Particle-hole State Densities Research Articles

Importance of the residual nucleus level density in the calculation of the 239Pu(n, 2n)238Pu excitation function

In parallel with the recent measurement by Méot et al. [1] of the 239Pu(n, 2n)238Pu reaction cross section in the [7-10] MeV neutron-incident energy range, a modeling work was undertaken to support the new measurement. This theoretical framework is here dedicated to study the impact of the type of level density for the 239Pu residual nucleus (RN) on the final shape of the low-energy tail of the (n, 2n) excitation function. For this purpose, the AVXSF-LNG program [4] has been upgraded to model second-chance reactions and coupled to the TALYS-ECIS06 nuclear reaction codes [3] that are used to provide the compound nucleus (CN) cross section to AVXSF-LNG as a function of the selected pre-equilibrium model . The spin-dependent population of the RN obtained after CN emission is commonly advertised when using an exciton-based model for calculating the total pre-equilibrium cross section. On the other hand a population based on a particle-hole state density is rather expected when using the MSD/MSC quantum mechanical description in place of the exciton-based model. As far as a spectrum of a few number of individual excitations is favored, the combinatorial Quasi-Particle-Vibrational-Rotational Level Density method [4] implemented in the AVXSF-LNG computer program is well suited to model the various alternatives. Several RN level densities have been tested in this work. It includes the testing of the Quasi-particle Random Phase Approximation-based result of Ref. [6]. As a guide line, the amount of pre-equilibrium flux in the total (n,2n) excitation function is being studied as a function of the type of level density selected for the residual nucleus.

Pre-Equilibrium Differential and Double Differential Cross Sections of Neutrons Induce Nuclear Reactions in 64Zn Nucleus

The pre - equilibrium and equilibrium differential and double differential cross sections have been calculated for 2MeV neutrons induced reactions in 64Zn nucleus using Kalbach Systematic approach in terms of Exciton model with Feshbach, Kerman and Koonin (FKK) statistical theory. The two-component exciton model and some corrections have been implemented in the calculation the particle-hole state density. In this work, Isospin, finite well depth, and shell effects are considered. The obtained results were compared with the available published experimental and theoretical data that in the international libraries such as TENDL, ENDF, and JEFF. The comparisons with these available data showed an acceptable agreement especially for the reactions: 64Zn(n,n)64Zn, 64Zn(n,p)64Cu, 64Zn(n,D)63Cu, 64Zn(n,T)62Cu, 64Zn(n,3He)62Ni, and 64Zn(n,4He)61Ni at different emission angles.

The pre-equilibrium and equilibrium double differential cross sections for the nucleons and light nuclei induce nuclear reactions on 27Al nuclei

The pre - equilibrium and equilibrium double differential crosssections are calculated at different energies using Kalbach Systematicapproach in terms of Exciton model with Feshbach, Kerman andKoonin (FKK) statistical theory. The angular distribution of nucleonsand light nuclei on 27Al target nuclei, at emission energy in the centerof mass system, are considered, using the Multistep Compound(MSC) and Multistep Direct (MSD) reactions. The two-componentexciton model with different corrections have been implemented incalculating the particle-hole state density towards calculating thetransition rates of the possible reactions and follow up the calculationthe differential cross-sections, that include MSC and MSD models.The finite well depth, isospin, shell effects, Pauli effect, chargeeffect, pairing, surface, angular and linear momentum distributionscorrections are considered in this work. The nucleons (n and p) andlight nuclei (2D and 3T) have been employed as projectiles at thetarget 27Al nuclei and at different incident energies (4MeV, 14 MeVand 14.8MeV). The results have been compared with the availableexperimental and theoretical published work. The comparisons showan acceptable agreement with the TALAYS code (Tendel 2014) forthe reactions: 27Al (n, n) 27Al, 27Al (p, n) 63Zn, 27Al (p, D) 62Cu, 27Al(p, p) 63Cu and 27Al (p, 4He)60Ni and at different emission energiesand angles.

