Nuclear level densities and level spacing distributions from 20F to 244Am

The parameters of nuclear level density formulae have been determined from extensive and complete level schemes and neutron resonance densities for 24 nuclei between ²⁰F and ²⁴⁴Am. The constant temperature level-density formula and the Bethe formula are compared with the experimental level densities. Both formulae reproduce experimental densities equally well. The same nuclei have been used to obtain an A-dependent spin cut-off parameter for low-lying levels. The spacing distribution of levels with equal spins and parities at lower excitation energies is found to be much closer to an exponential distribution than to the Wigner distribution especially for even-even nuclei. This is at variance with previous theoretical expectations and interpretations of nuclear data compilations. It gives evidence for a further good quantum number at low excitation energies in addition to spin and parity or for very different structures.     

Nuclear level density with fixed exciton number

An expression for the nuclear level density has been obtained for fixed number of particlesp and holesh taking into account the energy dependence of the single-particle state density for a gas model of noninteracting particles. The applicability of the equidistant spacing model (g=const.) is analyzed for the casesp?h andp? h?1. The possibility of factorization of the angular momentum dependence is studied provided g≠const. A semi-classical expression for the dependence of the single-particle density on angular momentum projection has been obtained without specifying the nuclear potential. Them-dependence ofg(?, m) is shown to be close to a Gaussian for the infinite square well potential. A simple expression which takes into account the effect of the Pauli exclusion principle is proposed for the angular momentum dispersion of the level density distribution.     

Parity dependence of the nuclear level density at high excitation

The basic underlying assumption ϱ(l + 1, J) = ϱ(l, J) in the level density function π(U, J, π) has been checked on the basis of high quality data available on individual resonance parameters (E₀, Γ_n, J^π) for s- and p-wave neutrons in contrast to the earlier analysis where information about p-wave resonance parameters was meagre. The missing level estimator based on the partial integration over a Porter-Thomas distribution of neutron reduced widths and the Dyson-Mehta Δ₃ statistic for the level spacing have been used to ascertain that the s- and p-wave resonance level spacings D(0) and D(1) are not in error because of spurious and missing levels. The present work does not validate the tacit assumption π(l + 1, J) = π(l, J) and confirms that the level density depends upon parity at high excitation. The possible implications of the parity dependence of the level density on the results of statistical model calculations of nuclear reaction cross sections as well as on pre-compound emission have been emphasized.     

Evolution of Distribution of Dislocations in Ni3Ge Single Crystals During Deformation

The evolution of the distribution of dislocations in Ni₃Ge single crystals subjected to deformation in uniaxial compression is studied. The dislocation ensemble in the material under review is found to be of a chaotic homogeneous type. Contact interactions between dislocations prevail, and a linear relation of the spacing between dislocations to the length of dislocation segments is observed for stoppers of an arbitrary type. An equation is derived for the probability density function of the fraction of mobile dislocation segments. The solution to the equation is the normal distribution law. This solution can be extended to parameters that are functions of the dislocation density or spacing between dislocations. The experimental histograms of the spacing between dislocations and of that between arbitrary stoppers with a high significance level obey the lognormal law for all degrees of reduction studied.     

Systematics of nuclear level density parameters of nuclei deficient in one neutron

Photoneutron mean energies of 38 elements were measured as a function of peak bremsstrahlung energy for elements with 23 ≦ Z ≦ 83. Results are compared with neutron mean energies calculated from statistical theory, using for nuclear level densities modified Fermi gas formulae with and without pairing corrections and a constant temperature formula. Except near closed shells the Fermi gas formula with pairing corrections gives reasonable to good correlation between experimental and theoretical data. Derived values of the nuclear level density parameter a-except near Z = 82-are in quantitative agreement with those from recent neutron resonance data.     

Effect of vibrational states on nuclear level density

Simple methods to calculate a vibrational enhancement factor of a nuclear level density with allowance for damping of collective state are considered. The results of the phenomenological approach and the microscopic quasiparticle—phonon model are compared. The practical method of calculation of a vibrational enhancement factor and level density parameters is recommended. The text was submitted by the authors in English     

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.     

