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  1. Central Michigan Shell Model Code (CMichSM): Present and Future Applications Mihai Horoi, Physics Department, Central Michigan University Mt.Pleasant, MI 48859, horoi@phy.cmich.edu Support from NSF grants PHY-0070911 and DMR-9977582 is acknowledged M Horoi - Central Michigan University

  2. Why CMichSM (YASMC)? • Old codes got reached some of their limits (OXBASH) • New codes not shared • Exponential Convergence Method (ECM) requires truncation on partitions and dimensions of the order of 10M – 100M • Applications: • Energies of yrast states • Shell model densities • Spectroscopic factors for astrophysics reaction rates • Properties of g.s. generated by random interaction M Horoi - Central Michigan University

  3. M-Scheme Shell Model • Shell model Hamiltonian: two body forces • Basis states: Slater determinants • Eigenvalue problem M Horoi - Central Michigan University

  4. How the Code Works • Matrix elements: • Hash Table: • It works with a selected collection of partitions (s.p. configurations): REQUIRED for truncation with Exponential Convergence Method (ECM) • Separation of protons and neutron not yet implemented: will reduce memory requirement M Horoi - Central Michigan University

  5. Salient Features of m-scheme • Number of matrix elements per row is quasi constant: e.g. in fp is ~ 500 in average • Amount of work is O(N) and not (N^2) as is in projected methods • Matrix file size increases linearly with the m-scheme dimension • Lanczos vectors can be stored in SINGLE PRECISION M Horoi - Central Michigan University

  6. Effective JT Projection • Start with basis |M=J T_z=T> • Diagonalize the modified Hamiltonian • Calculate modified energies • Similar to center-of-mass “purification” M Horoi - Central Michigan University

  7. “Observables” • s.p. occupation probabilities • One-Body Transition Densities • Spectroscopic factors M Horoi - Central Michigan University

  8. Harwdare • Dual Alpha 833 MHz /UP2000 /2GB RAM • Dual Intel / AMD 2 GHz + /4 – 8 GB DDR • PCI 64/32-66 MHz • PCI-X 64/133 MHz available for Prestonia (Pentium 4 Xeon) • SCSI Ultra 160 – Ultra 320 is coming soon • 5-10 x 73 GB SCSI HDs 10K RPM M Horoi - Central Michigan University

  9. Software • Linux alpha : free • Compaq Fortran for Linux alpha : • Free • OpenMP not available • Linux x386 : free • Intel Fortran 95 compiler for Linux : • Free • OpenMP available • RAID 0 (stripping) M Horoi - Central Michigan University

  10. Performance: single processor • Most of the time is spent in Lanczos • Time per iteration depends mostly of the I/O throughput • 48Cr : dimension 2M / matrix file size 5 GB / time per iteration starts at 1min 20 sec • 56Ni : truncated dimension 34M / matrix file size 87 GB / time per iteration starts at 32 min M Horoi - Central Michigan University

  11. Parallelization • Amdahl’s Law • Extended Amdahl’s Law M Horoi - Central Michigan University

  12. Distributed Memory : MPI • Simplest way is keeping a copies of the Lanczos vector on each processor • Collective communications: • MPI_Reduce • MPI_AllGatherV • MPI_ScatterV • Caveat: waste of memory M Horoi - Central Michigan University

  13. Shared Memory: OpenMP • Efficient use of memory • One has to update Lanczos vector atomically • GS160: • 16 alpha 1GHz / • up to 128 GB mem / • 32 I/O channels / • v2=Hv1 in10 minutes for dimension of 1 billion (no I/O) M Horoi - Central Michigan University

  14. Application: Shell Model Binding Energies of 1f7/2 Nuclei Relative to 40^Ca • Use the exponential convergence method (Phys. Rev. Lett. 82, 2064 (1999)) • Fully test FPD6 interaction (Nucl. Phys. A523, 325 (1991)) • Similar study for KB3 interaction (Phys. Rev. C 59, 2033 (1999)) - only even-even and odd-odd (J=0) above A=52 • Present study: lowest T_z (0 or 1/2) M Horoi - Central Michigan University

  15. Exponential Convergence Method M Horoi - Central Michigan University

  16. Interaction • FPD6 - W.A. Richter, et al., Nucl. Phys. A523, 325 (1991) • scales with number of valence particles: • better describes the gap around • KB3 - E. Caurier at al., Phys. Rev. C 59, 2033 (1999) • Coulomb correction M Horoi - Central Michigan University

