Nuclear level densities and level spacing distributions: Part II

Extensive and complete nuclear level schemes and neutron resonance densities of 75 nuclides have been used to determine parameters of level density formulae, the spacing distribution of levels with equal spins and parities, and spacing correlations. The dependences of the level density parameters on A, on pairing energies and on shell effects are discussed. The spacing distribution is found to lie between a Poisson and Wigner distribution indicating a transitional character between regular and chaotic properties of the nuclear quantum system near the ground state.     

Universality of level spacing distributions in classical chaos

We suggest that random matrix theory applied to a matrix of lengths of classical trajectories can be used in classical billiards to distinguish chaotic from non-chaotic behavior. We consider in 2D the integrable circular and rectangular billiard, the chaotic cardioid, Sinai and stadium billiard as well as mixed billiards from the Limaçon/Robnik family. From the spectrum of the length matrix we compute the level spacing distribution, the spectral auto-correlation and spectral rigidity. We observe non-generic (Dirac comb) behavior in the integrable case and Wignerian behavior in the chaotic case. For the Robnik billiard close to the circle the distribution approaches a Poissonian distribution. The length matrix elements of chaotic billiards display approximate GOE behavior. Our findings provide evidence for universality of level fluctuations—known from quantum chaos—to hold also in classical physics.     

Nuclear level densities in self-consistent field approximation

The effect of two-body nature of the nuclear shell model potential on the recent numerical calculations of the nucleai level density has been examined. For the two most widely used single particle energy level schemes based on harmonic oscillator and Woods-Saxon potential, this effect is shown to significantly modify the excitation energy dependence of the level densisties.     

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.     

Nuclear level densities in the constant-spacing model

A new method to calculate level densities for non-interacting Fermions within the constant-spacing model with a finite number of states is developed. We show that asymptotically (for large numbers of particles or holes) the densities have Gaussian form. We improve on the Gaussian distribution by using analytical expressions for moments higher than the second. Comparison with numerical results shows that the resulting sixth-moment approximation is excellent except near the boundaries of the spectra and works globally for all particle/hole numbers and all excitation energies.     

Taylor and Padé analysis of the level spacing distributions of random-matrix ensembles

We construct the distributionP(S) of nearest-neighbor level spacings for the orthogonal, unitary, and symplectic ensembles of (Hermitian and unitary) random matrices in the limit of large dimension. The Taylor expansion ofP(S) aroundS=0 is given explicitly to arbitrarily high orders. By employing a diagonal Padé approximation we interpolate between the small-S behavior given by the Taylor expansion and the rigorously known asymptotic form at largeS.     

Level spacing distributions and the Bessel kernel

Scaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others'as well, the kernel one obtains by scaling in the “bulk” of the spectrum is the “sine kernel” $\frac{{\sin \pi (x - y)}}{{\pi (x - y)}}$ . Rescaling the GUE at the “edge” of the spectrum leads to the kernel $\frac{{Ai(x)Ai'(y) - Ai'(x)Ai(y)}}{{x - y}}$ , where Ai is the Airy function. In previous work we found several analogies between properties of this “Airy kernel” and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painlevé transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely $$\frac{{J_\alpha (\sqrt x )\sqrt y J'_\alpha (\sqrt y ) - \sqrt x J'_\alpha (\sqrt x )J_\alpha (\sqrt y )}}{{2(x - y)}},$$ , whereJ _α(z) is the Bessel function of order α. In the cases α=?1/2 these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels.     

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.     

Nuclear charge and matter distributions

The experimental methods used to determine nuclear charge and matter distributions are described, together with their theoretical interpretation in terms of nuclear structure.     

Bivariate distributions in statistical spectroscopy studies: Fixed-J level densities,fixed-J averages and spin cut-off factors

Analytical studies with the TBRE ensemble, supplemented by numerical calculations with realistic interactions show that for large particle numbers, the bivariate cumulants (k _rs ) defined for the Hamiltonian operator (H) and theJ _z operator, are very small; ∥k _rs ∥?0.3 for 3≦r+s≦6. As a result the expansions around a bivariate normal density are meaningful for the fixed-M densities (ρ(E,M)). We adopt a bivariate Edgeworth expansion forρ(E,M) and give a compact form for the same. Finally using thisρ(E,M) (which also define fixed-J densities uniquely), new series expansions are given for fixed-M (and hence for fixed-J) averages of the powers ofH and also for the spin cut-off factors.     

Exact removal of the center-of-mass spurious states from level densities

We give recursive formulae for the exact removal of the contribution of the center-of-mass spurious states from the fixed-spin and parity nuclear level density found in shell-model calculations, provided the total level density for restricted configurations is known. The method is valid for a large class of problems using a harmonic oscillator basis. Using our earlier methods based on statistical spectroscopy that utilize the centroids and widths for a restricted class of fixed-spin configurations, such as Nvariant Planck's over 2piomega excitations, one can calculate very accurately level densities free of spurious states. The approach is applicable to other fermion and boson systems trapped by an oscillator potential.     

Fixed angular momentum moments and level densities

Approximate fixed-J energy moments are evaluated by orthogonal polynomial expansion and parametric derivative methods. Comparisons with exact values, both directly in terms of moments and in terms of level densities calculated from energy moments, show that sufficiently accurate values can be calculated to produce good fixed-J level densities.     

Spin- and parity-resolved level densities from the fine structure of giant resonances

Level densities of J pi=2+ and 2- states extracted from high-resolution studies of E2 and M2 giant resonances in 58Ni and 90Zr are used to test recent predictions of a possible parity dependence. The experimental results are compared to a combinatorial approach based on the Hartree-Fock-Bogoliubov model and to shell-model Monte Carlo calculations including both spin and parity projection. No parity dependence is observed experimentally, which is in agreement for 90Zr but in contrast with the model predictions for 58Ni.     

Rapidity distributions and energy densities from hydrodynamical studies at 5–200 GeV/n

3+1 dimensional relativistic calculations of the space-time evolution of heavy ion collisions at bombarding energies from 5 to 200 GeV/n are presented. Collisions with heavier projectiles seem to be more rewarding to form extended regimes of highly excited nuclear matter containing enough baryons for a sufficient time span to enable a transition of the hadron matter into a quark gluon plasma. A strong impact parameter dependence has to be taken into account when comparing the final baryon rapidity distributions with experimental results. Experimental results of the reactions¹⁶O(60, 200 GeV/n)→Pb are compared with hydrodynamical calculations.     

Spin densities in the transverse plane and generalized transversity distributions

We show how generalized quark distributions in the nucleon describe the density of polarized quarks in the impact parameter plane, both for longitudinal and transverse polarization of the quark and the nucleon. This density representation entails positivity bounds including chiral-odd distributions, which tighten the known bounds in the chiral-even sector. Using the quark equations of motion, we derive relations between the moments of chiral-odd generalized parton distributions of twist two and twist three. We exhibit the analogy between polarized quark distributions in impact parameter space and transverse momentum dependent distribution functions.Received: 25 April 2005, Published online: 27 July 2005     

Errors due to sampling in community noise level distributions

At the present time the most commonly used schemes for sampling community noise levels involve collection of one “x-minute” sample, once each hour, over a 24-hour period, where “x-minutes” is less than 60 minutes. This paper is concerned with the validity of this method of sampling. An alternative sampling scheme is presented and discussed, which reduces the probability that significant differences between the sample estimates and the time (60-minute) values will result.