Partial Level Density Research Articles

A Comparison between Ericson's Formulae Results and Experimental Data Using New Formulae of Single Particle Level Density

The partial level density PLD of pre-equilibrium reactions that are described by Ericson’s formula has been studied using different formulae of single particle level density . The parameter was used from the equidistant spacing model (ESM) model and the non- equidistant spacing model (non-ESM) and another formula of are derived from the relation between and level density parameter . The formulae used to derive are the Roher formula, Egidy formula, Yukawa formula, and Thomas –Fermi formula. The partial level density results that depend on from the Thomas-Fermi formula show a good agreement with the experimental data.

Quantification of the 35Cl[formula omitted] reaction channel

The evaluated 35Cl(n,p) cross sections in the ENDF/B-VIII.0 nuclear data library remain uncertain because of the lack of measured data in the 100–600 keV neutron energy range. In this energy region, the R-matrix analysis relies heavily on estimated (n,p) partial widths and level density information. This resulted in significant differences for the evaluated cross sections and predicted reactivity in nuclear reactor applications with respect to previous nuclear data libraries. This paper presents a methodology to quantify the cross sections for the 35Cl(n,p) reaction channel from the combination of measured total cross sections and model calculations.

Extracting model-independent nuclear level densities away from stability

The nuclear level density (NLD) is a fundamental measure of the complex structure of atomic nuclei at relatively high energies. Here we present the first model-independent measurement of the absolute partial NLD for a short-lived nucleus. For this purpose we adapt the recently introduced ``shape method'' for $\ensuremath{\beta}$-decay experiments, providing the shape of the $\ensuremath{\gamma}$-ray strength function for exotic nuclei. In this work, we show that combining the shape method with the $\ensuremath{\beta}$-Oslo technique allows for the extraction of the NLD of the populated states without the need for theoretical input. This development opens the way for the extraction of experimental NLDs far from stability with major implications in astrophysical and other applications. We benchmark our approach using data for the stable $^{76}\mathrm{Ge}$ nucleus, finding excellent agreement with previous experimental results. In addition, we present new experimental data and determine the absolute partial level density for the short-lived $^{88}\mathrm{Kr}$ nucleus. Our results suggest a fivefold increase in the NLD for the case of $^{88}\mathrm{Kr}$, compared to the recommended values from semimicroscopic Hartree-Fock Bogoliubov calculations recommended by the RIPL3 nuclear data library. However, our results are in good agreement with other semimicroscopic level density models. We demonstrate the impact of our method on the $^{87}\mathrm{Kr}(n,\ensuremath{\gamma}$) neutron capture rate and show that our experimental uncertainties for NLDs fulfill the requirements needed for astrophysical calculations predicting $r$-process abundances.

A Comparison Between the Theoretical Cross Section Based on the Partial Level Density Formulae Calculated by the Exciton Model with the Experimental Data for (_79^197)Au nucleus

In this paper, the theoretical cross section in pre-equilibrium nuclear reaction has been studied for the reaction at energy 22.4 MeV. Ericson’s formula of partial level density PLD and their corrections (William’s correction and spin correction) have been substituted in the theoretical cross section and compared with the experimental data for nucleus. It has been found that the theoretical cross section with one-component PLD from Ericson’s formula when doesn’t agree with the experimental value and when . There is little agreement only at the high value of energy range with the experimental cross section. The theoretical cross section that depends on the one-component William's formula and on-component corrected to spin PLD formula doesn't agree with the experimental cross section. But in case of theoretical cross section based on two-component Ericson's and William's PLD formulae it has been found that there is acceptable agreement when the exciton number is taken .

A Comparison Between the Theoretical Cross Section Based on the Partial Level Density Formulae Calculated by the Exciton Model with the Experimental Data for (_79^197)Au nucleus

في هذا البحث، تمت دراسة المقطع العرضي النظري لتفاعلات قبل التوازن للتفاعل بطاقة 22.4 MeV. حيث تم تعويض صيغة اركسون للمركبة الواحدة لكثافة الحالات الجزئية PLDوتصحيحاتها ( تصحيح وليم, وتصحيح البرم) بالمقطع العرضي النظري وقورنت النتائج مع النتائج العملية لنواة الذهب . حيث وجد ان المقطع العرضي النظري مع صيغة اركسون لمركبة واحدة لا يتفق مع للمقطع العرضي العملي عندما وعندما هناك اتفاق بسيط عند القيم الكبيرة للطاقة. المقطع العرضي النظري المعتمد على صيغة كثافة الحالات الجزئية PLD المصححة بتصحيح وليم لحالة المركبة الواحدة و صيغة المركبة الواحدة المصححة للزخم الزاوي لا يتطابقان مع القيم العملية. ولكن في حالة المقطع العرضي النظري المعتمد على صيغتي اركسون و وليم ذواتا المركبتين وجد ان هناك تطابق جيد لصيغتي اركسون و وليم عندما n=5.

