Beyond-mean-field study of elastic and inelastic electron scattering off nuclei

Abstract

Background: Electron scattering provides a powerful tool to determine charge distributions and transition densities of nuclei. This tool will soon be available for short-lived neutron-rich nuclei. Purpose: Beyond-mean-field methods have been successfully applied to the study of excitation spectra of nuclei in the whole nuclear chart. These methods permit determination of energies and transition probabilities starting from an effective in-medium nucleon-nucleon interaction but without other phenomenological ingredients. Such a method has recently been extended to calculate the charge density of nuclei deformed at the mean-field level of approximation [J. M. Yao et al., Phys. Rev. C 86, 014310 (2012)]. The aim of this work is to further extend the method to the determination of transition densities between low-lying excited states. Method: The starting point of our method is a set of Hartree-Fock-Bogoliubov wave functions generated with a constraint on the axial quadrupole moment and using a Skyrme energy density functional. Correlations beyond the mean field are introduced by projecting mean-field wave functions on angular momentum and particle number and by mixing the symmetry-restored wave functions. Results: We give in this paper detailed formulas derived for the calculation of densities and form factors. These formulas are rather easy to obtain when both initial and final states are 0 + states but are far from being trivial when one of the states has a finite J value. Illustrative applications to 24 Mg and to the even-mass 58-68 Ni have permitted an analysis of the main features of our method, in particular the effect of deformation on densities and form factors. An illustrative calculation of both elastic and inelastic scattering form factors is presented. Conclusions: We present a very general framework to calculate densities of and transition densities between low-lying states that can be applied to any nucleus. Achieving better agreement with the experimental data will require improving the energy density functionals that are currently used and also introducing quasiparticle excitations in the mean-field wave functions.The reduced transition density ρ J f μ f J i μ i ,L (r) with angular momentum transfer L is therefore given byBy substituting the expression for ρ α f α i (r) into Eqs. (A8)-(A10), one finds as an expression for the reduced transition density024301-14 BEYOND-MEAN-FIELD STUDY OF ELASTIC AND . . . PHYSICAL REVIEW C 91, 024301 (2015) With the help of the orthogonality relation of spherical harmonics, dr Y LM (r) Y λν (r) = (−1) −M δ Lλ δ M−ν , and the symmetry relation J f M f L − M|J i M i = (−1) 2L−M+J i −J f J î J f J i M i LM|J f M f of the Clebsch-Gordan coefficients, the reduced transition density can be simplified to ρ J f μ f J i μ i ,L (r) = (−1) J i −J fĴ 2 f J 2

Semimicroscopic modeling of heavy-ion fusion reactions with multireference covariant density functional theory

“…On the other hand, the 0 states. A similar conclusion has been reached also with the non-relativistic DFT [11].…”

“…3. One can see that the main feature of the energy spectrum and the E2 transition strength from 2 relativistic Skyrme SLy4 interaction [11], but quantitatively the present calculations with the relativistic DFT reproduce the experimental data slightly better. For instance, the B(E2) value from the 2 Fig.…”

“…These nuclei have been studied recently in Ref. [11] using the non-relativistic MR-DFT method. Here we repeat similar calculations with the relativistic framework, in order to check the dependence of the conclusions on a choice of energy density functional.…”

“…For example, the 58 Ni nucleus, which has usually been considered to be a typical vibrational nucleus, does not exhibit a level spectrum characteristic to the harmonic vibration, that is, the degeneracy of the two-phonon triplet is considerably broken. Moreover, a recent theoretical calculation based on a multi-reference density functional theory with the Skyrme interaction also indicates that the B(E2) strengths among the collective levels in 58 Ni deviate largely from what are expected from a simple harmonic oscillator [11]. It is therefore of considerable interest to investigate the role of anharmonicity, that is, the deviation from the harmonic limit, in subbarrier fusion of 58 Ni.…”

Ab initio-driven nuclear energy density functional method

“…First, it involves an "horizontal" expansion in terms of non-orthogonal symmetry-breaking product states that provides an efficient way to resum so-called non-dynamical 19 correlations through the MR mixing. The associated off-diagonal kernels N(Ω ) and h(Ω ) are themselves "vertically" expanded in terms of particle-hole excitations, which corresponds to the treatment of so-called 18 Care has to be taken for non-scalar operators such that the associated MR expression must be properly adapted from the one appropriate to the calculation of the energy [53]. 19 Dynamical and non-dynamical refer to the wording at play in quantum chemistry.…”