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Application of DFT to the Spectroscopy of Odd Mass Nuclei

http://unedf.org. 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

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Application of DFT to the Spectroscopy of Odd Mass Nuclei

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  1. http://unedf.org 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 onBulk Nuclear PropertiesMichigan State University, November 19-22, 2008

  2. Outline • Motivations • Energy Density Functional theory with Skyrme Interactions • Computational Aspects • Results • Conclusions

  3. Introduction 1 Motivations G. Bertsch et al, Phys. Rev. C 71, 054311 (2005) Nuclear DFT Principle The energy of the nucleus is a function, to be found, of the density matrix and pairing tensor. • Confidence gained from success of phenomenological functionals built on Skyrme and Gogny interactions M. Kortelainen et al., Phys. Rev. C 77, 064307 (2008) Cf. Talks by M. Stoitsov, S. Bogner, W. Nazarewicz • 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 Theory: Symmetry-unrestricted Skyrme DFT + HFB method

  4. Theory (1/3) Fields 2 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 Skyrme Energy Functional Interaction Picture Constants C related to parameters of the interaction Functional Picture Constants C free parameters to be determined

  5. Theory (2/3) 3 Odd Nuclei in the Skyrme HFB Theory Time-odd part of the functional becomes active in odd nuclei • 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 excitationon a fully-paired vacuum = blocking approximation • Equal Filling Approximation (EFA): • Average over blocking time-reversal partners: “⌈EFA〉 = ⌈〉 + ⌈〉” • Conserves time-reversal symmetry • 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 • Quality of the EFA approximation • Impact of time-odd fields

  6. Theory (3/3) 4 Symmetries and Blocking • 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. Codes (1/2) 5 DFT Solver: HFODD • 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 Nshell, 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

  8. Codes (2/2) 6 Terascale Computing in DFT Applications • 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. Results (1/4) 7 Equal Filling Approximation Blocked states in 163Tb in EFA (HFBTHO) and Exact Blocking (HFODD) [SIII Interaction, 14 full HO shells, spherical basis, mixed pairing] In axially-symmetric systems, EFA is valid within 10 keV (maximum)

  10. Results (2/4) 8 Comparison With Experiment SLy4 Exp. • 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 Ho Isotopes (Z=67) Z=65, N=96 – SLy4 Interaction Scale: 10,000+ processors for 5 hours: ~30 blocked configurations, number of interactions 1 ≤ N ≤ 24, ~100 isotopes, optional scaling of time-odd fields • Overall Trend: • Right q.p. levels and iso-vector trend • Wrong level density

  11. Results (3/4) 9 Effect of Time-Odd Fields • 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 ETOdd≠0 – ETOdd=0 Varying Cs and Cs by 50% and 150%: do we have the right order of magnitude here ?

  12. Results (4/4) 10 Effect of Time-Odd Fields • 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. Conclusions (1/2) 11 Summary and conclusions • 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 ?

  14. Conclusions (2/2) 12 Outlook “Golden Set” • 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... ! http://orph02.phy.ornl.gov/workshops/lacm08/UNEDF/database.html

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