Level Density Formula Research Articles

Comparison of neutron resonance spacings with microscopic theory for nuclei with static deformation

A level density formula including low-energy rotational levels for nuclei with axially symmetric deformation is tested with neutron resonance data for lanthanide and actinide nuclides. The calculations with the microscopic theory including nuclear pairing utilize deformed single particle levels of Nilsson et al. The experimental data for the actinide nuclei and the lanthanide nuclei are consistent with a theory which includes collective rotational levels. The derivation and applicability of a level density formula which includes collective rotational levels are discussed.

Statistical Calculation of Fission

Calculation of fission fragment distributions based on Fong's statistical theory was carried out for the case of 241 Am thermal neutron fission using Myers and Swiatecki's mass formula and a phenomenological level density formula, originally developed by Gilbert and extended for deformed nuclei by the present authors. The results show that Fong's statistical theory with the now attainable knowledge on mass and level density, can only account for narrow asymmetric mass peaks centred around A ≃108 and 134. The present results are in good agreement with those calculated by Ignatyuk with a microscopic approach. Other physical quantities associated with fission reaction, such as, kinetic energy distribution, charge distribution and total excitation energy distribution at scission configuration are given. Comparison was also made among various approaches for level density formulation.

Liquid-drop description of the nuclear excitation energy

An extended Fermi-gas model is presented to describe volume, surface and asymmetry contributions to the nuclear excitation energy. The potential part of the nuclear energy is treated within Brueckner's energy-density formalism; appropriate corrections due to surface and asymmetry effects are included. An excited nucleus is interpreted as a metastable superheated liquid. The excitation energy as a function of the temperature is given by a sum of volume, surface and asymmetry terms. The corresponding level-density formula is in reasonable agreement with experimental values for a suitable choice of the nuclear incompressibility modulus. It is shown that the surface contribution is appreciable.

Level densities from spectra of different reactions populating the same nucleus

Level densities of 56Fe, 59Co, 60Mi, 62Ni, 63Cu and 65Cu are determined from analyses of several charged-particle spectra populating the same residual nucleus. Experimental spectra are reported for the following reactions: 56Fe(α, α′) 56Fe, 59Co(p, α) 56Fe, 59Co(p, p′) 59Co, 59Co (α, α′) 59Co, 56Fe(α, p) 59Co, 62Ni(p, α) 59Co, 60Ni(α, α′) 60Ni, 63Cu(p, α) 60Ni, 62Ni(p, p′) 62Ni, 62Ni(α, α′) 62Ni, 59Co(α, p) 62Ni, 65Cu(p, α) 62Ni, 63Cu(p, p′) 63Cu, 63Cu(α, α′) 63Cu, 60Ni(α, p) 63Cu, 65Cu(p, p′) 65Cu, 65Cu(α, α′) 65Cu and 62Ni(α, p) 65Cu. Each of these experimental spectra is analyzed with an exact theory giving the angular and energy-dependent differential cross sections for compound-nucleus reactions including explicitly the angular momentum. From these analyses we have determined the Fermi-gas level density parameters: 56Fe, a = 5.7 MeV −1, Δ = 0.7 MeV; 59 Co, a = 6.2 MeV −1, Δ = − 0.8 MeV; 60 Ni, a = 6.4 MeV −1, Δ = 1.3 MeV; 62 Ni, a = 6.4 MeV −1, Δ = 0.5 MeV; 63 Cu, a = 6.8 MeV −1, Δ = − 0.5 MeV; and 65 Cu, a = 6.6 MeV −1 and Δ = − 0.5 MeV . All of the spectra were also analyzed with the conventional slope technique which utilizes the Weisskopf expression for the differential cross sections. The results from such an analysis depend on the form of the pre-exponential term in the level-density formula and may give sizeable errors in the Fermi-gas level-density parameter a. A number of theoretical spectra are computed with the exact theory which explicitly includes angular momentum and these spectra are analyzed with the conventional methods to illustrate the errors introduced when angular momentum is not treated properly for such reactions. An estimate of precompound proton emission is made for several (p, p') reactions.