Combinatorial level densities by the real-time method

Levels densities of independent-particle fermionic systems can be calculated easily by using the real-time representation of the evolution operator together with the fast Fourier transform. We describe the method and implement it with a set of Python programs. Examples are provided for the total and partial levels densities of a heavy deformed nucleus (164Dy). The partial level densities that may be calculated are the projected ones on neutron number, proton number, azimuthal angular momentum, and parity. Program summaryProgram title: rt_levels.pyCatalog identifier: AEUI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUI_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 6515No. of bytes in distributed program, including test data, etc.: 263821Distribution format: tar.gzProgramming language: Python.Computer: PCs.Operating system: Unix, Apple OSX.RAM: 1 MBClassification: 17.16.External routines: NumpyNature of problem: Compute many-body level densities of finite systems of fermions described by their single-particle spectraSolution method: Fast Fourier transform of the real-time Green’s functionRunning time: A few minutes to one hour

Investigation of the Appropriate Partial Level Density Formula for Pre-Equilibrium Nuclear Exciton Model

Ericson formula represents the first formula, which was suggested to describe the partial level density (PLD) formula in pre-equilibrium region of the nuclear reactions. Then a number of corrections were added to this formula in order to make it more suitable to physical meaning. In this paper, there are two aims to be done: the first aim is to study the correspondence between one and two-components formulae in Ericson, Pauli, and pairing corrections; the second aim is to compare and study the results of Comprehensive formula, which contents with all corrections, with Ericson, Pauli, and pairing formulae. The Comprehensive formula was suggested to simulate the reality. To achieve these aims the 56Fe and 90Zr nuclei were chosen and the results showed that the difference between one and two-components formulae was too small which can be neglected. Furthermore, the results strongly recommended that for cross section calculations of the nuclear reaction, one must use Comprehensive formula rather than Pauli formula.

Electronic structures of AlB2-type superconducting YbGa1+xSi1−x alloys probed by NMR and Seebeck coefficient

We report the electronic properties of the AlB2-type compounds YbGa1+xSi1−x (x=0, 0.15, and 0.3) studied by means of the nuclear magnetic resonance (NMR) and the Seebeck coefficient measurements. These materials are of current interest due to the presence of superconductivity with Yb element. From the analysis of G69a NMR spin-lattice relaxation rates, we deduce the Ga 4s partial Fermi level density of states Ns(EF) for these compounds. The result indicates a gradual increase in Ns(EF) with increasing x in YbGa1+xSi1−x. In addition, the evolution of the Seebeck coefficient feature can be understood well within the band-filling scenario. From the Seebeck coefficient analysis, we find that the variation in the total Fermi level density of states N(EF) is not consistent with the trend of superconducting temperature Tc which shows a gradual decrease with Ga content. These observations support the hypothesis that the electronic Fermi level density of states is not the key factor in determining the superconducting transition temperature of YbGa1+xSi1−x.

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

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.

NMR and Seebeck coefficient studies of the electronic band structure in Nb1−xB2

With the aim of providing experimental information for the Nb deficiency enhanced superconducting temperature (TC) in the Nb1−xB2 samples, we carried out a study on Nb1−xB2 (x=0, 0.13, 0.20, and 0.26) by means of nuclear magnetic resonance (NMR) and Seebeck coefficient (S) measurements. From the N93b NMR spin-lattice relaxation rates, we can deduce the Nb 4d partial Fermi level density of states (DOS) Nd(EF) for each individual composition. The result indicates that Nb0.74B2 and Nb0.80B2 possess large Nd(EF) while the lowest one appears in Nb0.87B2. The Seebeck coefficient also shows smaller absolute values in Nb0.74B2 and Nb0.80B2, associated with higher Fermi level DOS in both compounds. Interestingly, the temperature variation in S exhibits a broad minimum in Nb0.87B2, which is attributed to the presence of a pseudogap near the Fermi level. These observations were found to be in good agreement with the prediction from band structure calculations based on the appearance of the Nb vacancies in Nb1−xB2. In addition, the present study clearly reveals that the observed Tc enhancement by Nb deficiency has no direct relevance to their electronic Fermi level DOS.

Partial level density of n-quasiparticle excitations, radiative strength functions and new experimental information on the dynamics of nuclear structure change in the B n range

The most reliable at present values of the level density in the fixed spin window and the sums of radiative strength functions of cascade gamma transitions are obtained from analysis of intensities of two-step cascades excited upon thermal neutron capture for approximately 40 nuclei in the mass range 40 ≤ A ≤ 200. The maximal reliability of these data is provided by the experimental conditions—minimum possible propagation error coefficients and practically unique solution of the problem of determination of gamma decay parameters from measured spectra. The experimental data are approximated by the sum of partial level densities corresponding to excitation of n quasiparticles. Steplike structures in the level density at excitation energies smaller than 3–4 MeV are described with good accuracy as the superposition of two-quasiparticle (three-quasiparticle in odd A nuclei) and vibrational excitations with the coefficient of collective density enhancement K coll ≈ 10−20. They correspond to excitation-energy-correlated maximum enhancement of the radiative strength functions of primary gamma transitions. The level density at larger excitation energies is well reproduced if the breakup of at least two more Cooper pairs of nucleons is taken into account. The increase in the number of excited quasiparticles in the nucleus corresponds to unconditional reduction of the radiative strength functions of primary gamma transitions of the compound state decay. However, the maximum possible value of partial widths of primary transitions increases regularly with decreasing energy. Some ambiguity in the results of approximation and divergence from existing theoretical ideas of the energy dependence of nucleon correlation functions in an excited nucleus point to the possibility of direct extraction from experiment of fundamentally new information on the structure of excited nuclear levels in the range of the neutron binding energy. These are, first of all, the parameters of dependence of nucleon correlation functions on the excitation energy of the nucleus.

