1. 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 http://unedf.org 5th ANL/MSU/JINA/INT FRIB Workshop on Bulk Nuclear Properties Michigan State University, November 19-22, 2008

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

3. 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, 054311 (2005) M. Kortelainen et al., Phys. Rev. C 77, 064307 (2008) Cf. Talks by M. Stoitsov, S. Bogner, W. Nazarewicz Confidence gained from success of phenomenological functionals built on Skyrme and Gogny interactions

4. 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)

5. 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)

6. 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

7. 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)

8. 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 Jaguar@ORNL: Cray XT4, 7,832 quad-core, 2.1 GHz AMD Opteron (31,328 cores) Franklin@LBNL: Cray XT4, 9,660 dual-core, 2.6 GHz AMD Opteron (19,320 cores)

9. 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]3/2+-1318.935-1318.942-1318.900 [ 4, 2, 0]1/2+-1320.090 -1319.963 [ 4, 1, 3]5/2+-1322.151-1322.150-1322.103 [ 4, 1, 1]3/2+-1323.490-1323.495-1323.420 [ 4, 1, 1]1/2+-1322.322 -1322.279 [ 4, 0, 4]9/2+-1319.851 -1319.730 [ 5, 4, 1]3/2--1321.357-1321.353-1321.310 [ 5, 4, 1]1/2--1321.771-1321.785-1321.766 [ 5, 2, 3]7/2--1322.415-1322.414-1322.350 [ 5, 3, 2]5/2--1322.648-1322.647-1322.595 [ 5, 3, 0]1/2--1320.762-1320.761-1320.725 [SIII Interaction, 14 full HO shells, spherical basis, mixed pairing] Results (1/4) In axially-symmetric systems, EFA is valid within 10 keV (maximum)

10. 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

11. 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 ?

12. 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 ?

13. 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)

14. 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 ? http://orph02.phy.ornl.gov/workshops/lacm08/UNEDF/database.html Comments and suggestions are welcome... ! “Golden Set” Conclusions (2/2)