Photoneutron cross-section measurements in the Bi209 ( γ,xn ) reaction with a new method of direct neutron-multiplicity sorting

Photoneutron cross sections were measured in the $^{209}\mathrm{Bi}$($\ensuremath{\gamma}, xn$) reaction with $x=1--4$ at energies from the neutron threshold to 40 MeV using quasimonochromatic laser Compton-scattering $\ensuremath{\gamma}$-ray beams. A novel technique of direct neutron-multiplicity sorting with a flat-efficiency detector was used. Results are compared with the positron in-flight annihilation data of Livermore and Saclay as well as with bremsstrahlung data. The present neutron yields are consistent with the latest photoactivation measurement with bremsstrahlung. The giant dipole resonance component of the total photoneutron cross section is reasonably reproduced by model calculations of the Hartree-Fock Bogoliubov plus quasiparticle random phase approximation based on the Gogny D1M interaction for both E1 and M1 components. In contrast, the quasideuteron component of the total photoneutron cross section requires a factor of $\ensuremath{\sim}3$ enhancement of the Levinger parameter. Furthermore, a survival of large $(\ensuremath{\gamma},n)$ cross sections above 20 MeV may indicate the surface effect on the two-component particle-hole state density. The threshold behavior of photoneutron emission is discussed in terms of the average neutron kinetic energy.

Quantum oscillations in aπ-striped superconductor

Within Bogoliubov--de Gennes theory, a semiclassical approximation is used to study quantum oscillations and to determine the Fermi surface area associated with these oscillations in a model of a $\ensuremath{\pi}$-striped superconductor, where the $d$-wave superconducting order parameter oscillates spatially with period 8 and zero average value. This system has a nonzero density of particle-hole states at the Fermi energy, which form Landau-like levels in the presence of a magnetic field $B$. The Fermi surface is reconstructed via Andreev-Bragg scattering, and the semiclassical motion is along these Fermi surface sections as well as between them via magnetic breakdown. Within the approximation, oscillations periodic in $1/B$ are found in both the positions and widths of the lowest Landau levels. The area corresponding to these quantum oscillations for intermediate pairing interaction strength is similar to that reported for experimental measurements in the cuprates. A comparison is made of this theory to data for quantum oscillations in the specific heat measured by Riggs et al.

Description of cross sections for photonuclear reactions in the energy range between 7 and 140 MeV

A combination of the exciton and evaporation models is used to describe photonuclear reactions induced in light, medium-mass, and heavy nuclei by photons of energy in the range 7 ≤ Eγ ≤ 140 MeV. Two mechanisms of the photoexcitation of nuclei are considered. These are the formation of a giant dipole resonance at energies in the range Eγ ≲ 30 MeV and quasideuteron photoabsorption, which is dominant at energies in the region Eγ ≳ 40 MeV. The density of particle-hole states, which appears in the exciton model, is calculated on the basis of the Fermi gas model. The emission of two preequilibrium particles is taken into account in describing the quasideuteron reaction channel. The effect of isospin conservation on giant-dipole-resonance decay accompanied by photonucleon emission is examined. The model in question is used to describe cross sections for photon-induced reactions on 26Mg, 54Fe, 112,118,119,124Sn, and 181Ta nuclei.

Fast-neutron induced pre-equilibrium reactions on 55Mn and 63,65Cu at energies up to 40 MeV

The experimental cross sections for all neutron-induced reactions on the stable isotopes of Mn and Cu at incident energies up to 40 MeV are compared with results of pre-equilibrium (PE) and statistical model calculations. Taking advantage of the large sensitivity of statistical model analysis to residual nuclei parameters, for energies up to 20 MeV, our work was then focused on assessing PE model assumptions by data analysis in the energy range 20–40 MeV. The PE model reliance on the nuclear potential finite-depth correction for description of particle–hole state densities (PSD), nuclear-shell effects within the PSD formula, and single-particle level density is thus discussed with regard to the whole data basis. The related variance of the model results has been found as large as the differences between the global and local model calculations, while the data basis available between 20 and 40 MeV for the Mn and Cu isotopes is however too scarce in this respect.

