Towards a Universal Energy Towards a Universal Energy Density FunctionalDensity Functional

Study of Odd-Mass Nuclei in EDF Theory

N. SchunckUniversity of Tennessee, 401 Nielsen Physics , Knoxville, TN-37996, USAOak Ridge National Laboratory, Bldg. 6025, MS6373, P.O. Box 2008, Oak Ridge, TN-37831, USA

Together with:

J. Dobaczewski, W. Nazarewicz, N. Nikolov and M. Stoitsov

• Construct density fields in normal, spin, isospin space

• Use Local Density Approximation (LDA)• Express the energy density as a scalar made of these Express the energy density as a scalar made of these

fields and their derivatives up to 2fields and their derivatives up to 2ndnd order order • Examples:

• Compute the total energy (variational principle):

Nuclear EDF in a NutshellNuclear EDF in a Nutshell

EDF and Effective InteractionsEDF and Effective Interactions• Skyrme effectiveeffective interaction:

• Zero-range (Skyrme) or finite-range (Gogny)

• Many-body Hamiltonian reads:

• Express total energy as function of densities

• Energy density is realreal, scalarscalar, time-even,time-even, iso-scalariso-scalar but constituting fields are not (necessarily) and therefore need be computed…

• Time-evenTime-even part (depends on time-even fields)

• Time-oddTime-odd part (depends on time-odd fields)

• Scalar, vector and tensor terms depend on scalar, vector or tensor fields

• Iso-scalar (t = 0), iso-vector (t = 1)• Parameters C may be related to (t,x) set of Skyrme force but

this is not necessarynot necessary

Nuclear Energy Density FunctionalNuclear Energy Density Functional

Pairing correlationsPairing correlations

• Pairing functional a prioria priori as rich as mean-field functional: the choice of the pairing channel is not finalized yet

• Form considered here: surface-volume pairing

• Cut-off, regularization procedure• (Some form of) projection on the number of particles is

required– Breaking down of pairing correlations– Fluctuations of particle number

• Current Difficulties:– Constrained/unconstrained calculations with Lipkin-Nogami

prescription– Blocking in odd nuclei with Lipkin-Nogami prescription

Determination of VDetermination of V00

Neutron Pairing Gap in 120Sn in Skyrme HFB Calculations

Computational AspectsComputational Aspects

• Variational principle

gives a set of equations: HF (no pairing) or HFB (pairing)• Self-consistency requires iterative procedureiterative procedure• Modern super-computers allow large-scale calculations

– Symmetry-restricted codes: one mass table in a couple of hours couple of hours (Mario)

– Symmetry-unrestricted codes: one mass table in a couple of dayscouple of days

• HFODD = HFB solver– Basis made of 3D3D, deformeddeformed, cartesiancartesian, y-simplex conservingy-simplex conserving

harmonic oscillatorharmonic oscillator eigenfunctions– All symmetries broken (including time-reversal)

• MPI-HFODD: about 1.2 Gflops/core on Jaguar: HFODD core and parallel interface

Treatment of Odd NucleiTreatment of Odd Nuclei

• Odd nuclei neglected in previous fit strategies– Provide unique handle on time-odd fieldstime-odd fields (half of the functional !)– Will help constrain high-spinhigh-spin physics better

• Practical difficulties:– Break time-reversal symmetrytime-reversal symmetry (HF and HFB)– Treatment of the odd particleodd particle– Proton-neutron pairingProton-neutron pairing for odd-odd nuclei (not addressed here)

• Blocking in HFB

• Several states around the Fermi level need to be blocked to find the g.s.

Normalized Q.P. SpectrumNormalized Q.P. Spectrum

SLy4 ExperimentSLy4 + Tensor

Rare Earth RegionRare Earth Region

ConclusionsConclusions

• Progress:– Computing core (HFODD) optimized

– Parallel architecture ready (can/will be improved)

– Information on deformation properties and odd nuclei spectra vs. functional

• Difficulties:– Remaining problems with approximate particle number projection for odd

nuclei

– Stability problems of calculations with full time-odd terms

• Needs:– Discussion of treatment of pairing and correlations

– Interface symmetry-restricted vs. symmetry-unrestricted codes

– Highly optimized minimization routines for fit of functional

• Small number of function evaluations

• Parallelized (or …-able)

– Automatic handling of divergences of self-consistency procedure