M. Matsuo

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 γ 1 · E γ 2 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 168 Yb, as well as by analyzing resolved bands in 168 Yb and its neighbors. Consecutive γ-rays emitted from the region of rotational damping spread out more in the E γ 1 · E γ 2 plane than those associated with transitions between members of discrete rotational bands, and are studied most clearly in the central valley ( E γ 1 = E γ 2 ). 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.

Shell model for warm rotating nuclei

Abstract In order to provide a microscopic description of levels and E2 transitions in rapidly rotating nuclei with internal excitation energy up to a few MeV, use is made of a shell model which combines the cranked Nilsson mean-field and the residual surface delta two-body force. The damping of collective rotational motion is investigated in the case of a typical rare-earth nucleus, namely 168Yb. It is found that rotational damping sets in at around 0.8 MeV above the yrast line, and the number of levels which form rotational band structures is thus limited. We predict at a given rotational frequency the existence of about 30 rotational bands of various lengths, in overall agreement with the experimental findings. The onset of the rotational damping proceeds quite gradually as a function of the internal excitation energy. The transition region extends up to around 2 MeV above yrast and it is characterized by the presence of scars of discrete rotational bands which extend over few spin values and stand out among the damped transitions, and by a two-component profile in the Eγ-Eγ correlation. The important role played by the high-multipole components of the two-body residual interaction is emphasized.

THE DISTRIBUTION OF THE ROTATIONAL TRANSITION STRENGTH IN WARM NUCLEI STUDIED THROUGH GAMMA-RAY CORRELATIONS

Abstract The study of damping of rotational motion applying te rotational plane mapping (RPM) method is presented and discussed. The aim of this technique is to extract the distribution of the rotational transition strength from an analysis of the shape of the “central valley” of two- and three-dimensional γ-ray spectra. The method is applied to a triple γ-coincidence data set of 162,163Tm nuclei formed in 37Cl+130Te reactions. The rotational transition strength is obtained as a function of rotational frequency for selected regions of entry states, and the width is found to be rather constant and approximately equal to 80 keV. This value is significantly smaller than the value predicted theoretically for the rotational damping width Γrot. Also the ratio between the observed depth and width of both the 2D and 3D valleys does not agree with the simple model adopted in the RPM method. These discrepancies point to the presence of both a wide and a narrow component in the distribution of rotational strength as extracted by the RPM method. The analysis of simulated spectra obtained on the basis of realistic band-mixing calculations, including residual interactions, confirms this behaviour.

Recent studies of the high-spin quasi-continuum

Abstract Two new methods, the Rotational Plane Mapping, and the Fluctuation Analysis Method are reviewed, and applied to the study of high spin quasi-continuum states in 167,168 Yb and 163 Tm nuclei. New results are given for the rotational damping width Г rot and the number of different 2-step-paths below and above U 0 , defined to be the energy of the borderline between undamped regular rotational motion for bands along the yrast line and the region of heated nuclei with damped rotational motion. The results indicate that at least 30 bands with regular rotational structure exist in the I = 40 region before damping becomes dominating. The fundamental concept of rotational damping may be questioned, and has never been verified experimentally. The analysis support the damping picture, giving evidence for a strong branching of the E2 decay, showing 10 times as many 2-step-paths through the continuum than expected if only regular rotational E2 decay, and E1 statistical branching were the decay modes.

Level statistics of near-yrast states in rapidly rotating nuclei

Abstract The nearest neighbour level spacing distribution and the Δ 3 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.

A finite number of regular rotational bands in the superdeformed well of 143Eu

Abstract The number of excited superdeformed bands in 143 Eu is measured by use of the Fluctuation Analysis Method. Between 10 and 40 rotational bands, displaying typical rotational energy correlations over two consecutive transitions, are populated within a rather narrow range in transition energy, E γ ≈ 1300–1500 keV. These numbers are close to the values found for normally deformed nuclei and agree with microscopic cranking + band mixing calculations for the specific superdeformed nucleus, which predict the onset of rotational damping to occur at the excitation energy U 0 = 1.3–1.6 MeV above the yrast line.

