Can Shell-Model Predict Cluster-Decay Q-Values?

The general belief is that all the information concerning α and heavy-cluster decay is practically included in the barrier penetration because it depends exponentially on the Q-value and on the other hand the formation probability has a smooth behaviour versus the mass number. In all calculations the Q-values are taken as experimental numbers. To our knowledge up to now no attempts were done to connect the Q-value with the microscopic cluster formation probability. In this lecture we show that this connection can be performed using the fact that any cluster-decaying state is a resonance. It is shown that, by including the continuum part of the single particle spectrum, the shell model is able to reproduce cluster-decay widths. The diagonalization of the mean field is performed using a relative small single particle basis with two harmonic oscillator parameters. The first part of the basis is connected with the bound states while the second one with the states in continuum. The microscopic cluster formation amplitude and outgoing Coulomb wave function should have the same logarithmic derivative beyond the nuclear surface, for a given energy. Therefore this condition predicts in principle cluster-decay Q-value. This is equivalent to the 'plateau condition': the independence of the cluster decay width versus the channel radius. The dependence of the asymptotic tail of single particle states on mass number along a neutron chain, necessary to fulfil the plateau condition, can be induced by the second expansion ho parameter. This is based on the observation that Q-value decreases with increasing neutron number. Therefore the second term of the expansion, connected with the states in continuum, which mainly influences the cluster formation, should be more bound. This procedure is performed without changing the standard Woods-Saxon potential and the first ho expansion parameter, connected with the spectroscopic properties. The dependence of the second ho parameter versus the mass number can be approximated with a good accuracy by a parabola for α-decays along Os, Pt, Hg neutron chains. This function is able to predict the experimental Q-values within 300 keV error. A similar conclusion we made by analysing two cold ternary fission configurations of 252 Cf, with emissions of the α and 10 Be clusters, respectively. Our calculations showed that the total decay widths fulfil the plateau condition since their values are practically independent on the matching radius within a range of about 4 fm. This behaviour corresponds to a resonant state and is obtained only for a definite value of the cluster kinetic energy. This allowed us to assign the status of long living ternary nuclear molecule to both systems. In particular, the 10 Be-molecule is about five orders of magnitude more stable than the α-molecular configuration, which in turn has a half-life three orders of magnitude longer than the characteristic nuclear time. Therefore the standard shell model describing nuclear spectroscopic properties is not able to predict cluster-decay Q-values but the enlarged shell-model, properly taking care on single particle states in continuum, is able to do this. (authors)