Asymmetric Two-Center Oscillator Model for Fission

This Hamiltonian is axially symmetric along the z axis, p denotes the cylindrical coordinate perpendicular to the symmetry axis [71]. The shapes described by this Hamiltonian are those of two semispheroids (either prolate or oblate) connected by a smooth neck (which is specified by the term Vneck( ))- 0 and Z2 0 are the centers of these semispheroids. For the smooth neck, the following 4th-order expression [70] was adopted, namely 

The frequency copi in Eq. (36) must be z-dependent in order to interpolate smoothly between the values of the lateral frequencies associated with the left (i = 1) and right (/ = 2) semi spheroids, which are not equal in asymmetric cases. The frequencies (/ = 1, 2) characterize the latteral harmonic potentials associated with the two semispheroids outside the neck region. In the implementation of such an interpolation, we follow Ref. [70]. 

The cluster shapes associated with the spatial-coordinate-dependent single-particle potential V(p, z) in the Hamiltonian (36) (i.e., the second, third and fourth terms) are determined by the assumption that the cluster surface coincides with an equipotential surface of value Vo. namely, from the relation V(p, z) = Vq. Subsequently, one solves for p and derives the cluster shape p = p(z). For the proper value of Vq, we take the one associated with a spherical shape containing the same number of atoms, Ny as the parent cluster, namely, Vo = mcColR, where Ticoq = 49r N eV, R = and rs is the 

Wigner-Seitz radius in atomic units (monovalent metals have been assumed). Volume conservation is implemented by requiring that the volume enclosed by the fissioning cluster surface (even after separation) remains equal to 47tR /3. 

The cluster shape in this parameterization is specified by four independent parameters. We take them to be the separation d = Z2 — Z of the semispheroids, the asymmetry ratio = o)°p2l(o p y and the deformation ratios for the left (1) and right (2) semispheroids qi = co i/coli (/ = 1, 2). 

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