IN THE CLUSTER RADIOACTIVITY

We investigate the influence of the deformation on the decay rates of the cluster emission process 224Ra + 2'0Pb + ''C. The interaction between the daughter and the cluster is given by a double folding potential, containing a nilclear repulsive core, with account of the quadrupole and hexadecupole deformed densities of both fragments. Upon comparison with the experimental value of the decay rate, the results obtained in this paper point out the importance of such deformations especially for the daughter nucleus.


I. INTRODUCTION
The theoretical study of the heavy-cluster emission and the super-asymmetric fission started at the end of seventies in Dubna by the romanian physicist A.SPndulescu and its collaborators [l]. Since the beginning this phenomenon was recognized to be a consequence of the shell closure of one or both fragments because of its cold nature, i.e. the low excitation energy involved in the process. Later on Rose and Jones confirmed experimentally the existence of this new phenomenon [2]. Since then many theoretical and experimental studies have been carried out (for a review see [3]). Recently it was advocated that the cluster radioactivity is not an isolated phenomenon, and must be related to other processes like the cold fusion or cold fission [4], where the closed shell effects play a dominant role. A still opened problem in the study of the cluster radioactivity is represented by the question of the existence of only the spherical or both the spherical and the deformed closed shells. Although both daughter and emitted cluster have in many cases, at least for even-even nuclei, a spherical shape in the ground state, according to the liquid drop model [5], nothing prevents us, in the cluster radioactivity process, to deal also with deformed shapes.
Until now there are no experimental data available for deformed daughters. The first theoretical study of the cluster deformation effects on the WKB penetrabilities have been carried out by SPndulescu et al. [6] using the double folded Michigan-3 Yukawa (M3Y).
In this paper we extend the study of the deformation effects in cluster radioactivity by accounting also for the deformation of the daughter nucleus and including higher multipole deformations, like the hexadecupole one. The interaction between the daughter nucleus and the cluster, in the region of small overlap and throughout the barrier is computed by means of a double folding potential. The nuclear part includes a repulsive core at small distances. In this way our deformed cluster approach 1 supposes a cluster already formed in the potential pocket coming from the interplay between the Coulomb and the repulsive nuclear core on one hand and the attractive nuclear force on the other hand. The depth and the wideness of this pocket will determine the assault frequency of the cluster on the barrier, through which it will eventually tunelate. In its turn, the penetrability will depend on the height and wideness of the barrier. Since all these geometrical characteristics depend sensitively on the shape of the fragments we will investigate in this paper the modification induced by the quadrupole and hexadecupole deformations of the fragments on the pocket and the barrier and finally compute decay rates for the disintegration reaction

CLUSTER-DAUGHTER DOUBLE-FOLDING POTENTIAL
The nuclear interaction between the daughter and the cluster can be calculated as the double folding integral of ground state one-body densities ~~(~) ( t ) of heavy ions as follows where v is the N N effective interaction and the separation distance between two interacting nucleons is denoted by s = r1 + R -T Z and R is the centre-to-centre distance. In the past a G-matrix M3Y effective interaction was used to discuss light and heavy cluster radioactivity. This interaction contains isoscalar and isovector

CALCULUS OF DECAY CONSTANTS
We adopt a modified Gammow approach (31 which is based on the idea that the cluster is pre-born, with a certain probability Po, in the pocket of the Migdal+Coulomb potential and later on it tunnels through an essentially onedimensional barrier. Consequently the decay rate X will be defined as follows: where vo is the assault frequency with which the cluster bombards the walls of the potential pocket. It is given by the inverse of the classical period of motion where is the reduced mass of the cluster-daughter pair and rll and r t 2 are the inner turning points, where the potential curve intersects the &-value (see Fig.1). Thus, in our model, vo depends sensitively on the size of the potential pocket. The barrier penetrability is given by the well known WKB formula 4 where rt3 is the outer turning point.
The calculus of the preformation probability Po is usually based on elaborated microscopic'models. Since its calculation is beyond the purpose of this material, we In what follows we consider the 14C-decay of 224Ra.
In figure Fig.1  The same is done in the next figure but for cluster deformations (see Fig.6). Here we compare the dependencies of the decay constant on @ and p:. The difference in slope between the two curves is much evident than in the precedent figure. This fact is easily understood by recalling the observations made earlier (see figures 1 and a), on the modification of the barrier due to the quadrupole and hexadecupole deformations.
The hexadecupole deformation of the cluster increases the pocket depth (see Fig.1) and consequently uo will increase too, while the increase of hexadecupole deformation of the cluster is accompanied by the rise of the bottom of the potential pocket and the lowering of the barrier height is partly compensated by the diminution of vo.
In table I we selected some of the most favorable cases for our calculated decay 6 rates. From hew w e infer the importance of daughter's deformation. The value corresponding to case 1 is obtained when considering both nuclei in their ground state deformations. As we expected, prolate deformations (see cases 7, 11-18) favor the decay. Deformations around 0.04 either in /j2 or in p4 of the daughter nucleus give us decay rates close to tlie experimental one. One can notice that acceptable values of A are reached more convenient through deformations of the daughter nucleus, than through deformations of the cluster (see cases 15-18). Case 5 shows that even an oblate shape for the emitted cluster can be taken into discussion. If we remind that the mother nucleus 224Ra has a prolate deformed ground state & = 0.17, one might suppose that such a pictureprolate daughter and an oblate clusteris intuitively acceptable.

IV. CONCLUSIONS
The aim of this paper was to extend previous studies of deformation effects in cluster radioactivity by considering also the deformation of the daughter nucleus and to include the next higher even deformation, the hexadecupole one. Considering thai the cluster is pre-born in the potential pocket produced by the interplay between repulsive and attractive forces we investigated the modifications induced by deformations on the specific potential that we employ. The computed decay rates depends 011 the assault frequency, which varies with the pocket depth, and on the penetrability, which changes with the barrier height. We showed that the experimental values can be reproduced for several selections of the deformations. If we maintain the cluster spherical and vary the quadrupole and/or hexadecupole deformations of the daughter nucleus we may reach the experimental value within a reasonable range of deformations parameters. Another interesting result is that even for an oblate deformation of the cluster we may obtain decay rates close to the experimental value.  A deformed state of the daughter is needed in order to reach the experimental X within reasonable values of cluster deformation. The daughter nucleus is taken in its ground state deformation.