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Renormalized Interactions for CI constrained by EDF methods Alex Brown, Angelo Signoracci and Morten Hjorth -Jensen. Wick’s theorem for a Closed-shell vacuum filled orbitals. Closed-shell vacuum filled orbitals. EDF (Skyrme Phenomenology). Closed-shell vacuum filled orbitals.
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Renormalized Interactions for CI constrained by EDF methods Alex Brown, Angelo Signoracci and MortenHjorth-Jensen
Wick’s theorem for a Closed-shell vacuum filled orbitals
Closed-shell vacuum filled orbitals EDF (Skyrme Phenomenology)
Closed-shell vacuum filled orbitals NN potential with V_lowk EDF (Skyrme) phenomenology
Closed-shell vacuum filled orbitals “tuned” valence two-body matrix elements EDF (Skyrme) phenomenology
Closed-shell vacuum filled orbitals Monopole from EDF EDF (Skyrme) phenomenology
Closed-shell vacuum filled orbitals Monopole from EDF A3 A2 A 1
Aspects of evaluating a microscopic two-body Hamiltonian (N3LO + Vlowk+ core-polarization) in a spherical EDF (energy-density functional) basis (i.e. Skyrme HF) TBME (two-body matrix elements): Evaluate N3LO + Vlowk with radial wave functions obtained with EDF. TBME: Evaluate core-polarization with an underlying single-particle spectrum obtained from EDF. TBME: Calculate monopole corrections from EDF that would implicitly include an effective three-body interaction of the valence nucleons with the core. SPE for CI: Use EDF single-particle energies – unless something better is known experimentally.
Why use energy-density functionals (EDF)? Parameters are global and can be extended to nuclear matter. Effort by several groups to improve the understanding and reliability (predictability) of EDF – in particular the UNEDF SciDAC project in the US. This will involve new and extended functionals. With a goal to connect the values of the EDF parameters to the NN and NNN interactions. At this time we have a reasonably good start with some global parameters – for now I will use Skxmb – Skxm from [BAB, Phys. Rev. C58, 220 (1998)] with small adjustment for lowest single-particle states in 209Bi and 209Pb.
What do we get out of (spherical) EDF? Binding energy for the closed shell Radial wave functions in a finite-well (expanded in terms of harmonic oscillator). gives single-particle energies for the nucleons constrained to be in orbital (n l j)a where BE(A) is a doubly closed-shell nucleus. 4) gives the monopole two-body matrix element for nucleons constrained to be in orbitals (n l j)a and (n l j)b
EDF core energy and single-particle energy EDF two-body monopole
Theory (ham) from Skxmb with parameters adjusted to reproduce the energy for the 9/2- state plus about 100 other global data.
x = experiment CI with N3LO CI (ham) N3LO with EDF constraint EDF (or CI) with no correlations 208Pb 218U
Skyrme (Skxmb) + Vlow-k N3LO (second order) 210Po
Skyrme (Skxmb) + Vlow-k N3LO (first order) 210Po
Skyrme (Skxmb) + Vlow-k N3LO (second order) 213Fr
214Ra Skyrme (Skxmb) + Vlow-k N3LO (second order)
EDF core energy and single-particle energy EDF two-body monopole
Theory (ham) from Skxmb with parameters adjusted to reproduce the energy for the 9/2+ state plus about 100 other global data.
210Pb Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Bi Skyrme (Skxmb) + Vlow-k N3LO (second order)
212Po Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Pb Skyrme (Skxmb) + Vlow-k N3LO (second order)
210Pb Skyrme (Skxmb) + exp spe Vlow-k N3LO (second order)
Skyrme (Skxmb) for 208Pb (closed shell) + Vlow-k N3LO (second order)