Chaos and Regularity in the Doubly Magic NucleusPb208

Abstract

High resolution experiments have recently lead to a complete identification (energy, spin, and parity) of 151 nuclear levels up to an excitation energy of Ex = 6.20 MeV in 208 Pb [Heusler et al., Phys. Rev. C 93, 054321 (2016)]. We present a thorough study of the fluctuation properties in the energy spectra of the unprecedented set of nuclear bound states. In a first approach we grouped states with the same spin and parity into 14 subspectra, analyzed standard statistical measures for short-and long-range correlations, i.e., the nearest-neighbor spacing distribution, the number variance Σ 2 , the Dyson-Mehta ∆3 statistics, and the novel distribution of the ratios of consecutive spacings of adjacent energy levels in each energy sequence and then computed their ensemble average. Their comparison with a random matrix ensemble which interpolates between Poisson statistics expected for regular systems and the Gaussian Orthogonal Ensemble (GOE) predicted for chaotic systems shows that the data are well described by the GOE. In a second approach, following an idea of Rosenzweig and Porter [Phys. Rev. 120, 1698Rev. 120, (1960] we considered the complete spectrum composed of the independent subspectra. We analyzed their fluctuation properties using the method of Bayesian inference involving a quantitative measure, called the chaoticity parameter f , which also interpolates between Poisson (f = 0) and GOE statistics (f = 1). It turns out to be f ≈ 0.9. This is so far the closest agreement with GOE observed in spectra of bound states in a nucleus. The same analysis has also been performed with spectra computed on the basis of shell model calculations with different interactions (SDI, KB, M3Y). While the simple SDI exhibits features typical for nuclear many-body systems with regular dynamics, the other, more realistic interactions yield chaoticity parameters f close to the experimental values.

Spectral analysis of molecular resonances in erbium isotopes: Are they close to semi-Poisson?

“…The NNSD in this situation, with some approximations,is given in terms of a single parameter f [43,44],…”

Structure of negative parity states at $${\mathbf {6}\,\lesssim \, \mathbf {E}_\mathbf {x} <{\mathbf {7.0}} }$$ MeV in $$\mathbf {^{208}}{} \mathbf{Pb}$$

“…The distance between two states with the same spin and parity is larger than 30 keV. It is explained by the chaoticity [62][63][64]. The state contributing dominantly to a certain level in the MPIK data could be identified by the high resolution Q3D data (Table 15).…”

“…(2)] for the 2 − states at E x > 6.0 MeV (Table 7). The changed mean spacing between the 2 − states would affect the chaoticity only little [62][63][64].…”

“…The rather firm completeness of the sequence of states for most spins at E x < 6.2 MeV allowed to study the chaoticity among bound states in a heavy nucleus [62][63][64]. The rather complete orthonormal transformation matrices from one-particle one-hole configurations to states [34,49,50] for Table 1 Resonance energy E res and total width Γ tot for IARs L J in 209 Bi…”

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“…4, they are used for the statistical analysis of the obtained experimentally [34 -39] collective-excitation spectra, in contrast to the s.p. spectra considered in [2,7] and [30] in Sec. 4.…”