Semi-microscopic calculation of the level density in spherical nuclei

The level density at the neutron binding energy for 90 spherical nuclei in the interval 50 < A < 205 is calculated by a method of direct counting of the number of states taking into account collective vibrational excitations. The results of calculations are in satisfactory agreement with the experimental data. The difference in the level density of doubly even and odd-A nuclei is correctly described. The effect of nuclear vibrations on the level density is studied, and it is shown that the account of them leads to an increase in the density by a factor of 1.5–10 and to a decrease in the density fluctuations. It is also studied how the level density depends on excitation energy. With increasing excitation energy, our results come nearer the corresponding values obtained by the statistical model. It is found that the density fluctuations decrease with increasing excitation energy but remain still strong at the neutron binding energy for nuclei with A = 50–70 and for nuclei around closed shells. The density ρ(I^π) is studied as a function of spin and parity. It is shown that at the neutron binding energy the ratio $\text{ρ(I}{}^{\mathrm{+}}\text{)}\text{ρ(I}{}^{\mathrm{?}}\text{)}$ is different from unity for the majority of nuclei. This difference is especially striking for ⁵⁷Fe and ⁵⁸Fe nuclei.     

Effects of grain size on dislocation organization and internal stresses developed under tensile loading in fcc metals

The relationship between deformation and dislocation properties has been studied for pure polycrystalline nickel and austenitic stainless steel AISI 316L in stage III. Special care was taken to study statistically the effects of the grain size and grain orientation on dislocation densities and distribution. It is shown that the nature of dislocation cells depends on grain size and crystallographic orientation. The dimensional parameters, which depend on grain size, i.e. the inter-boundary spacing (λ) and boundary thickness (e), define three domains of crystallographic orientation and depend on the grain size. Scaling hypotheses reveal two physical mechanisms which, at this level of plastic strain, are correlated to a specific value of the noise, associated with distribution functions. Similarities between structural parameters and dislocation densities in each phase (walls and inter-walls spacing) are identified and discussed in terms of kinetic equations describing dislocation density evolution and fluctuations of certain physical parameters. This similarity provides physical signification of the scaling distribution obtained on λ and e in terms of a stochastic approach to dislocation distribution. The origin of Hall–Petch behaviour observed at large strain is interpreted in terms of an interaction between inter- and intra-granular long-range internal stresses, which depends on grain size. We conclude that, at high strain, the Hall–Petch phenomenological relationship is a consequence of plastic strain history and strain gradient in grains. From this last point, a length scale arises naturally, which depends on stacking fault energy.     

Nuclear level densities with collective rotations included

The level density is calculated from the single particle energies in a Woods-Saxon potential with pairing included in the BCS approximation. The collective rotations are included by addition of a rotational band on top of each of the intrinsic levels. The nuclei investigated have mass numbers in the region 100 ≦ A ≦ 253. At the ground state deformation and at the neutron separation energy for the nucleus in question we compare calculated and observed level densities. The dependence on the parameters in the model are investigated. Considering the uncertainties in these parameters the calculated results are believed accurate to within a factor of 3. The rotations contribute typically a factor of 40. They must be included for deformed and not for spherical nuclei. We underestimate systematically the level density by a factor of 4 with fluctuations around the average value by a factor of 3. The nuclei lighter than ¹³⁸Ba are an exception. We obtain around a factor 100 too few levels in the calculation.     

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.     

Systematic for parity distribution in nuclear level density near neutron separation energies

The correct form of nuclear level density function ρ(U,J,π)$\rho (U,J,\pi )$ is required to calculate nuclear cross-sections needed for various applications ranging from reactor designing, nuclear astrophysics, etc., to transmutation of nuclear waste. The asymmetrical statistical distribution of parity of nuclear levels at low energies poses an intriguing problem leading to larger uncertainties in the calculated values of cross-sections. On the basis of high resolution data available on individual resonance parameters (Eo,Jπ,Γn)$(E_o,J^{\pi },{\Gamma }_n)$ for s- and p-wave neutrons, mass and energy dependence formulae for the parity distribution in the nuclear level density have been proposed which supports the fact of equipartition of parities at high excitation energies.     

