Renormalized Interactions for CI constrained by EDF methods Alex Brown, Angelo Signoracci and Morten Hjorth-Jensen.

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Presentation transcript:

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 EDF (Skyrme) phenomenology NN potential with V_lowk

Closed-shell vacuum filled orbitals EDF (Skyrme) phenomenology “tuned” valence two-body matrix elements

Closed-shell vacuum filled orbitals EDF (Skyrme) phenomenology Monopole from EDF

Closed-shell vacuum filled orbitals A 3 A 2 A 1 Monopole from EDF

Aspects of evaluating a microscopic two-body Hamiltonian (N3LO + V lowk + core-polarization) in a spherical EDF (energy- density functional) basis (i.e. Skyrme HF) 1)TBME (two-body matrix elements): Evaluate N3LO + V lowk with radial wave functions obtained with EDF. 2)TBME: Evaluate core-polarization with an underlying single-particle spectrum obtained from EDF. 3)TBME: Calculate monopole corrections from EDF that would implicitly include an effective three-body interaction of the valence nucleons with the core. 4)SPE for CI: Use EDF single-particle energies – unless something better is known experimentally.

Why use energy-density functionals (EDF)? 1)Parameters are global and can be extended to nuclear matter. 2)Effort by several groups to improve the understanding and reliability (predictability) of EDF – in particular the UNEDF SciDAC project in the US. 3)This will involve new and extended functionals. 4)With a goal to connect the values of the EDF parameters to the NN and NNN interactions. 5)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 209 Bi and 209 Pb.

Calculations in a spherical basis with no correlations

What do we get out of (spherical) EDF? 1)Binding energy for the closed shell 2)Radial wave functions in a finite-well (expanded in terms of harmonic oscillator). 3) 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.

218 U 208 Pb x = experiment CI (ham) N3LO with EDF constraint EDF (or CI) with no correlations CI with N3LO

Skyrme (Skxmb) + V low-k N 3 LO (second order) 210 Po

Skyrme (Skxmb) + V low-k N 3 LO (first order)

213 Fr Skyrme (Skxmb) + V low-k N 3 LO (second order)

214 Ra Skyrme (Skxmb) + V low-k N 3 LO (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.

Skyrme (Skxmb) + V low-k N 3 LO (second order) 210 Pb

Skyrme (Skxmb) + V low-k N 3 LO (second order) 210 Bi

Skyrme (Skxmb) + V low-k N 3 LO (second order) 212 Po

Skyrme (Skxmb) + V low-k N 3 LO (second order) 210 Pb

Skyrme (Skxmb) + exp spe V low-k N 3 LO (second order) 210 Pb

Skyrme (Skxmb) for 208 Pb (closed shell) + V low-k N 3 LO (second order)

“ab-initio” calculation for absolute energies of 213 Fr

Energy of first excited 2 + states