Renormalized Interactions with EDF Single-Particle Basis Statesand NuShellX@MSU

Alex Brown, Angelo Signoracci, Morten Hjorth-Jensen and Bill Rae

Closed-shell vacuumfilled orbitals

Closed-shell vacuumfilled orbitals

Skyrme phenomenology

Closed-shell vacuumfilled orbitals

Skyrme phenomenology

NN potential with V_lowk

Closed-shell vacuumfilled orbitals

Skyrme phenomenology

“tuned” valence two-body matrix elements

Closed-shell vacuumfilled orbitals

“tuned” valence two-body matrix elements

A3 A2 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 (N3LO + Vlowk+ core-polarization) .

Not realistic for the nuclei near the drip line.

No three-body interactions.

Aspects of evaluating a microscopic two-body Hamiltonian (N3LO + Vlowk+ core-polarization) in a spherical EDF (energy-density functional) basis (i.e. Skyrme HF)

1)TBME (two-body matrix elements): Evaluate N3LO + 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 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 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 - 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 (g7/2) and neutron (f7/2) orbitalsN3LO – Vlowk (lambda=2.2)

TBME for the lowest proton (g7/2) and neutron (f7/2) orbitalsN3LO – Vlowk (lambda=2.2) - 4hw

TBME for the lowest proton (g7/2) and neutron (f7/2) orbitalsN3LO – Vlowk (lambda=2.2) - 4hw

TBME for the lowest proton (g7/2) and neutron (f7/2) orbitalsN3LO – Vlowk (lambda=2.2) - 4hw

134Sn

134Sb

134Te

136Te

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 - 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(146Gd) – BE(132Sn)] (MeV) theory: filled g7/2 and d5/2

101.585 experiment

117.232 using ea and M(a,b) from N3LO for all

98.573 Skxtb applied to 146Gd and 132Sn

97.925 using ea and M(a,b) from Skxtb

100.452 Skxtb + 2p-2h from N3LO

134Te

Experiment Skxtb134Sb

Experiment “adjusted to exp”133Sb

134Te

Experiment Skxtb133Sn

sdpn

fppn

jj44pn

jj44 means f5/2, p3/2, p1/2, g 9/2 orbits for protons and neutrons

Recent results from Angelo Signoracci

SDPF-U: Nowacki and Poves, PRC79, 014310 (2009).

Energy of first excited 2+ states

What is NuShellX@MSU?

1)NuShellX - Nathan-type pn basis CI code implemented by Bill Rae (Garsington).

2)NuShellX@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 NuShellX.

6)Windows version now – linux version being finished - maybe someday a Mac version.

HamHamiltonian Input programs

Shell wrapper for NuShellX

ToiNuclear Data

*.sp *.int

library of tuned Hamiltonians*.int files (sps folder)

*.sp model space files*.int Hamiltonian files

*.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 basiswithin the next 5-10 years with M-basis dimensions up to 1014

Test case for speed of NuShellX - 48Cr 0+ J-dim=41,355 M-dim=1,963,461 10 eigenstates to 1 keV precision

Chip RAM cpu speed time cost GB GHz sec $Intel i7 Quad (8GB) (2.8)x(4) = 11.2 23 (1,400)

Intel i7 2xQuad (48GB) (3.3)x(8) = 26.4 11 (10,000) How far can we go - number of cores and speed?

Now – transfer from ifort to Portland compilersNext – test replacement of OpenMP with MPITry out GPU