  17. Exponential Convergence Method for fp-nuclei M Horoi - Central Michigan University

  18. Exponential Convergence Method for fp-nuclei M Horoi - Central Michigan University

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  22. Application: Shell Model Analysis of the 45V(p,gamma) Thermonuclear Reaction Rate Relevant to 44Ti Production in Core-Collapse Supernovae Radioactive 44Ti isotope, produced in core collapsed supernovae is of great astrophysical interest Observed effects: High abundance of 44Ca Large excess of 44Ca in silicon carbide meteoritic samples Direct observation of 44Ti 1.157 MeV gamma-ray decay from supernova remnant M Horoi - Central Michigan University

  23. No levels (except g.s.) are known in 46Cr • Proton separation energy in 46Cr is 4.89 MeV • Gamow window for T=5.5x10^9 K is 1-2 MeV • Isobar analog states in 46Ti in the energy range 4.89 – 7.0 MeV could be used • Proton excitation energy high enough to consider p and f waves ( l = 1,3 ) M Horoi - Central Michigan University

  24. Single resonance S-factor (Rolfs and Rodney, Nucl. Phys. A235, 450 (1974)) • Thomas-Ehrman shift calculations (Phys. Rev. 88, 1109 (1952)) M Horoi - Central Michigan University

  25. Shell Model Calculations of Bound States in 46Ti • Using fp major shell: • Tractable dimension • Describes only positive parity states • Alternative sd-fp: (- parity states states and s waves included)/(intractable shell model dimension) • FPD6 interaction: W.A. Richter, M.G. van der Merwe, R.E. Julies and B.A. Brown, Nucl.Phys. A 523, 325 (1991) • Very good description of nuclear structure around A=46 M Horoi - Central Michigan University

  26. fp Shell Model States for 46Ti M Horoi - Central Michigan University

  27. Spectroscopic Factors • Brussard and Glaudemans, Shell Model Applications in Nuclear Spectroscopy, 1977 M Horoi - Central Michigan University

  28. Shell Model Results M Horoi - Central Michigan University

  29. Astrophysical S-Factor M Horoi - Central Michigan University

  30. Reaction Rate • Two resonance interference (Rausher and Raimann, Phys.Rev. C 53, 2496(1996) M Horoi - Central Michigan University

  31. Reaction Rate M Horoi - Central Michigan University

  32. Application: Random Interaction • Recent suggestion that ensembles of random shell model interactions can describe the quantum numbers and gaps of the low-lying states in even-even nuclei (Johnson et al, Phys. Rev. Lett. 80, 2749 (1998)) • Pairing contributes to this effect (C.W. Johnson et al, Phys. Rev. C61, 014311 (2000)) • Effect of “geometric chaoticity” (D. Mulhall et al, PRL 85, 4016 (2000)) M Horoi - Central Michigan University

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  34. Random vs Realistic Interaction Horoi, Brown, Zelevinsky, PRL 87, 062501 (2001) • 3 single particle energies and 63 rotational and isospin invariant matrix elements • Realistic interaction W (Ann.Rev.Nucl.Part. Sci. 38, 29(1988)): matrix elements fitted to the data • 8 particles in the sd-shell corresponding to 24Mg nucleus • Overlap of Random (R) and Realistic (W) interaction w.f. - |<W|R>| and B(E2) values M Horoi - Central Michigan University

  35. Models of Two Body Random Ensemble (TBRE) of Interaction • (a) s.p. energies set to zero and 63 two body matrix elements (m.e.) random in (-1,1) • (b) s.p. energies taken from W and 63 m.e. randomly generated in (a-s,a+s) • (c) same as (b), but the matrix elements with JT=01 (pairing) from W were kept fixed (a=-0.616 MeV, s=3.03 MeV) • (d) same as (a), but only the six two body pairing m.e. were randomly generated M Horoi - Central Michigan University

  36. Random Interaction Models M Horoi - Central Michigan University

  37. Even-Even Case: SD-8 M Horoi - Central Michigan University

  38. SD-8 Dimensions M Horoi - Central Michigan University

  39. Odd-Odd Case: SD-10 M Horoi - Central Michigan University

  40. Odd-Even Case: SD-9 M Horoi - Central Michigan University

  41. Comparison with Experiment M Horoi - Central Michigan University

  42. What’s Next • Next 6-12 month goal: 400M dimension using a dual Intel Xeon Prestonia in few days (cost ~ $10-15K) • Superdeformed bands in 56Ni and 40Ca • sdpf interaction • ECM in sdpf • 45V(p,gamma)46Cr – negative parity states in 46Cr up to 7 MeV • Random interaction in fp M Horoi - Central Michigan University