Isomer Ratios from Low Primary Excitation of Residual Nuclei

A calculational procedure based on the Huizenga-Vandenbosch formalism for the determination of isomer ratios has been developed which is applicable also at low primary excitation energies of the residual nuclei, in contrast to earlier methods. This has been accomplished by introducing level densities which are based on the shell-model level calculations of Hillman and Grover rather than a level-density formula. Furthermore, the assumption that all the neutrons are emitted with the average energy $2T$ (where $T$ is the nuclear temperature) has been replaced by a more realistic neutron spectrum. In order to test this procedure, the isomer ratio of ${\mathrm{Y}}^{87}$ from the photonuclear reaction ${\mathrm{Y}}^{89}(\ensuremath{\gamma},2n)$, which has its reaction threshold at 20.8 MeV, was measured for bremsstrahlung of end-point energies 23, 25.6, 28.6, and 50 MeV. The results of our method are in much better agreement with the experiment than the original Huizenga-Vandenbosch approach. The analysis indicates that a relatively large fraction of the product nuclei is formed directly in the ground state and that quandrupole transitions are an important decay mode near the threshold.

Statistical Model Calculations of Iron-56 Neutron Cross Sections

Abstract : The present work is part of an Air Force initiated program to calculate certain neutron cross sections which have not been measured. The first paper serves the purpose of determining the free parameters in the calculational scheme while at the same time extending and improving the results for Fe56. The new element in the calculation is the use of a statistical level density formula to describe unknown high energy states of the daughter nucleus. This allows the extension of the calculation to higher neutron energies and in particular to multiple particle emissions. The success with which this new computational system provides one with a thorough and self-consistent picture of the neutron reactions on Fe56 over a large range of neutron energies, and gives one confidence that he may now proceed to estimate theoretically neutron cross-sections for other nuclei of interest. (Author)

A shell-model nuclear level density

A nuclear level-density formula was derived by introducing a stepwise single-particle level structure. It was found to be formally identical to the well-known Fermi gas level density, when the level-density parameter a was regarded as a function of the excitation energy E instead of a constant. The computed level-density parameter was compared with that deduced from neutron resonance data.

Density of particle-hole states in the equidistant-spacing model

For the most simple model of the nucleus, a system of independent fermions with equally spaced single-particle levels, the density of particle-hole states is calculated. The exact result, obtained by combinatorial methods, is compared with a level-density formula which is derived by means of the statistical approximation in the conventional way. Even simplifying approximations of this formula are in good agreement with the exact result.

Theory of Nuclear Level Density for Periodic Independent-Particle Energy-Level Schemes

Accurate and perspicuous formulas are derived for the density of states of a degenerate system of any number of types of fermions moving in arbitrary periodic single-particle energy-level schemes. The results are of the standard form with the modification that the excitation energy is to be replaced by an "effective energy." The effective energy contains an additive correction which depends explicitly on the structure and ground-state occupation of the periodic level sequences. The spin-dependent level-density formula for two kinds of particles should be useful in the correlation of observed nuclear level densities with shell-model level sequences. The present work generalizes, unifies, and simplifies earlier treatments of related problems.

