Renormalized Interactions with EDF SingleParticle Basis States and

Renormalized Interactions with EDF Single-Particle Basis States and Nu. Shell. X@MSU Alex Brown, Angelo Signoracci, Morten Hjorth-Jensen and Bill Rae

Closed-shell vacuum filled orbitals

Closed-shell vacuum filled orbitals Skyrme phenomenology

Closed-shell vacuum filled orbitals NN potential with V_lowk Skyrme phenomenology

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

Closed-shell vacuum filled orbitals “tuned” valence two-body matrix elements A 3 A 2 A 1

Typically one uses an harmonic-oscillator basis for the evaluation of the microscopic two-body matrix elements used in shell-model configuration mixing (N 3 LO + Vlowk+ corepolarization). Not realistic for the nuclei near the drip line. No three-body interactions.

Aspects of evaluating a microscopic two-body Hamiltonian (N 3 LO + Vlowk+ core-polarization) in a spherical EDF (energydensity functional) basis (i. e. Skyrme HF) 1) TBME (two-body matrix elements): Evaluate N 3 LO + Vlowk 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: 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) Large effort by several groups to improve the understanding and reliability (predictability) of EDF – in particular the UNEDF Sci. DAC 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 Skxtb (Skyrme with tensor) [BAB, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, 061303(R) (2006)}.

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) ea = - [BE(A+1, a) – BE(A)] 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) M(a, b) = - [BE(A+2, a, b) – BE(A)] - ea gives the monopole two-body matrix element for nucleons constrained to be in orbitals (n l j)a and (n l j)b

TBME for the lowest proton (g 7/2) and neutron (f 7/2) orbitals N 3 LO – Vlowk (lambda=2. 2)

TBME for the lowest proton (g 7/2) and neutron (f 7/2) orbitals N 3 LO – Vlowk (lambda=2. 2) - 4 hw

134 Sn

134 Sb

134 Te

136 Te

What do we get out of (spherical) EDF? 1) ea = - [BE(A+1, a) – BE(A)] 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. 2) M(a, b) = -[BE(A+2, a, b) – BE(A)] - ea gives the monopole two-body matrix element for nucleons constrained to be in orbitals (n l j)a and (n l j)b 3) [BE(146 Gd) – BE(132 Sn)] (Me. V) theory: filled g 7/2 and d 5/2 101. 585 experiment 117. 232 using ea and M(a, b) from N 3 LO for all 98. 573 Skxtb applied to 146 Gd and 132 Sn 97. 925 using ea and M(a, b) from Skxtb 100. 452 Skxtb + 2 p-2 h from N 3 LO

134 Te

Experiment 134 Sb Skxtb

Experiment 133 Sb “adjusted to exp”

134 Te

Experiment 133 Sn Skxtb

jj 44 pn fppn sdpn jj 44 means f 5/2, p 3/2, p 1/2, g 9/2 orbits for protons and neutrons

Recent results from Angelo Signoracci SDPF-U: Nowacki and Poves, PRC 79, 014310 (2009).

Energy of first excited 2+ states

What is Nu. Shell. X@MSU? 1) Nu. Shell. X - Nathan-type pn basis CI code implemented by Bill Rae (Garsington). 2) Nu. Shell. X@MSU - developments at MSU that includes wrapper code for input, Hamiltonians, output and comparison to data. Three parts: 3) Toi - connection with nuclear data base (175 MB) 4) Ham - connections with the codes of Morten Hjorth-Jensen together with EDF to generate new Hamiltonians. 5) Shell – implementations of Nu. Shell. X. 6) Windows version now – linux version being finished maybe someday a Mac version.

Toi Nuclear Data *. sp model space files *. int Hamiltonian files Hamiltonian Input programs *. sp *. int library of tuned Hamiltonians *. int files (sps folder) Shell wrapper for Nu. Shell. X *. eps Outputs for energies *. lpt <|a+|> *. lsf <|a+ a|> *. obd <|a+ a+|> *. tna postscrip (*. eps) (pdf) figures

Shears Bands

Energy of first excited 2+ states

What might be possible to consider in the spherical CI basis within the next 5 -10 years with M-basis dimensions up to 1014

Test case for speed of Nu. Shell. X - 48 Cr 0+ J-dim=41, 355 M-dim=1, 963, 461 10 eigenstates to 1 ke. V precision Chip RAM cpu speed GB GHz Intel i 7 Quad (8 GB) (2. 8)x(4) = 11. 2 time sec 23 Intel i 7 2 x. Quad (48 GB) (3. 3)x(8) = 26. 4 11 How far can we go - number of cores and speed? Now – transfer from ifort to Portland compilers Next – test replacement of Open. MP with MPI Try out GPU cost $ (1, 400) (10, 000)