Fluctuation analysis of rotational spectra

Abstract

It is well established that nucleons in the atomic nucleus can organize their motion, leading to quadrupole deformed shapes of the average field and to rotations of the nucleus as a whole. At an excitation energy of few MeV above the yrast line, rotational bands become very closely spaced in energy. Any single band can be viewed as a collective sequence of related states embedded in a dense background of other (more or less complicated) states, to which it will couple by residual interactions. This coupling leads to stationary states of the system (the compound nucleus) which are complicated mixtures of unperturbed configurations. The rotational degree of freedom is “damped” in these compound states in the sense that the electric quadrupole decay of a single quantum state with angular momentum I will not go to a unique final state with spin I − 2 (as for the unperturbed bands) but will exhibit a spectrum of final states all having spin I − 2. In other words, for each compound nucleus formed in an experiment, the cascade of ∼20 γ-rays, which eventually will cool the system, will find many transitions through which to proceed in the regions where bands mix strongly (damped region) and only few in the region of discrete bands. In actual experiments, the cascade of γ-rays associated with each of the members of the ensemble of compound nuclei will use each of the “discrete” transitions many more times than the “continuum” transitions. Relatively large and small fluctuations in the recorded coincidence spectrum will ensue. respectively. The analysis of the fluctuations will be shown to be instrumental to gain insight into the phenomenon of rotational damping. For this purpose, two- and higher-fold coincidence spectra emitted from rotating nuclei are analyzed with respect to the count fluctuations. The coincidences from consecutive γ-rays emitted from discrete rotational bands generate ridges in the E γ1 · E γ2 spectrum, and the fluctuation analysis of the ridges is based upon the ansatz of a random selection of transition energies from band to band. This ansatz is supported by a cranked mean-field calculation for the nucleus 168Yb, as well as by analyzing resolved bands in 168Yb and its neighbors. Consecutive γ-rays emitted from the region of rotational damping spread out more in the E γ1 · E γ2 plane than those associated with transitions between members of discrete rotational bands, and are studied most clearly in the central valley ( E γ1 = E γ2 ). The fluctuation analysis of the valley is based upon the ansatz of fluctuations in the intensity of the transitions of Porter-Thomas type superposed on a smooth spectrum of transition energies. This ansatz is again supported by a mixed-band calculation. The mathematical treatment of count fluctuations is formulated in general terms, and the connection to earlier treatments of one-fold spectra of high level density is established. The statistical assumptions underlying the fluctuation analysis imply the existence of a principal uncertainty, which is examined in detail. In the experimental section, the fluctuation analysis is applied to two-dimensional γ-spectra, the only available data at present with sufficient intensity to warrant a meaningful analysis. Large fluctuations are observed in the ridge structures from the four cases analyzed, showing that only a rather low number (≈ 30) of discrete rotational bands exist. In contrast, only weak fluctuations are found along the central valley, revealing that the spectrum in the valley effectively contains different (of the order of 10 5) coincidence combinations. This number is considerably larger than what is found assuming that the rotational decay leads to a unique final state, showing that the transition strength through each decay step is spread over many states within a given energy interval, the damping width, and thus providing fairly direct evidence of the rotational damping picture.