Level Densities from Excitation Functions of Isolated Levels

Thick-target excitation functions were measured for the reactions ${\mathrm{Co}}^{59}(p,{\ensuremath{\alpha}}_{0}){\mathrm{Fe}}^{56}$, ${\mathrm{Co}}^{59}(p,{\ensuremath{\alpha}}_{1}){\mathrm{Fe}}^{56}$, ${\mathrm{Co}}^{59}(p,{\ensuremath{\alpha}}_{2}){\mathrm{Fe}}^{56}$, ${\mathrm{Mn}}^{55}(p,{\ensuremath{\alpha}}_{0}){\mathrm{Cr}}^{52}$, ${\mathrm{Mn}}^{55}(p,{\ensuremath{\alpha}}_{1}){\mathrm{Cr}}^{52}$, and ${\mathrm{Ni}}^{62}(p,{\ensuremath{\alpha}}_{0}){\mathrm{Co}}^{59}$ at two or more angles for proton bombarding energies 6-13.5 MeV. The ${\mathrm{Fe}}^{56}(\ensuremath{\alpha},{p}_{0}){\mathrm{Co}}^{59}$ reaction was studied at $\ensuremath{\alpha}$-particle bombarding energies of 12-18.5 MeV. These measurements were used to determine the parameters in three different level-density formulas which were thought to be reasonable candidates to describe the level densities of the residual nuclei. In addition, the experimental values of the cross sections to isolated levels were used in conjunction with available values of the level width $\ensuremath{\Gamma}$ from cross-section fluctuation measurements to determine level densities of compound nuclei at about 20-MeV excitation energy. The absolute values and energy dependence of the nuclear level density in the excitation energy range 0-20 MeV investigated for several nuclei agree with a back-shifted Fermi gas model. The constant-temperature model gives a reasonable fit to the level density in the energy range 0-10 MeV but fails at higher excitation energies. The conventional shifted Fermi gas model does not reproduce the energy dependence and absolute values of the various experimental level densities for any value of the level-density parameter $a$.

Excitation functions of (n, p) and (n, 2n) reactions for some isotopes of K, Mn, Zn and Cu

The excitation functions of the (n, p) reactions for 41K and 64Zn and of the (n, 2n) reactions for 55Mn, 63Cu, 65Cu, 64Zn and 66Zn were measured in the energy region 13–18 MeV by the activation method. The experimental results are compared with statistical-theory predictions based on optical-model absorption cross sections and the Fermi gas model nuclear level density formula. Three different proposals to account for the pairing effect in the nuclear level density are compared. The dependence of (n, 2n) cross sections on the neutron excess of the target nucleus is discussed.

Protons from Zn64 and Zn66 Bombarded with 14.1 MeV Neutrons

Energy spectra and angular distributions were measured for protons emitted from Zn 64 and Zn 66 bombarded with 141.1 MeV neutrons. Any of level density formulae fits equally well the energy spectrum for backward angles. The constant temperature formula gives 1.22±0.07 MeV and 1.35±0.05 MeV as the nuclear temperatures of Cu 64 and Cu 66 respectively. Using the Lang level density formula, the level density parameters for Cu 64 and Cu 66 are given as 8.26±0.78 MeV -1 and 5.39±0.56 MeV -1 respectively. The proton angular distributions are concave and symmetric around 90°. Anisotropy is greater for higher energy protons. The anisotropy of lower-energy protons emitted through the ( n , p ) reaction is found to be comparable to the anisotropy of those emitted through the ( n , n p ) reaction. The cross sections for the ( n , p ) and ( n , n p ) reactions for Zn 64 through compound process are 127±9 mb and 169±26 mb and those for Zn 66 30.0±1.1 mb and 40.5±8.3 mb respectively. The direct ( n , p ) reaction for Zn...