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.

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.

Preequilibrium escape widths of giant resonances

In this work we present a calculation of the $2p\ensuremath{-}2h$ preequilibrium escape width of giant resonances for the nuclei ${}^{40}\mathrm{Ca},$ ${}^{90}\mathrm{Zr},$ and ${}^{208}\mathrm{Pb}.$ The problem studied here involves an excited nucleus in the $1p\ensuremath{-}1h$ configuration, evolving to the $2p\ensuremath{-}2h$ configuration with the $1p$ in the continuum. The theoretical approach used for our calculations is based on the statistical multistep compound theory of Feshbach, Kerman, and Koonin (FKK) and on the particle-hole state densities given by Oblo\ifmmode \check{z}\else \v{z}\fi{}insk\'y. Our calculations show that although different state densities supply a similar result for the damping width, the escape width is strongly dependent on the nuclei, on the binding energy of the emitted nucleon, and the excitation energy of the giant resonance.

Particle-hole state densities with nonequidistant single-particle levels

The correct use of energy-dependent single-particle level (s.p.l.) densities within particle-hole state densities based on the equidistant spacing model (ESM) is analysed. First, an analytical expression is obtained following the convolution of energy-dependent excited-particle and hole densities. Next, a comparison is made with results of the ESM formula using average s.p.l. densities for the excited particles and holes, respectively. The Fermi-gas model (FGM) s.p.l. densities calculated at the corresponding average excitation energies are used in both cases. The analysis concerns also the density of particle-hole bound states. The pairing correlations are taken into account while the comparison of various effects includes the exact correction for the Pauli exclusion principle. Quantum-mechanical s.p.l. densities and the continuum effect can also match a corresponding FGM formula, suitable for use within the average energy-dependent partial state density in multistep reaction models.

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.

Microscopic two-component multistep direct theory for continuum nuclear reactions

We present a multistep direct reaction theory for analyzing nucleon-induced reactions to the continuum for incident energies up to 200 MeV. Two principal advances in multistep direct theory are studied. (1) A microscopical approach is given for calculating distorted-wave Born approximation (DWBA) transitions to the continuum, where transitions to all accessible 1$p$1$h$ shell model states are explicitly determined. These states, obtained from a simple noninteracting Nilsson model, are assumed to be spread according to Gaussian distributions. In this approach, therefore, state densities are not used. We also provide a link with more conventional methods that utilize particle-hole state densities, and present a more accurate technique for sampling the DWBA strength. (2) A two-component formulation of multistep direct reactions is given, where neutron and proton excitations are explicitly accounted for in the evolution of the reaction, for all orders of scattering. We show that the attractive convolution structure for multistep processes persists within a two-component formalism, and conveniently automatically generates the many reaction pathways that can occur in the Feshbach-Kerman-Koonin expansion of the multistep cross section when neutron and proton excitations are followed. This formalism is particularly important for the simultaneous analyses of neutron and proton emission spectra. The multistep direct theory is applied, along with theories for multistep compound, compound, and collective reactions, to analyze experimental emission spectra for a range of targets and energies. Particular attention is paid to a complete and comprehensive analysis of all important decay channels and reaction mechanisms. We show that the theory correctly accounts for measured neutron and proton emission angle-integrated spectra, as well as angular distributions. Additionally, we note that these microscopic and two-component developments facilitate more fundamental studies into effective nucleon-nucleon interactions in multistep calculations.

