Giant Modes in Cold  The Nuclear Response to  Probing the nucleus at

Onset of Chaos in Warm Nuclei

Low lying nuclear states are characterised by quantum numbers, appropriate to the mean field description. Their decay modes are governed by selection rules based on these quantum numbers. A completely different situation is encountered at the rather modest excitation energy, corresponding to the neutron separation energy, of approximately 6 MeV. Random matrix theory, developed to describe the properties of these neutron resonance states, now constitutes the basis for the general concept of quantum chaos. Nuclei at excitation energies between the regular ground-state region and the chaotic neutron resonance region can be characterised as warm. The temperature in this region is rather low, and the gross properties, such as the deformation, are governed by the shell structure. The transition between order and chaos occurs in this energy region, and it is of fundamental importance to find out where it occurs in energy, as well as to investigate experimentally the observable consequences of this transition. The warm excitation energy region can be studied particularly well in deformed nuclei, where long sequences of rotational transitions proceed with only a little cooling. Already for the lowest lying rotational bands one encounters deviations from the rotating mean field description in terms of band interactions, leading to level repulsion as well as a bifurcation of the rotational strength. Statistical studies of nearest neighbour energy level distances of the lowest rotational bands, over the whole range of rare earth nuclei, have revealed the most Poisson-like distributions observed so far in nuclear physics. Recently the first small steps have been taken to obtain statistically sound information on the interactions between rotational bands above the yrast line. Systematic investigations at higher excitation energies require much more experimental information, and must await the next generation of detector systems. The two-body interaction producing the coupling between the rotational bands has only a minor influence on the rotational pattern close to the yrast line, whereas it implies drastic changes in the appearance of the rotational transitions when going up in energy, due to the rapidly increasing level density. Statistical analysis of fluctuations in the experimental $\gamma$-ray spectra of decay cascades shows that the rotational strength function for most of the states is highly fragmented. In other words, the rotational motion is damped, that is the nucleus is rotating in excited states with a distribution of rotational frequencies, in analogy with phenomena in other areas of physics when a periodic motion is influenced by thermal fluctuations. For rare earth nuclei, it has been found that the first 20 to 40 states above the yrast line form regular rotational bands, and rotational damping sets in smoothly at around 1 MeV of excitation energy, see figure .  

Figure: Perspective plot of a part of the two-dimensional energy-energy spectrum of gamma rays emitted from the rapidly rotating nucleus ¹⁶⁸Yb. Known discrete transitions have been removed. The ridges are composed of cascades along so far unresolved rotational bands. Most of the damped transitions generate a smooth background.

 

The rotational strength function contains microscopic information, for example about the alignment of angular momentum vectors and the interaction strength. An extreme situation of motional narrowing of such damping widths can in principle occur for rotational nuclei, if the intrinsic states are chaotic in nature, while at the same time the rotational strength is not fragmented. Such ergodic bands pose a challenge for future experiments. Selection rules derived from the symmetries of the nuclear field are expected to gradually lose their validity with increasing excitation energy in the warm region. However, it has been suggested that the K quantum number, associated with the axial symmetry of a deformed nuclear shape, may only be partially broken around the neutron separation energy. This is based on the observation of gamma transitions from neutron resonance states. A statistical analysis of unresolved spectra shows that it is hard to break the K quantum number in the excited bands despite the rapid rotation. States in the neutron resonance region in medium mass and heavy nuclei $S_n \cong$ 6 MeV have chaotic properties. Systematic studies of neutron resonances in nuclei approaching the neutron drip line (where S_n = 0 MeV) might provide a tool to reveal the mechanism for the onset of chaos, and the dependence on excitation energy. The recently discovered decay path out of some superdeformed bands displays a strong fragmentation into many final states in the normal deformed well. These observations support a picture in which the decay occurs via the coupling to a compound state, as illustrated in figure .

 

Figure: Decay of superdeformed states by coupling to compound states at smaller deformation by a tunnelling phenomenon.

 

Here one has the advantage of having a well defined, superdeformed, regular state at considerable excitation energy embedded in thousands of normal deformed compound states. The coupling occurs via tunnelling with very small coupling matrix elements. Measuring the distribution of the decay strength provides a unique opportunity to study both the chaotic nature of the excited normal deformed states and the tunnelling process. Comparing superdeformed states in different mass regions one finds a considerable variation in the excitation energy, the angular momentum, and the barrier height as well as the structure content relative to the normal deformed states. This presents the prospect of investigating chaos-assisted tunnelling as well as the chaotic nature of the normal deformed states under different conditions. Both fission and the high spin cascades provide examples of dynamical processes, where the coupling between quantum chaos and dissipation may be investigated.