Isotopic-Spin Mixing in Direct and Compound-Nuclear Reactions

The semidirect mechanism of isospin violation in deuteron-induced nuclear reactions, which was presented in an earlier article, is recapitulated. A number of reactions involving the same basic mechanism are discussed and the available data on them are briefly reviewed. This mechanism also leads to much stronger isospin mixing in certain levels in light nuclei than is predicted by the usual perturbative treatment of Coulomb interactions. Such mixing is discussed in terms of a simple two-level model. A suggestion by Levinson has led to a reexamination of the Wilkinson hypothesis that isospin-violating reactions proceed via formation of isotopically impure compound-nuclear levels at "intermediate" excitation energies. It is shown that the most important factor in determining the contribution of these levels is not their degree of isospin mixing, but rather the experimental energy resolution. If the resolution $I$ is poor compared to the average spacing $〈{D}_{J}〉$ of levels of given ${J}^{P}$, then in general the contribution of these levels is reduced by the ratio $\frac{〈{D}_{J}〉}{I}$. When compound levels are so correlated as to give rise to intermediate structure, or a giant resonance, or an isobaric analog state, their contribution to the isospin-nonconserving amplitude need not be small. This implies that isospin-violating reactions are ideally suited to the study of intermediate structure. Because the actual level density in light nuclei at intermediate excitation is not well known, it is not clear whether Levinson's suggestion is relevant to the studies so far conducted on (a) ${\mathrm{C}}^{12}(d,\ensuremath{\alpha}){\mathrm{B}}^{10*}(1.74,T=1)$, and (b) ${\mathrm{O}}^{16}(d,\ensuremath{\alpha}){\mathrm{N}}^{14*}(2.31,T=1)$. It is surely relevant to heavier compound nuclei, however, and estimates based on a statistical level-density formula suggest that Levinson's idea is even relevant for the ${\mathrm{N}}^{14}$ and ${\mathrm{F}}^{18}$ compound nuclei entering reactions (a) and (b) above. Finally, the data of Meyer-Sch\"utzmeister et al. are analyzed for evidence of compound-direct interference. It is conjectured on the basis of a crude but suggestive calculation that the broad ${3}^{\ensuremath{-}}$ (with some possible ${1}^{\ensuremath{-}}$ admixture) resonance seen by them in ${\mathrm{N}}^{14}$ at ${E}_{x}\ensuremath{\sim}18$ MeV is actually an intermediate-structure resonance whose "doorway state" is a "single-particle cluster resonance" in either the entrance or exit channels, or in both. A brief theoretical discussion of compound-direct interference is included for completeness.

Isomeric cross-section ratios and total cross sections for the 59Co(n, 2n) 58g, mCo, 58Ni(n, p) 58g, mCo and 59Co(γ, n) 58g, mCo reactions

The isomeric ratios and excitation functions were measured for the 59Co(n, 2n) 58Co reaction in the neutron energy range 12–18 MeV and for the 58Ni(n, p) 58Co reaction in the neutron range 2–18 MeV. The isomeric ratio is also presented for the 59Co(γ, n) 58Co reaction induced by photons from 30 MeV electron bremsstrahlung. The results are compared with the statistical-model predictions. Use was made of modified nuclear level density formulae based on the superconductivity theory, which account for the pairing and shell effects. In all cases investigated, the calculated isomeric ratios show good agreement with experimental data, whereas the calculated total cross sections do not agree with the measured values. The disagreement may be due to the fact that no account was taken of the gamma emission in competition with the neutron emission from the residual nucleus.

Protons from 40Ca bombarded with 14.1 MeV neutrons

Energy spectra and angular distributions were measured for protons emitted from 40Ca bombarded with 14.1 MeV neutrons. The constant-nuclear-temperature formula for level density fits the energy distribution for backward angles and yields a nuclear temperature of 1.34 ± 0.04 MeV for 40K. The angular distributions of the protons emitted through compound process are nearly symmetric around 90°. The spin cut-off parameter is estimated to be about 2.0 from the angular distribution of protons emitted by the (n, p) reaction. The cross sections for the (n, p) and (n, np) reactions through compound process are 471 ± 21 mb and 180 ± 32 mb, respectively. The direct (n, p) reactions has a cross section of about 11 mb.

Level Density of Light Nuclei

The level density of light nuclei ($A\ensuremath{\le}70$) is studied by analyzing statistical reaction spectra, slow-neutron resonances, and level widths at high excitation energy. Further evidence seems to indicate that, as proposed in a preceding work, the effective excitation energy of excited nuclei, appearing in the Lang-Le Couteur level-density formula, must be calculated by means of the following expression: $U=(E\ensuremath{-}\ensuremath{\Delta}+\frac{70}{A})$ $E$ is the usual excitation energy, $\ensuremath{\Delta}$ is the pairing energy, and $\frac{70}{A}$ MeV is a term that shifts, in a smooth manner, the zero of the energy scale of different nuclei. In this case the Lang-Le Couteur formula, assuming $R\ensuremath{\cong}1.5{A}^{\frac{1}{3}}$ F, $\mathcal{I}=0.7{\mathcal{I}}_{\mathrm{rig}}$, $a=(0.127A)$ Me${\mathrm{V}}^{\ensuremath{-}1}$, gives, without further changes of its parameters, a correct estimation of the slope and absolute value of the level densities of light nuclei for energies ranging from ($1+\ensuremath{\Delta}$) up to \ensuremath{\sim}20 MeV. It is also shown that the preceding choice of the parameters of Lang-Le Couteur level-density expression allows one to reproduce very well the experimental distributions of low-energy levels observed with reactions which proceed, at least partially, through the formation of a compound nucleus.

