excesses of $\ensuremath{-}58.3\ifmmode\pm\else\textpm\fi{}0.2 \mathrm{MeV}$ for ${}^{81}\mathrm{Zr}$ and of $\ensuremath{-}59.1\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{MeV}$ for ${}^{85}\mathrm{Mo},$ respectively. Combining the measured half-lives with the calculated partial ones yields branches of $\ensuremath{\beta}$-delayed proton decay for ${}^{81}\mathrm{Zr}$ and ${}^{85}\mathrm{Mo}$ of $(1.2\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and $(1.4\ifmmode\pm\else\textpm\fi{}0.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3},$ respectively.

Shell model Monte Carlo level densities for nuclei with A∼50

We have calculated level densities for 48−50Cr and 50,51Mn using the Shell Model Monte Carlo method. As residual interaction we use a pairing+quadrupole force supplemented by an additional isospin repulsion which is treated in perturbation theory. The SMMC results compare favorably with empirical backshifted Fermi gas level densities.

Nuclear level density and the determination of thermonuclear rates for astrophysics

The prediction of cross sections for nuclei far off stability is crucial in the field of nuclear astrophysics. In recent calculations the nuclear level density -- as an important ingredient to the statistical model (Hauser-Feshbach) -- has shown the highest uncertainties. We present a global parametrization of nuclear level densities based on the back-shifted Fermi-Gas formalism. Employment of an energy-dependent level density parameter $a$ and microscopic corrections from a recent FRDM mass formula by M\"oller et al.\ leads to a highly improved fit of level densities at the neutron-separation energy in the mass range $20\le A \le 245$. The importance of using proper microscopic corrections from mass formulae is emphasized. The resulting level description is well suited for astrophysical applications. The level density can also provide clues to the applicability of the statistical model which is only correct for a high density of excited states. Using the above description one can derive a ``map'' for the applicability of the model for reactions of stable and unstable nuclei with neutral and charged particles.

Applicability of the Hauser-Feshbach approach for the determination of astrophysical reaction rates

Nuclear Astrophysics requires the knowledge of reaction rates over a wide range of nuclei and temperatures. In recent calculations the nuclear level density — as an important ingredient to the statistical model (Hauser-Feshbach) — has shown the highest uncertainties. In a back-shifted fermi-Gas formalism utilizing an energy-dependent level density parameter and employing microscopic corrections from a recent FRDM mass formula, we obtain a highly improved fit to experimental level densities. The resulting level density is used for determining criteria for the applicability of the statistical model.

207,208Pb(n,xn gamma ) reactions for neutron energies from 3 to 200 MeV.

High-resolution \ensuremath{\gamma}-ray spectra from the interaction of neutrons in the energy range from 3 to 200 MeV with $^{207,208}\mathrm{Pb}$ were measured with the white neutron source at the weapons neutron research (WNR) facility at Los Alamos National Laboratory. From these data, excitation functions for prominent \ensuremath{\gamma} transitions in $^{200,202,204,206,207,208}\mathrm{Pb}$ were derived from threshold to 200 MeV incident neutron energy. These \ensuremath{\gamma}-production cross sections reflect the excitation cross sections for the respective residual nuclei. The results are compared with the predictions of nuclear reaction calculations based on the exciton model for precompound emission, the Hauser-Feshbach theory for compound nucleus decay, and coupled channels calculations to account for direct excitation of collective levels. Good agreement was obtained over the entire energy range covered in the experiment with reasonable model parameters. The results of this work clearly demonstrate that multiple preequilibrium emission has to be taken into account above about 40 MeV, and that the level density model of Ignatyuk, which accounts for the gradual disappearance of shell effects with increasing excitation energy, should be used instead of the Gilbert-Cameron and backshifted Fermi-gas models if excitation energies exceed about 30 MeV. No indication for a reduction of the nuclear moment of inertia below the rigid body value was found.

Density of discrete levels in 116Sn.

Model fits were made to the cumulative distribution of all known levels (with spins [ital J][le]10) in [sup 116]Sn below an excitation energy of 4.1 MeV identified in a variety of experiments, and simultaneously to the density of [ital J]=0 and [ital J]=1 resonances inferred from neutron capture and total cross-section measurements in [sup 115]Sn+[ital n]. Neither the back-shifted Fermi-gas model nor the constant-temperature model provides a satisfactory fit over the entire energy range. The generalized superfluid model, with a level-density parameter and a spin-cutoff factor that are both energy dependent, does give a good fit, and the resulting best-fit parameter values are consistent with those found in other applications of this model.

Test of photon strength functions by a method of two-step cascades.

The applicability of sum-coincidence measurements of two-step cascade \ensuremath{\gamma}-ray spectra for the determination of photon strength functions is discussed. The method is sensitive, not only to primary transitions, but also to secondary transitions proceeding from states of intermediate excitation. An experiment based on thermal neutron capture in a Nd target was undertaken at the Brookhaven National Laboratory High Flux Beam Reactor to test various hypotheses of E1 and M1 photon strength function in the $^{144}\mathrm{Nd}$ nucleus. The results were compared to a series of model calculations, in which the back-shifted Fermi gas model is used to represent the level density. Models derived from the Fermi theory of liquids are strongly preferred.

A Semiclassical Theory of Multistep Nuclear Reaction Processes

The master equation theory of precompound and compound nuclear reactions has been generalized with inclusion of the conservation of angular momentum and parity. This improved semiclassical theory has been extended for application as an evaluation tool of the calculations of nucleon induced reaction cross sections and double differential cross sections. For structural materials at incident neutron energies below , it is demonstrated that the constructed model contains the Hauser-Feshbach, Weisskopf-Ewing as well as the exciton models as limiting cases. The unified treatment of pre-equilibrium processes includes a number of interesting features, such as the exciton state densities with the exact Pauli exclusion correction which are renormalized to the back-shifted Fermi-gas formula, the introduction of formation factors of composite particle in calculations of pick-up type composite particle emission and the double differential cross sections (DDCS) for all kinds of particles in terms of the leading particle model.

