From Regular to Chaotic States in Atomic Nuclei

An interesting aspect of nuclear dynamics is the co–existence, in atomic nuclei, of regular and chaotic states [1]. This is a vast subject and, for reasons of space, in this paper we limit ourselves to a few examples only. In order to highlight the difference between the regular and chaotic states, it is perhaps useful to remember that the low energy states (0 ∼ 4 MeV above the ground state), termed regular, are described by a variety of models: the shell–model, the collective model and their various extensions [2–8]. By means of these models, all the properties of nuclear levels such as excitation energies, transition probabilities, magnetic and quadrupole momenta, etc. may be accurately calculated. When the excitation energy increases, the level density rises, making the calculation of nuclear properties in terms of single levels both of no physical interest and impossible. Instead, a statistical description, called Statistical Nuclear Spectroscopy (SNS), is used, based on the division into global and local nuclear properties [9–17]. A typical example of this is the separation of the level density into a global component, the secular variation, and a local component, the fluctuations, which are well described by the random matrix ensembles [18–21]. In recent years, the study of quantum levels in classically chaotic regions has shown that they have the same fluctuation properties as those predicted by random matrix ensembles [28–30] in a large energy range; we term these states chaotic. The physical significance of statistical concepts in atomic nuclei is therefore understood through the link with chaotic motion in hamil-2 tonian dynamics [22–25]. In the first part of the present work, we review the state of the art of nuclear dynamics and use a schematic shell model to show how a very simple and schematic nucleon–nucleon interaction can produce an order→chaos transition. The second part is devoted to a discussion of the wave function behaviour and decay of chaotic states using some simple models. 2. Regular states in atomic nuclei: shell and collective models As is well known, the regular states of atomic nuclei are described, roughly speaking, by the mean field approximation of the shell model and by the oscillations about the mean field which give rise to collective excitations. Nuclear potential is essentially symmetric; nucleons move in regular orbits and, besides the energy, there are other constants of motion and other quantum numbers [6,7]. For the sake of completeness we …