Onset of rotational damping in superdeformed nuclei

Abstract We discuss damping of the collective rotational motion in A ∼ 150 superdeformed nuclei by means of a shell-model combining the cranked Nilsson mean field and the surface and volume delta two-body residual forces. It is shown that, because of the shell structure associated with the superdeformed mean field, onset energy of the rotational damping becomes E x ∼ 1.5–3 MeV above the yrast line, with significant variation for different neutron and proton numbers. The mechanism of the shell structure effect is investigated through detailed analysis of level densities in superdeformed nuclei. The variation in onset of damping is associated with variation in the single-particle structure at the Fermi surface.

Shell effects on rotational damping in superdeformed nuclei

Damping of rotational motion in superdeformed Hg and Dy-region nuclei is studied by means of cranked shell model diagonalization. It is shown that a shell oscillation in single-particle alignments affects significantly properties of rotational damping. Onset properties of damping and damping width for Hg are quite different from those for Dy-region superdeformed nuclei.Abstract Damping of collective rotational motion in A ∼ 190 and A ∼ 150 superdeformed nuclei is studied by means of the cranked shell model with two-body residual force. Numerical calculations predict that in a typical A ∼ 190 superdeformed nucleus, 192 Hg, the rotational damping width is significantly small, Γ rot ∼ 30 keV, and that the number of superdeformed bands in the offyrast region amounts up to 150 at a given rotational frequency. These features are quite different from the prediction for A ∼ 150 superdeformed nuclei and rare-earth normally deformed nuclei. It is shown that the single-particle alignments of the cranked Nilsson orbits have strong shell oscillation. It affects significantly the properties of rotational damping in superdeformed 192 Hg.

Covariance analysis of selection rules governing the γ−decay cascades of the rotational nucleus 164Yb

The γ-cascades feeding specific intrinsic configurations of 164Yb are studied making use of a statistical analysis of counts fluctuations in γ–γ coincident spectra. In particular, variance and covariance spectra are obtained starting from the measured (0,+), (0,−) and (1,−) gate-selected matrices. The analysis of the first ridge of the spectra yields about 25 excited bands carrying rotational energy correlations for the total spectrum, and about 10 in coincidence with each of the (α,π) configurations. The covariance of count fluctuations between the different (α,π) gates is expressed by an experimental correlation coefficient, r, which measures the degree of sharing of decay paths through the excited bands, and thereby the extent to which selection rules are obeyed by the decay. The correlation coefficient is around r≈0.3 for the ridge, and also in general rather small for valley fluctuations. Only at the highest valley energies, a tendency to larger correlation coefficients is seen, indicating some weakening of the selection rules over longer decay cascades. The results of the experimental analysis are compared to rotational damping model predictions based on cranked shell model plus residual interactions, and rather good agreement is found.

The rotational γ-continuum in the mass region A≈110

Abstract Unresolved γ transitions of 114 Te and of 112 Sn sorted into one-dimensional and two-dimensional spectra have been studied. The reaction 64 Ni + 54 Cr at bombarding energies 230, 240, 250, 260, 270 MeV was used and the γ -rays were detected with the EUROBALL array. In the case of the nucleus 114 Te the values of the multiplicity as a function of bombarding energy and of the moment of inertia were obtained. The effective moment of inertia was found to be almost constant in the interval I=20 – 40 ℏ , in contrast to the decreasing behaviour of the dynamic moment of inertia for the terminating yrast band. The ridge valley structures in E γ 1 ×E γ 2 spectra of 114 Te and of 112 Sn were analysed with the fluctuation analysis technique. The analysis of the two nuclei are compared to simulations based on microscopic cranking calculations with residual interactions included. A rather good agreement is found between data and predictions.