ABUNDANCE ANOMALIES PRODUCED BY NUCLEAR REACTIONS IN STELLAR SURFACE LAYERS: I. NUCLEAR-REACTION RATES

A preliminary investigation of the effects on abundances in stellar surfaces of extensive nuclear bombardment required the calculation of more than 105 nuclear-reaction cross sections. It was necessary to develop simplified methods for using the statistical theory of nuclear reactions to make these calculations in order that the computer time should not be prohibitive. These methods are described here and the results are compared with experiment. The accuracy of the calculations is, in general, about as good as, or somewhat better than, that obtained in previous applications of the statistical theory, probably because the use of an accurate level density formula outweighed the crudity of other approximations.

Nuclear-level densities from cross-section fluctuations

The value of the level density ϱ(U, J) at ∼7 MeV of excitation energy, for nuclei with 20 ≲A ≲ 70, obtained by analysing slow-neutron resonances, is compared with the value at ∼ (15 ÷ 20) MeV excitation energy, extracted from the coherence energies characterizing fluctuating excitation functions. The comparison seems to indicate that the Lang and Le Couteur level density formula gives a correct estimate of the energy dependence of the level density ϱ(U, J) with the following choice of the parameters:R=1.5A1/3 fm, J=0.7Jrig,a=(0.127A) (MeV)−1, and if the quantity ΔU=(70/A) MeV is added to the effective excitation energy, evaluated by subtracting the pairing energy from the usual excitation energy. The method followed in the analysis and the results obtained are discussed.

Absolute level densities from neutron inelastic scattering

The spectra of neutrons inelastically scattered from Na, Al, K, Cr, Fe, 58Ni and 60Ni have been measured for 7 MeV incident neutrons. For the nuclei 56Fe and 60Ni, statistical-model calculations were made to predict the complete spectrum of scattering to the known discrete levels below about 3.5 MeV together with a continuum of higher levels described by the level density formula ω( U) = KU −2 exp 2√ aU. By fitting the theoretical spectrum to the experimental one, the constants K and a were both obtained, thus determining the absolute level density in the continuum in addition to its variation with excitation energy. The results indicate that the nuclear moments of inertia of 56Fe and 60Ni have approximately the rigid-body value. Evidence is also presented for strong direct components in the scattering to the first levels in 23Na and 27Al and to the first 2 + level and 3 − collective states in 52Cr, 56Fe, 58Ni and 60Ni.

Dissipation of Energy and Angular Momentum by Emission of Neutrons and Gamma Rays

We examine quantitatively some of the consequences of the statistical theory of nuclear de-excitation for neutron and $\ensuremath{\gamma}$-ray emission from highly excited nuclei. Special attention is given to the dissipation of angular momentum. To a first approximation, the nuclear thermal energy is removed first, initially by neutron and then by dipole $\ensuremath{\gamma}$-ray emission. The rotational energy is dissipated afterwards as a cascade of lowenergy $\ensuremath{\gamma}$ rays which may be composed of a high proportion of quadrupole radiation. The meager experimental data available are consistent with the theoretical expectations, but do not as yet permit a crucial test. Further possible experiments are pointed out. The role of the lowest-energy excited state at every angular momentum (yrast levels) is taken into account explicitly. The importance of using correct level densities at energies near the yrast levels, where the conventional level-density formulas may be very inaccurate, is stressed.