Total cross section and neutron resonance spectroscopy for n+40Ar.

The neutron total cross section for $^{40}\mathrm{Ar}$ has been measured over the incident neutron energy range 0.007--50 MeV. R-matrix analysis of the cross section from 0.007 to 1.52 MeV provides resonance parameters which provide a complete description of the neutron scattering functions for the ${\mathit{s}}_{1/2}$, ${\mathit{p}}_{1/2}$, and ${\mathit{p}}_{3/2}$ scattering channels and less nearly complete scattering functions for the ${\mathit{d}}_{3/2}$ and ${\mathit{d}}_{5/2}$ channels. The back-shifted Fermi-gas model is used to model the level densities for s-, p-, and d-wave resonances.

Nuclear level densities below 40 MeV excitation energy in the mass regionA ≃ 50

Consistent pre-equilibrium emission and statistical model calculations of fast neutron induced reaction cross sections are used to validate nuclear level densities for excitation energies up to 40 MeV in the mass regionA ≃50. A “composed” level density approach has been employed by using the back-shifted Fermi gas model for excitation energies lower than 12 MeV and a realistic analytical formula for higher excitations. In the transition region from the BSFG model range to that of full applicability of the realistic formula, an interpolation between the predictions of the two models is adopted. The interpolation rule, suggested by microscopic level density calculations, has been validated through the comparison of the calculated and experimental cross sections.

Beta +-electron-capture decay of 69Se.

The ${\ensuremath{\beta}}^{+}$--electron-capture decay of $^{69}\mathrm{Se}$, produced via the $^{40}\mathrm{Ca}$${(}^{32}$S, 2pn${)}^{69}$Se reaction, has been studied by \ensuremath{\beta}-delayed proton and \ensuremath{\gamma}-ray emissions using the recoil transport helium jet technique. The established decay scheme for $^{69}\mathrm{Se}$ involves 13 up to now unreported \ensuremath{\gamma} emitting levels in $^{69}\mathrm{As}$. The total proton branching ratio has been measured to be (4.5\ifmmode\pm\else\textpm\fi{}1.0)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$. The autocorrelation analysis of the delayed proton spectra indicates a level spacing in agreement with the back shifted Fermi gas model estimates. The Gamow-Teller beta strength distribution has been measured in 95% of the ${Q}_{\ensuremath{\beta}}$ window, up to 6.5 MeV excitation energy in $^{69}\mathrm{As}$.

55Mn(

The energy and angular distributions of the \ensuremath{\alpha} particles from the $^{55}\mathrm{Mn}$(n,x\ensuremath{\alpha}) and $^{59}\mathrm{Co}$(n,x\ensuremath{\alpha}) reactions for 14.1 MeV incident neutrons were measured by means of a multitelescope system. The results concerning total helium production are in good agreement with existing helium accumulation measurements, whereas the angular distributions reported in the literature for the $^{59}\mathrm{Co}$(n,\ensuremath{\alpha}) reaction could not be confirmed. The results are analyzed in the framework of the statistical model of nuclear reactions. Comparison of such calculations with the measured \ensuremath{\alpha}-emission cross sections allows the extraction of nuclear level densities for $^{52}\mathrm{V}$ and $^{56}\mathrm{Mn}$, the residual nuclei reached in the reactions studied, in the energy region up to the neutron binding energy, and for $^{55}\mathrm{Mn}$ and $^{59}\mathrm{Co}$, the residual nuclei reached by neutron emission, for energies around 11 MeV. The level densities derived in this way agree very well with the prediction of the back-shifted Fermi-gas model for the level density parameters of Strohmaier and Uhl. For $^{52}\mathrm{V}$ and $^{56}\mathrm{Mn}$ there is also excellent agreement with the level densities derived from average s-wave neutron resonances. The spin cutoff factors derived from the \ensuremath{\alpha}-particle angular distributions indicate that there is no reduction of the effective moment of inertia below the rigid body value.

A new code system for the calculation of double differential reaction cross sections

Abstract The statistical exciton model following the master-equation approach has been improved and extended for application as an evaluation tool of double-differential reaction cross sections at incident nucleon energies of 5 to 50 MeV. For this purpose the code system GRAPE, centered around the exciton-model code GRYPHON has been developed, which combines a number of interesting features such as: unified treatment of pre-equilibrium and equilibrium processes, renormalized exciton state-densities summing up to the back-shifted Fermi gas formula, a new model for the internal transition rates based upon the nucleon mean free path in nuclear matter, angle-energy distributions based on intra-nuclear scattering in nuclear matter, account of discrete-level excitations, a new model for γ-ray competition, inclusion of multi-particle emission, and various sorting options with code output in the new ENDF-VI format. A major characteristic of the proposed model is that consistency with equilibrium models has been i...

New results obtained by the number theoretical description of the level densities of nuclei

Abstract The number theoretical method has been developed further towards a detailed application to realistic nuclei. Temperature dependent shell- and pairing-effects are obtained by evaluation of the Dirichlet series, occurring in the partition function. For nuclear temperatures small against the Fermi energy an analytic closed-form level density expression has been rigorously derived. It differs significantly from the Gilbert-Cameron as well as from the back-shifted Fermi gas formula. In particular the description of the case of small excitation energies in the neighbourhood of closed shells has been substantially improved.