Pre-equilibrium–Emission Surface Effects in Activation Reactions

Description of the high-energy side of activation excitation functions, most sensitive to pre-equilibrium processes modeling, is reported in this work within the geometry–dependent hybrid (GDH) model. Improved cross-section calculations are reported for fast-neutron induced reactions on 51V and 92, 94–98, 100Mo based on partial level density including surface effects as well as energy–dependent single–particle level densities.

Partial level densities for nuclear data calculations

The main formalisms of partial level densities (PLD) used in preequilibrium nuclear reaction models, based on the equidistant spacing model (ESM), are considered. A collection of FORTRAN77 functions for PLD calculation by using 14 formalisms for the related partial-state densities is provided and 28 sample cases (73 versions) are described. The results are given in graphic form too. Composite (recommended) formulas, which include the optional use of various corrections, i.e. the advanced pairing and shell correction in addition to the Pauli effect, and average energy-dependent single-particle level densities for the excited particles and holes are also given. The formalism comprises the density of particle—hole bound states, and the effects of an exact correction for the Pauli-exclusion principle are considered.

Single particle level density in a finite depth potential well

We consider the single particle level density $g(\ensuremath{\epsilon})$ of a realistic finite depth potential well, concentrating on the continuum $(\ensuremath{\epsilon}>0)$ region. We carry out quantum-mechanical calculations of the partial level density ${g}_{l}(\ensuremath{\epsilon})$, associated with a well-defined orbital angular momentum $l<~40$, using the phase-shift derivative method and the Green's-function method and compare the results with those obtained using the Thomas-Fermi approximation. We also numerically calculate $g(\ensuremath{\epsilon})$ as a $l$ sum of ${g}_{l}(\ensuremath{\epsilon})$ up to a certain value of ${\mathcal{l}}_{\mathrm{max}}<~40$ and determine the corresponding smooth level densities using the Strutinsky smoothing procedure. We demonstrate, in accordance with Levinson's theorem, that the partial contribution ${g}_{l}(\ensuremath{\epsilon})$ to the single particle level density from continuum states has positive and negative values. However, $g(\ensuremath{\epsilon})$ is nonnegative. We also point out that this is not the case for an energy-dependent potential well.

Approximation for shell-model level densities: Ergodicity

Using the supersymmetry method, we show that the maximum entropy approach for the calculation of nuclear shell-model partial and total level densities, developed in a previous paper, is ergodic.

Exciton level densities with spin and parity based on random matrix model

Level densities for fixedJ π and the exciton number are evaluated for closed shell nuclei40Ca and208Pb. The single particle spectra and wave functions are generated by Woods-Saxon potentials. The effects of the residual interaction are taken into account statistically by the method of generating function and Grassmann integral. The matrix elements for the residual interaction are assumed to be random variables with Gaussian distributions whose second moments are calculated by using a zero range interaction. The second moments are evaluated for fixedJ π by ignoring the Pauli principle between active nucleons and the spectator. This approximation is shown numerically to be very good. The partial level densities are calculated using the second moments as well as independent particle model spectra. The resulting level densities spread over wider energy ranges, have a smoother energy dependence and are enhanced at low energies compared with the independent particle model densities, although the total level densities do not differ by much.

Transmission coefficients in the presence of multistep-direct reactions

Formal expressions for transmission coefficients governing the multistep-compound process and introduced in an earlier paper, are explicitly worked out, using the idea of a multistep-direct process populating the compound nucleus. A perturbative treatment of inelastic scattering, combined with a “never come back” and the sudden approximation, and a statistical average over exit channels lead to simple, intuitively appealing expressions which contain shell-model type matrix elements and partial level densities and are therefore amenable to numerical evaluation.

Average partial level density based on the random matrix model. Inclusion of realistic one-body spectrum and effect of particle escape

The effect of the realistic one-body spectrum is included in calculating the average partial level density, which has been so far obtained assuming degenerate one-body spectrum and GDE for the residual interaction. We show that such partial level densities are obtained by solving the extended saddle point equations which need inputs of the realistic one-body spectrum as well as the second moment of the ensemble. It is seen that the average partial level density is expressed in a convolution form of the partial level density for realistic one-body spectrum with the distribution which describes the effect by the random residual interaction. Calculations numerically performed show the improvement over the previous results. The effect of particle escape on the level density is also investigated and turns out to be practically negligible in precompound reactions.

Partial level densities in precompound reactions

Results of a quantum-statistical theory of precompound reactions derived recently are recast into a form that allows their use in realistic calculations. Particular emphasis is put on the role of the partial level densities which replace the Ericson-Williams expressions for the density of states with fixed exciton number.