Work in Progress
We initiated systematic self-consistent calculations of the heavy and superheavy nuclei using modern nuclear energy density functionals. The calculations, carried out by Dr. Staszczak, are based on the Hartree-Fock+BCS code HFODD that solves the self-consistent HF equations by using a Cartesian (3D) harmonic oscillator finite basis.
We have studied the stability of our results with respect to the number of deformed harmonic oscillator states used in this basis. Figure 1 shows the calculated self-consistent potential energy curve (PES) as a function of the axial mass quadrupole moment Q20 for 240Pu. The solid line corresponds to Nosc=1000 stretched harmonic oscillator states included in the basis. The results with different values of Nosc=680 (14 spherical oscillator shells), 816 (15 shells), 969 (16), 1140 (17), 1330 (18), 1540 (19), and 1771 (20) are also shown at the extreme points (minima and maxima) of the PES. It is seen that the convergence weakly depends on the quadrupole deformation, i.e., the larger the elongation, the weaker the convergence. As shown in Fig. 1, reliable calculations can be carried out with Nosc=17, and the basis-stability error on the first and second barrier is less than 1 MeV.
Figure 1: Stability of the calculated HFB+BCS energy for 240Pu as a function of the harmonic oscillator basis size.
Having determined the basis size, we performed a set of calculations for the series of even-even fermium isotopes which are experimentally known to exhibit rapid variations of the spontaneous fission lifetimes and, in some cases, bimodal fission. In our calculations, the quadrupole and octupole mass moments were used as constraints. Two fission paths corresponding to bimodal fission have been investigated. The 'usual' reflection asymmetric path corresponds to two fission fragments with different mass. The reflection-symmetric path can be associated with division into symmetric, nearly spherical fragments with high kinetic energies. As seen in Fig. 2, the energy barriers calculated for 254-264Fm have fairly similar shapes, with a characteristic two-humped structure. Non-axial deformations are fairly small and they are present only in the region of the inner barrier. For the lighter isotopes, the reflection-asymmetric path is favored energetically. The situation changes when approaching 264Fm, where the fission paths become symmetric. Such a transition has previously been predicted in macroscopic-microscopic models, and in self-consistent calculations with the Gogny interaction.
Figure 2: Self-consistent HF+BCS potential energy surfaces for even-even fermium isotopes 254-264Fm calculated in this work.
Figure 3 shows the energy surface of the transuranic element 258Fm calculated within the self-consistent nuclear density functional theory with the SkM* functional as a function of two collective variables: the total quadrupole moment Q20 representing the elongation of nuclear shape, and the total octupole moment Q30 representing the left-right shape asymmetry. Indicated are the two static fission valleys: (i) asymmetric path aEF leading to asymmetric mass split of fission fragments and (ii) symmetric-compact path sCF corresponding to a division into nearly spherical fragments. Experimentally, a transition is observed from an asymmetric distribution of mass splits in neutron-deficient fermium isotopes to a more symmetric distribution when getting closer to 264Fm. The density functional theory calculations explain this phenomenon in terms of shell effects in the emerging fission fragments approaching the doubly-magic 132Sn nuclei. In calculations, all possible nuclear shapes, including triaxial and reflection-asymmetric (pear-like) shapes, are allowed. A detailed discussion of calculated static fission paths for the heaviest and superheavy nuclei is currently being prepared for publication in Nature.
Figure 3: Potential energy surface of 258Fm calculated in the symmetry-unconstrained HFB theory. Two fission valleys are indicated.
Recently, we have studied the influence of shape elongation on various contributions to the total energy of the liquid drop model (LDM) with parameters extracted from microscopic interactions (Phys. Rev. C 73, 014309 (2006)). While in the spherical case the surface-symmetry term is very poorly determined for nuclei close to the beta-stability valley, due to the presence of the large volume term, this is not the case for deformed configurations in unstable nuclei. We demonstrated that a large part of deformation energy in the neutron rich nuclei comes from the surface symmetry term (see Fig. 4 for a representative example). Currently, we are performing systematic multi-processor calculations using HFODD for shape isomers, superdeformed states, and fission barriers using the commonly used functionals. We want to determine whether the systematic dependence of the surface-symmetry term seen in the microscopic LDM will be reflected in self-consistent DFT results. This work will constitute the major part of a dissertation of Mr. Nikolov.
