Chaos in axially symmetric potentials with octupole deformation.

Classical and quantum mechanical results are reported for the single particle motion in a harmonic oscillator potential which is characterized by a quadrupole deformation and an additional octupole deformation. The chaotic character of the motion is strongly dependent on the quadrupole deformation in that for a prolate deformation virtually no chaos is discernible while for the oblate case the motion shows strong chaos when the octupole term is turned on. PACS Nos.: 21.10.-k, 05.45.+b on leave of absence from Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, 141980 Dubna, Russia The mean field approach is of central importance in any theoretical description of a many body system. In nuclear physics it is the basis for the shell model and its extensions such as collective states. Its success has been further demonstrated in the application to deformed nuclei and to metallic clusters where spherical symmetry is given up because of experimental evidence. Usually, quadrupole deformation is considered to be the major deviation from spherical symmetry. However, more recently a possible octupole contribution has been taken into account for a number of reasons [1, 2]. Inclusion of an octupole term in addition to a quadrupole term renders the classical single particle motion nonintegrable. In fact, the system turns out to be chaotic. This has been discussed by a number of authors with various degrees of simplification [3]. A nicely systematic approach is given in [4], where the motion in a quadrupole deformed cavity is analyzed and the terms that give rise to actual chaotic behaviour are clearly distinguished. In a recent investigation [5] the study of classical motion in a cavity with oscillating walls of even and odd higher order multipoles has led to interesting conclusions about elastic versus dissipative behaviour of a noninteracting gas depending on the integrability or nonintegrability of the equations of motion. Surfaces of section are used to discern the onset and degree of chaotic motion. This paper is similar in spirit in that we investigate a simplified model where we leave out terms, which, albeit physically important, are prone to blur the analysis when the interest is focussed on the essentials that give rise to chaotic behaviour. Since our aim is directed not only to the classical but also to the corresponding quantum mechanical motion, we leave out the spin-orbit term and the → l 2 -term pr Nilsson model to render as closely as possible the analogy between the classical and quantum case. The importance of the → l 2 -term is well known in nuclear physics and for metallic clusters[6]. To determine its role in the context of chaotic motion the corresponding classical case ought to be studied. We defer treatment of this term. Despite the simplifications, such a model allows to understand the main features of shell structure effects, for example, in super– (hyper) deformed nuclei[7, 8]. Recent experimental data of superdeformed K-isomers in nuclei [9] and electronic shell structure effects in metallic clusters [6] clearly underline the importance of oblate deformation. Therefore we investigate in the present paper the effect of the octupole term for prolate and oblate deformation; the oblate/octupole case has not been dealt with in ref.[5]. We analyse the case of zero temperature which is good for nuclei. For metallic clusters finite temperature should be considered[10]; however, our interest is focussed on shell structure whose character is unaffected by temperature except for the amplitude. Only axially symmetrical terms are taken into account which brings down to two the number of degrees of freedom of the motion.