0 Molecular Collective Vibrations in the Ternary Neutronless Fission of 252 Cf Ş

Based on a recent experimental finding which may suggest the existence of a tri-nuclear molecular structure before the cold ternary fragmentation of 252Cf takes place, we solved the eigenvalue problem of a certain class of vibrations which are very likely to occur in these molecules. These oscillations are the result of the joined action of rotations of the heavier fragments and the transversal vibrations of the lighter spherical cluster with respect to the fission axis. In the calculation of the interaction between the heavier fragments we took into account higher multipole deformations, including the hexadecupole one, and introduced a repulsive nuclear part to insure the creation of a potential pocket in which a few molecular states can be accommodated. The possibility to observe the de-excitation of such states is discussed in connection with the molecular life-time. PACS number : 21.60.Gx,24.75.+i,25.85.Ca; keywords : giant molecule, cold fission, collective motion, heavy-ion interaction [email protected] 1 The scope of this letter is to extend our recent investigations on the molecular configurations in the binary cold fission [1,2] and the ternary cold fission [3]. Growing interest aroused in the last year due to the experimental indication of a long-living (≥ 10s) structure in the Be accompanied ternary cold fission [4]. Very recently a molecular structure in which C plays the role of the light accompanying particle has been reported [5]. In this letter we make the suggestion that the ternary cold fission of Cf is a process consisting of two main stages : in the preformation stage a quasi-bound molecular structure is formed in which the heavier fragments are almost co-linear and the light particle (e.g. α, Li ,Be and C) which is responsible for the molecular bonding, is orbiting in the equatorial region. This is similar to the case encountered in Molecular Physics, where in a linear or nonlinear chain of three atoms, the central atom ensures two bondings with the eccentric atoms [6]. In the second stage the quasi-molecular state is decaying [7]. Our aim is to study the collective vibrations of the system before the decay takes place. Recently it has been advocated by us [1], based on the concept of nuclear molecule [8], that for fragments emitted in the binary cold fission, with almost no excitation energy, a collective vibrational spectrum will show up as a consequence of small non-axial fluctuations at scission. Such a molecular spectrum can be achieved if the interplay between the Coulomb and the repulsive nuclear core on one hand and the attractive nuclear part on the other hand will produce a pocket in the interaction potential between the fragments [9]. In the case of di-nuclear systems it was shown that possible molecular collective modes can be associated to the elongation variable and rotational vibrations taking place perpendicularly to the fission axis [10]. The last type of modes, e.g. butterfly (bending) and anti-butterfly (wriggling) is also believed to be responsible for the formation of angular momenta in fragments emerging in binary spontaneous fission [2,11,12]. In a previous paper the classical expression of the tri-nuclear Hamiltonian has been worked out for the case of the Sr+Be+Ba molecule in terms of the Jacobi variables R, ξ and the angular velocities ω of the molecular frame [3]. The equilibrium configuration was that of three aligned clusters, with the lighter in-between. In such a configuration the 2 interaction between the heavy fragments is almost entirely given by the Coulomb term. However the interaction between the lighter fragment and the heavy fragments consists also of a noticeable nuclear component, which in fact is responsible for the nuclear bond. Like in the case of binary molecules, butterfly modes can occur, in which the fragments rotates in phase while the lighter fragment is approximately preserving its pole-pole configuration with the heavier fragments. The classical expression of the kinetic energy of the three-body system, after removing the center of mass contribution, is expressed as a sum of translational and rotational degrees of freedom : T = 1 2 μ12Ṙ 2 + 1 2 μ(12)3ξ̇ 2 + 1 2 ω1J 1ω1 + 1 2 ω2J 2ω2 + 1 2 ω3J 3ω3 (1) The first term describes the relative motion of the di-nuclear sub-system (12) with reduced mass μ12 = m1m2/(m1 + m2), whereas the second one corresponds to the relative motion of the third cluster with respect to the heavier fragments center-of-mass with reduced mass μ(12)3 = (m1 +m2)m3/(m1 +m2 +m3). The vectors ω1,2,3 denote the angular velocities of the rotational motion of the three clusters, referred to the laboratory frame, ω being the transpose of ω. In this paper we consider a spherical light cluster and thence the last term in eq.(1) disappears. The inertia tensors J i are defined in the intrinsic frame such that the only non-vanishing components are the first two diagonal terms, (J i)11 = (J i)22 ≡ Ji, the quantum rotation around the symmetry axis of any of the two heavier fragments being discarded. In what follows we are interested in studying the collective spectrum which develops upon constraining the tri-nuclear molecule to perform an oscilllatory motion similar to the valence angle bending in molecular physics and the butterfly(bending) modes in di-nuclear molecules, i.e. to perform small displacements from the equilibrium position which result in the decrease of the angle between the two valence bonds, Φ = π − φ1 − φ2, attached to the spherical light fragment 3 (φi is the angle between the axis joining the two heavier fragments and the line joining the heavy fragment i with the light cluster). In the same time, since the 3 nuclear proximity forces have the tendency to keep constant the reciprocal distances and orientations of the heavy fragments with the light one, we exclude possible bond stretching vibrations. If the bond stretching is absent, then there will be a corresponding decrease in the distance between the heavy nuclei 1 and 2, when the bending angles φ1 and φ2 are increasing. The quantitative translation of the above mentioned considerations provides us with a set of constraints between the variables of interest in this problem : R-the distance between the centers of the two heavier fragments, ξ-the distance between the light cluster 3 and the center-of-mass of the heavy fragments ensemble, and the small bending angles φ1, φ2. These last two variables are related between them, due to the assumption on the constancy of the pole-pole configuration between the light cluster 3 and the heavy fragments 1 and 2 φ2 = R1 +R3 R2 +R3 ε (2) where ε = φ1. Consequently we obtain the following relations, which allows us to eliminate from the kinetic energy (1) the variables R and ξ in favor of ε R = (R1 +R2 + 2R3) ( 1− 1 2 R1 +R3 R2 +R3 ε ) (3) ξ = ξ0 + 1 2 ( ∂ξ ∂ε2 ) ε=0 ε (4) where ξ0 ≡ ξ(0) = A1(R1 +R3)−A2(R2 + R3) A1 + A2 (5)