Towards a Universal Energy Density Functional Study of Odd-Mass Nuclei in EDF Theory
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Transcript of Towards a Universal Energy Density Functional Study of Odd-Mass Nuclei in EDF Theory
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