INT Seattle 3/14/2002M Horoi - Central Michigan University 1 Central Michigan Shell Model Code (CMichSM): Present and Future Applications  Mihai Horoi, Physics Department, Central Michigan University Mt.Pleasant, MI 48859,  Support from NSF grants PHY and DMR is acknowledged

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M Horoi - Central Michigan University 3 M-Scheme Shell Model Shell model Hamiltonian: two body forces Basis states: Slater determinants Eigenvalue problem

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M 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”

INT Seattle 3/14/2002M Horoi - Central Michigan University 7 “Observables” s.p. occupation probabilities One-Body Transition Densities Spectroscopic factors

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M 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)

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M Horoi - Central Michigan University 11 Parallelization Amdahl’s Law Extended Amdahl’s Law

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M 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)

INT Seattle 3/14/2002M 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)

INT Seattle 3/14/2002M Horoi - Central Michigan University 15 Exponential Convergence Method

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M Horoi - Central Michigan University 17 Exponential Convergence Method for fp-nuclei

INT Seattle 3/14/2002M Horoi - Central Michigan University 18 Exponential Convergence Method for fp-nuclei

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INT Seattle 3/14/2002M Horoi - Central Michigan University 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 MeV gamma-ray decay from supernova remnant

INT Seattle 3/14/2002M 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 )

INT Seattle 3/14/2002M 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))

INT Seattle 3/14/2002M 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

INT Seattle 3/14/2002M Horoi - Central Michigan University 26 fp Shell Model States for 46Ti

INT Seattle 3/14/2002M Horoi - Central Michigan University 27 Spectroscopic Factors Brussard and Glaudemans, Shell Model Applications in Nuclear Spectroscopy, 1977

INT Seattle 3/14/2002M Horoi - Central Michigan University 28 Shell Model Results

INT Seattle 3/14/2002M Horoi - Central Michigan University 29 Astrophysical S-Factor

INT Seattle 3/14/2002M Horoi - Central Michigan University 30 Reaction Rate Two resonance interference (Rausher and Raimann, Phys.Rev. C 53, 2496(1996)

INT Seattle 3/14/2002M Horoi - Central Michigan University 31 Reaction Rate

INT Seattle 3/14/2002M 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, (2000)) Effect of “geometric chaoticity” (D. Mulhall et al, PRL 85, 4016 (2000))

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INT Seattle 3/14/2002M Horoi - Central Michigan University 34 Random vs Realistic Interaction Horoi, Brown, Zelevinsky, PRL 87, (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. - | | and B(E2) values

INT Seattle 3/14/2002M 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= MeV, s=3.03 MeV) (d) same as (a), but only the six two body pairing m.e. were randomly generated

INT Seattle 3/14/2002M Horoi - Central Michigan University 36 Random Interaction Models

INT Seattle 3/14/2002M Horoi - Central Michigan University 37 Even-Even Case: SD-8

INT Seattle 3/14/2002M Horoi - Central Michigan University 38 SD-8 Dimensions

INT Seattle 3/14/2002M Horoi - Central Michigan University 39 Odd-Odd Case: SD-10

INT Seattle 3/14/2002M Horoi - Central Michigan University 40 Odd-Even Case: SD-9

INT Seattle 3/14/2002M Horoi - Central Michigan University 41 Comparison with Experiment

INT Seattle 3/14/2002M 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