Shell-corrected particle-hole state densities for pre-equilibrium reaction calculations

Calculated particle-hole state densities in the shell-shifted equi-spacing model (s2-ESM) potentially provide a fast and simple way to consistently include shell structure effects in pre-equilibrium reaction calculations. Results from the existing state density formalism were compared with results from the exact counting of states, and the methodology for evaluating the effective single particle state densities in the s2-ESM state density formula was completely revised. Calculated state densities were then compared with literature results obtained from more realistic sets of single particle states, indicating the model's utility. Finally, model-independent estimates of the sizes of the shell gaps were obtained from a set of nuclear mass tables so that the s2-ESM can be used in reaction calculations without the introduction of any new free parameters.

Particle hole state densities

A new formula for the nuclear particle-hole state density is derived. Constant single particle spectra for neutrons and protons are assumed. The so-called Williams formula is extended to different numbers of particles and holes. Correction terms are also obtained. An infinite depth of the potential well is considered.

Conditional particle-hole state densities with exact Pauli exclusion effect

A formula is derived for the density of particle-hole states in the equidistant-spacing model. The effect of the Pauli exclusion principle is exactly taken into account in a more precise way. The pairing effects, as well as two energy constraints, are involved. Comparisons of the results imply that the Pauli effect and the pairing effect play an important role in the particle-hole state density.

Particle-hole state densities with good isospin.

Earlier results for two-component particle-hole state densities with good isospin have been rederived, removing a number of earlier approximations. The new results are shown to be consistent with the corresponding isospin-mixed state densities and are more suitable for light nuclei. Approximate results appropriate for reaction calculations are presented.

Particle-hole state densities with linear momentum and angular distributions in preequilibrium reactions.

We present two methods for the calculation of state densities with linear momentum. The first is exact, convoluting single-particle and hole densities in momentum space, and can be used for nuclear excitations with small numbers of particles and holes. The second, based on statistical arguments, is applicable for excitations of many particles and holes and leads to state densities with a Gaussian linear-momentum dependence. Together, these two techniques allow state densities with linear momentum to be determined for any particle-hole excitation. The relationship between linear- and angular-momentum state densities is discussed, and we show how the familiar Gaussian angular-momentum distribution of states can be obtained from state densities with linear momentum. We argue that these state densities provide the most elegant and straightforward means for obtaining angular distributions in semiclassical preequilibrium theories, avoiding inconsistencies inherent in the commonly adopted procedure of using a nucleon-nucleon scattering kernel. In our approach the angular distribution of preequilibrium particles emitted from a two-particle--one-hole state is identical to that found by Kikuchi and Kawai. Angular distributions from more complex particle-hole states do, however, differ from a convolution of Kikuchi-Kawai scattering kernels, since we do not make a leading-particle approximation. As an illustrative example to the use of the Gaussian form of state densities with linear momentum, we calculate the angular distributions of emitted particles in the preequilibrium reaction $^{165}\mathrm{Ho}$(\ensuremath{\alpha},p), ${\mathit{E}}_{\mathrm{\ensuremath{\alpha}}}$=109 MeV. Our calculations account for the experimental angular distributions well, even at large backward angles where many semiclassical approaches fail.

Linear momentum in the exciton model: Consistent way to obtain angular distributions.

We present an exciton model for preequilibrium emission in nucleon-induced reactions in which linear-momentum conservation is included. The particle emission contributions from the first two preequilibrium stages are calculated by determining exact particle-hole state densities with a specific energy and linear momentum in a Fermi-gas model of the nucleus. Angular distributions arise naturally from our treatment and do not have to be added in an ad hoc way. The angular distributions that we obtain from the first two preequilibrium stages are identical to those found using the Kikuchi-Kawai quasifree scattering kernel.

Exact formulation of particle-hole state densities in the equidistant spacing model with pauli and pairing corrections

A statistical method is used to derive an exact formula for the state density for a fixed number of particles and holes in the equidistant spacing model (ESM). The exact numerical value of the Pauli correction and the analytical formula for the state density are obtained. A way to take into account pairing correlations in the new formula, using Kalbach's method, is proposed.