Level statistics of near-yrast states in rapidly rotating nuclei

The nearest neighbour level spacing distribution and the Δ₃ statistic of level fluctuations associated with very high spin states (I ≳ 30) in rare-earth deformed nuclei are analysed by means of a cranked shell model. The many particle-many hole configurations created in the rotating Nilsson potential are mixed by the surface-delta two-body residual interaction. The levels in the near-yrast region show a Poisson-like level spacing distribution. As the intrinsic excitation energy U increases, the level statistics shows a gradual transition from order to chaos, reaching at U ≳ 2 MeV the Wigner distribution typical-of the Gaussian orthogonal ensemble of random matrices. This transition is caused by the residual two-body interaction. On the other hand, the level spacings between the yrast and the first excited state show a peculiar behaviour, displaying a Wigner-like distribution instead of the Poisson-like distribution seen for the other near-yrast rotational states. The lowest spacings reflect the properties of the single-particle orbits in the mean-field, and are only weakly affected by the residual two-body interaction.     

Spin-dependent level density in interacting boson-fermion-fermion model

The level density of the odd-odd nucleus¹³² Pr is investigated in the interacting boson-fermion-fermion model (IBFFM) which accounts for collectivity and complex interaction between quasiparticle and collective modes. The IBFFM total level density is fitted by a Gaussian and its tail is also fitted by Bethe formula and constant temperature Fermi gas model. The IBFFM spin-dependent level densities show high-spin reductions with respect to Bethe formula. This can be well accounted for by a modified spin-dependent level density formula.     

Road traffic sound level distributions

A microprocessor-controlled instrument has been used to form a traffic noise level histogram with a resolution better than 0·1 dB per channel. The instrument calculates the mean, standard deviation, skewness and kurtosis of the distribution, along with L_N and L_eq values. The results of over 200 measurements, of 400 s duration, are shown to be in disagreement with predictions based on the commonly assumed Gaussian distribution. Skewness values ranging from +1 to ?1 have been observed, while kurtosis can exceed 4. Measurements taken near freely flowing, pulsed and banked traffic have been used to describe the “typical” distribution shape, it being observed that banked traffic noise has markedly different characteristics from other types of traffic. Simultaneous measurements taken on each side of the road have been related to the position of the traffic on the road and these data have led to a simple model for estimating the distribution shape and statistical parameters.     

Oscillator parameters in nuclei

The dependence on nucleon numbers of the harmonic oscillator length parameter b or energy spacing ℏω in nuclei is determined, using an extensive tabulation of nuclear charge radii and an empirical expression for the difference between proton and neutron radii. Various possible constraints on the parameters or radii are discussed. Alternative parametrizations are compared and a preferred “universal” parametrization is proposed. It is argued that the established Blomqvist–Molinari formula is perfectly adequate for determining the oscillator parameter for specific applications.     

Shape transition of state density for bosonic systems

For a finite m boson system, the ensemble-averaged state density has been computed with respect to the body interaction rank k. The shape of such a state density changes from Gaussian to semicircle as the body rank of the interaction increases. This state density is expressed as a linear superposition of Gaussian and semicircular states. The nearest-neighbour spacing distribution (NNSD), which is one of the most important spectral properties of a system, is studied. The NNSDs are rather independent of body rank k and show a Wigner distribution throughout.     

Pseudointegrable systems in classical and quantum mechanics

We study particles moving in planar polygonal enclosures with rational angles, and show by several methods that trajectories in the classical phase space explore two-dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply-handled spheres. The quantum mechanics of one such ‘pseudointegrable system’ is studied in detail by computing energy levels using an exact formalism. This system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side L. We find that neighbouring levels avoid degeneracies as L varies, and that the probability distribution for the spacing S of adjacent levels vanishes linearly as S→0 (‘level repulsion’). The Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density. Oscillatory corrections to the mean level density are given as a sum over closed classical paths; for pseudointegrable systems these closed paths form families covering part of the phase-space invariant surfaces.