Application of DFT to the Spectroscopy of Odd Mass Nuclei N. Schunck Department of Physics  Astronomy, University of Tennessee, Knoxville, TN-37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN-37831, USA J. Dobaczewski, J. McDonnell, W. Nazarewicz, M. V. Stoitsov 5th ANL/MSU/JINA/INT FRIB Workshop on Bulk Nuclear Properties Michigan State University, November 19-22, 2008

Outline 1.Motivations 2.Energy Density Functional theory with Skyrme Interactions 3.Computational Aspects 4.Results 5.Conclusions

Motivations 1 Introduction The energy of the nucleus is a function, to be found, of the density matrix and pairing tensor. UNEDF: find a universal functional capable of describing g.s. and excited states with a precision comparable or better to macroscopic-microscopic models Odd nuclei allow to probe time-odd terms in g.s. systems Nuclear DFT Principle Theory: Symmetry-unrestricted Skyrme DFT + HFB method G. Bertsch et al, Phys. Rev. C 71, (2005) M. Kortelainen et al., Phys. Rev. C 77, (2008) Cf. Talks by M. Stoitsov, S. Bogner, W. Nazarewicz Confidence gained from success of phenomenological functionals built on Skyrme and Gogny interactions

Skyrme Energy Density Functional The DFT recipe:  Start with an ensemble of independent quasi-particles characterized by a density matrix  and a pairing tensor   Construct fields by taking derivatives of densities  and  up to second order and using spin and isospin degrees of freedom  Constructs the energy density  (r) by coupling fields together  Interaction-based functionals: couplings constants are defined by the parameters of the interaction (Ex.: Skyrme, Gogny, see Scott’s talk)  Apply variational principle and solve the resulting equations of motion (HFB)  Allow spontaneous symmetry breaking for success Fields Skyrme Energy Functional Interaction Picture Constants C related to parameters of the interaction Functional Picture Constants C free parameters to be determined 2 Theory (1/3)

Odd Nuclei in the Skyrme HFB Theory Standard fits of Skyrme functionals do not probe time-odd part: – How well do existing interactions ? – How much of a leverage do the time-odd part give us ? Odd particle described as a one quasi- particle excitation on a fully-paired vacuum = blocking approximation Equal Filling Approximation (EFA): – Average over blocking time-reversal partners: “ ⌈ EFA 〉 = ⌈〉 + ⌈〉 ” – Conserves time-reversal symmetry Time-odd part of the functional becomes active in odd nuclei Practical issues:  Dependent on the quality of the pairing interaction used (density- dependent delta-pairing here)  Blocked state is not known beforehand: warm-start from even- even core  Broken time-reversal symmetry + many configurations to consider = computationally VERY demanding 3 1.Quality of the EFA approximation 2.Impact of time-odd fields Theory (2/3)

Symmetries and Blocking 4 Theory (3/3) Skyrme functional (any functional based on 2-body interaction) gives time-odd fields ⇒ They break T-symmetry Definition of the blocked state: Criterion: quasi-particle of largest overlap with “some” single-particle state identified by a set of quantum numbers Quantum numbers are related to symmetry operators: Time-odd fields depend on choice of quantization axis – Example: Symmetry operator chosen to identify s.p. states must commute with the projection of the spin operator onto the quantization axis

DFT Solver: HFODD 5 Solves the HFB problem in the anisotropic cartesian harmonic oscillator basis Most general, symmetry-unrestricted code Recent upgrades include: − Broyden Method, shell correction, interface with HFBTHO (Schunck) − Isospin projection (Satuła) − Exact Coulomb exchange (Dobaczewski) − Finite temperature (Sheik) Truncation scheme: dependence of results on N shell, ħ , deformation of the basis (see NCSM, CC, SM, etc.) Reference provided by HFB-AX Error estimate for given model space  give theoretical error bars Codes (1/2)

