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Register a new userState Dependent Effective Interaction for the Hyperspherical Formalism
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Nir Barnea n ECT* Trento
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The method of effective interaction, traditionally used in the framework of an harmonic oscillator basis, is applied to the hyperspherical formalism of few-body nuclei (A=3-6). The separation of the hyperradial part leads to a state dependent effective potential. Undesirable features of the harmonic oscillator approach associated with the introduction of a spurious confining potential are avoided. It is shown that with the present method one obtains an enormous improvement of the convergence of the hyperspherical harmonics series in calculating ground state properties, excitation energies and transitions to continuum states.
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A different formulation of the effective interaction hyperspherical harmonics (EIHH) method, suitable for non-local potentials, is presented. The EIHH method for local interactions is first shortly reviewed to point out the problems of an extension to non-local potentials. A viable solution is proposed and, as an application, results on the ground-state properties of 4- and 6-nucleon systems are presented. One finds a substantial acceleration in the convergence rate of the hyperspherical harmonics series. Perspectives for an application to scattering cross sections, via the Lorentz transform method are discussed.
The Pauli rearrangement potential given by the second-order diagram is evaluated for a nucleon optical model potential (OMP) with $G$ matrices of the nucleon-nucleon interaction in chiral effective field theory. The results obtained in nuclear matter are applied for $^{40}$Ca in a local-density approximation. The repulsive effect is of the order of 5MeV at the normal density. The density dependence indicates that the real part of the microscopic OMP becomes shallower in a central region, but is barely affected in a surface area. This improves the overall resemblance of the microscopic OMP to the empirical one.
The effective-interaction theory has been one of the useful and practical methods for solving nuclear many-body problems based on the shell model. Various approaches have been proposed which are constructed in terms of the so-called $widehat{Q}$ box and its energy derivatives introduced by Kuo {it et al}. In order to find out a method of calculating them we make decomposition of a full Hilbert space into subspaces (the Krylov subspaces) and transform a Hamiltonian to a block-tridiagonal form. This transformation brings about much simplification of the calculation of the $widehat{Q}$ box. In the previous work a recursion method has been derived for calculating the $widehat{Q}$ box analytically on the basis of such transformation of the Hamiltonian. In the present study, by extending the recursion method for the $widehat{Q}$ box, we derive another recursion relation to calculate the derivatives of the $widehat{Q}$ box of arbitrary order. With the $widehat{Q}$ box and its derivatives thus determined we apply them to the calculation of the $E$-independent effective interaction given in the so-called Lee-Suzuki (LS) method for a system with a degenerate unperturbed energy. We show that the recursion method can also be applied to the generalized LS scheme for a system with non-degenerate unperturbed energies. If the Hilbert space is taken to be sufficiently large, the theory provides an exact way of calculating the $widehat{Q}$ box and its derivatives. This approach enables us to perform recursive calculations for the effective interaction to arbitrary order for both systems with degenerate and non-degenerate unperturbed energies.
We construct an effective shell-model interaction for the valence space spanned by single-particle neutron and single-hole proton states in $^{100}$Sn. Starting from chiral nucleon-nucleon and three-nucleon forces and single-reference coupled-cluster theory for $^{100}$Sn we apply a second similarity transformation that decouples the valence space. The particle-particle components of the resulting effective interaction can be used in shell model calculations for neutron deficient tin isotopes. The hole-hole interaction can be used to calculate the $N = 50$ isotones south of $^{100}$Sn, and the full particle-hole interaction describes nuclei in the region southeast of $^{100}$Sn. We compute low-lying excited states in selected nuclei southeast of $^{100}$Sn, and find reasonable agreement with data. The presented techniques can also be applied to construct effective shell-model interactions for other regions of the nuclear chart.
Predicting the properties of neutron-rich nuclei far from the valley of stability is one of the major challenges of modern nuclear theory. In heavy and superheavy nuclei, a difference of only a few neutrons is sufficient to change the dominant fission mode. A theoretical approach capable of predicting such rapid transitions for neutron-rich systems would be a valuable tool to better understand r-process nucleosynthesis or the decay of super-heavy elements. In this work, we investigate for the first time the transition from asymmetric to symmetric fission through the calculation of primary fission yields with the time-dependent generator coordinate method (TDGCM). We choose here the transition in neutron-rich Fermium isotopes, which was the first to be observed experimentally in the late seventies and is often used as a benchmark for theoretical studies. We compute the primary fission fragment mass and charge yields for 254 Fm, 256 Fm and 258 Fm from the TDGCM under the Gaussian overlap approximation. The static part of the calculation (generation of a potential energy surface) consists in a series of constrained Hartree-Fock-Bogoliubov calculations based on the D1S, D1M or D1N parameterizations of the Gogny effective interaction in a two-center harmonic oscillator basis. The 2-dimensional dynamics in the collective space spanned by the quadrupole and octupole moments is then computed with the finite element solver FELIX-2.0. The available experimental data and the TDGCM post-dictions are consistent and agree especially on the position in the Fermium isotopic chain at which the transition occurs. The main limitation of the method lies in the presence of discontinuities in the 2-dimensional manifold of generator states.
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