Terascale Computing in DFT Applications 6 Codes (2/2) MPI-HFODD: HFODD core plus parallel interface with master/slave architecture. About 1.2 Gflops/core on Jaguar and 2 GB memory/core Optimizations : Unpacked storage BLAS and LAPACK, Broyden Method, Interface HFBTHO To come: Takagi Factorization, ScaLAPACK and/or OpenMP for diagonalization of HFODD core Cray XT4, 7,832 quad-core, 2.1 GHz AMD Opteron (31,328 cores) Cray XT4, 9,660 dual-core, 2.6 GHz AMD Opteron (19,320 cores)

Equal Filling Approximation 7 Blocked states in 163 Tb in EFA (HFBTHO) and Exact Blocking (HFODD) Blocked StateEFA (HFBTHO)Exact (HFODD)HFODD (full) [ 4, 2, 2]3/ [ 4, 2, 0]1/ [ 4, 1, 3]5/ [ 4, 1, 1]3/ [ 4, 1, 1]1/ [ 4, 0, 4]9/ [ 5, 4, 1]3/ [ 5, 4, 1]1/ [ 5, 2, 3]7/ [ 5, 3, 2]5/ [ 5, 3, 0]1/ [SIII Interaction, 14 full HO shells, spherical basis, mixed pairing] Results (1/4) In axially-symmetric systems, EFA is valid within 10 keV (maximum)

Comparison With Experiment Z=65, N=96 – SLy4 Interaction 8 Results (2/4) SLy4Exp. Ho Isotopes (Z=67) Scale: 10,000+ processors for 5 hours: ~30 blocked configurations, number of interactions 1 ≤ N ≤ 24, ~100 isotopes, optional scaling of time-odd fields  Rare-earth region (A ~ 150)  Well-deformed mean-field with g.s. deformation about 2 ~ 0.3  HFB theory works very well and correlations beyond the mean- field are not relevant  Abundant experimental information, in particular asymptotic Nilsson labels Overall Trend: Right q.p. levels and iso-vector trend  Wrong level density

Effect of Time-Odd Fields 9 Results (3/4)  Effect of time-odd fields of ~ 150 keV (maximum) on q.p. spectra  Deformation, pairing, interaction more important for comparison with experiment  Induced effects such as triaxiality and mass-filters  Can only be accounted for by symmetry-unrestricted codes  Can significantly influence pairing fits and deformation properties E TOdd ≠ 0 – E TOdd=0 Varying C s and C  s by 50% and 150%: do we have the right order of magnitude here ?

Effect of Time-Odd Fields 10 Results (4/4) Single nucleon in odd nuclei can induce small tri-axial polarization Time-odd fields impact (slightly) the Odd-Even Mass (OEM) How can we constrain these time-odd fields ?

Summary and conclusions 11 Study of odd-mass quasi-particle spectra in the rare-earth region using fully- fledged, symmetry-unrestricted Skyrme HFB Equal Filling Approximation is of excellent quality Effect of time-odd fields: Weak impact on q.p. spectra. Induced effects, e.g. on OEM, are larger but still second-order  Skyrme functionals: time-odd terms determined automatically by parameters of the interaction. Are we sure this is the right order of magnitude ?  DFT “a la Kohn-Sham”: introduce terms dependent on s ( r,r’ ) and derivatives All standard Skyrme interactions agree poorly with experimental data – Ground-state offset of the order of a few MeV ( ⇒ bulk properties of EDF) – Excited states offsets of the order of a few hundreds of keV ( ⇒ largely dictated by effective mass m*) Proper description of pairing correlation is crucial – Underlying shell structure must be reliable – Pairing interaction/functional should be richer (Coulomb, isovector at the very least) – Are  (n) mass filters sufficient to capture all features of pairing functional ? Conclusions (1/2)

Outlook 12 What is the best data set to constrain time-odd terms ? More generally: how can we make sure to constrain each term of the functional ? Comments and suggestions are welcome... ! “Golden Set” Conclusions (2/2)