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Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory

Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory. David J. Dean Oak Ridge National Laboratory Nuclear Coupled-cluster Collaboration: T. Papenbrock, K. Roche, Oak Ridge National Laboratory P. Piecuch, M. Wloch, J. Gour, Michigan State University M. Hjorth-Jensen, Oslo

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Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory

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  1. Building Nuclei from the Ground Up: Nuclear Coupled-cluster Theory David J. Dean Oak Ridge National Laboratory Nuclear Coupled-cluster Collaboration:T. Papenbrock, K. Roche, Oak Ridge National LaboratoryP. Piecuch, M. Wloch, J. Gour, Michigan State University M. Hjorth-Jensen, Oslo A. Schwenk, Triumf Funding: DOE-NP, SciDAC, DOE-ASCR

  2. Supernova E0102-72.3 “Given a lump of nuclear material, what are its properties, and how does it interact?” • How do we describe nuclei we cannot measure? • Robust, predictive nuclear theory exists for structure and reactions. • Nuclear data needed to constrain theory. • Goal is the Hamiltonian and nuclear properties: – Bare intra-nucleon Hamiltonian. – Energy density functional. • Mission relevant to NP, NNSA. • Half of all elements heavier than iron produced in r-process where limited (or no) experimental information exits. • Nuclear reaction information relevant to NNSA and AFCI.

  3. Pushing the nuclear boundaries Thermal properties regions Nuclear DFT effort • The Leadership Computing Facility effort will • Enlarge ab-initio square to mass 100 • Enable initial global DFT calculations with restored symmetries Nuclear Coupled Cluster effort All Regions: Nuclear cross-section efforts (NNSA, SC/NP, Nuclear Energy)

  4. Nuclear interactions: Cornerstone of the entire theoretical edifice Depends on spin, angular momentum, and nucleon(proton and neutron) quantum numbers. Complicated interactions Real three-body interactionsderived from QCD-basedeffective theories Solved up to mass 12 with GFMC, converged mass 8 with diagonalization. We want to go much further!    Method of Solution: Nuclear Coupled-Cluster Theory

  5.   exp  … T T T T = + + + 1 2 3  ) = F - F E exp( T ) H exp T F - F = F F = ab… ab… ( ) exp( T ) H exp T H 0 ij… ij… Coupled-cluster theory: Ab initio in medium mass nuclei Reference Slaterdeterminant Correlated ground-statewave function Correlationoperator Energy Amplitude equations • It boils down to a set of coupled, nonlinear algebraic equations (odd-shaped tensor-tensor multiply). • Storage of both amplitudes and interactions is an issue as problems scale up. • Largest problem so far: 40Ca with 10 million unknowns, 7 peta-ops to solve once(up to 10 runs per publishable result). • Breakthrough science: Inclusion of 3-body force into CC formalism (6-D tensor)weakly bound and unbound nuclei.

  6. Early results -100 E* (0+) = 19.8 MeV CCSD CR-CCSD(T) -105 E = H = e-THeT -110 E* (3-) = 12.0 MeV ab…H = 0 E = (MeV) ij… -115 RH = E*R -120 *Eg.s. = –120.5 MeV -125 Wolch et al PRL 94, 24501 (2005) -130 4 6 8 infty Number of Oscillator Shells Coupled cluster theory for nuclei   exp(T) T = T1 + T2 + T3 + … R= excitation operator POLYNOMIAL SCALING!! (good)

  7. -16 -134 -18 N = 6 CCSD CCSD(T) FY 4He -136 -20 N = 7 -22 E(4He) [MeV] -138 ECCSD(16O) [MeV] -24 N = 8 -140 -26 N = 9 N = 9 N = 11 16O -28 -142 N = 12 N = 13 -30 2 4 5 6 10 12 14 16 -144 N=2n+1 12 14 16 18 20 22 24 ħ [MeV] 380 N = 3 Error estimate: << 1% < 1% 1% -400 -420 N = 4 ECCSD(40Ca) [MeV] -440 Fast convergence with cluster rank N = 5 -460 N = 6 40Ca N = 7 -480 N = 8 -500 ħ [MeV] 1063 many-bodybasis states 18 22 24 26 28 30 32 16 20 Ab initio in medium mass nuclei Hagen et al., Phys. Rev. C76,044305 (2007)

  8. 100 0 = + + + + + 10-1 + + + + + + 10-2 + + + 10-3 10-4 Inclusion of full TNF in CCSD: F-Y comparisons in 4He Solution at CCSD and CCSD(T) levels involve roughly 67 more diagrams… 2-body only 0-body 3NF 1-body 3NF -22  / CCSD <E>=-28.24 MeV +/- 0.1MeV (sys) estimated triples corrections -23 2-body 3NF -24 ECCSD(T) (MeV) -25 Residual 3NF -26 1 2 3 4 5 -27 N Challenge: Do we really need the full 3-body force, or just its density dependent terms? -28 3 4 5 6 N Hagen, Papenbrock, Dean, Schwenk, Nogga, Wloch, Piecuch, Phys. Rev. C76, 034302 (2007)

  9. lm(k) bound states L- k2 k3 k4 Re(k) L+ k1 antibound states capturing states decaying states Coupling of nuclear structure and reaction theory (microscopic treatment of open channels) Open QS Energy Correlation dominated Sn=0 n ~ n Sn Closed QS Neutron number Introduction of continuum basis states (Gamow, Berggren)

  10. 100 G-HF basis HO-HF basis 10-1 r2(r) [fm-1] Gamov states capture the halo structure of drip-line nuclei 10-2 4 5 6 7 8 r [fm] Ab initio weakly bound and unbound nuclei Single-particle basis includes bound, resonant, non-resonant continuum, and scattering states ENORMOUS SPACES….almost 1k orbitals. 1022 many-body basis states in 10He He Chain Results (Hagen et al) [feature article in Physics Today (November 2007)] Challenge: Include 3-body force

  11. tnew(ab, ij) = V (kl, cd)told (cd, ij)told (ab, kl) k,l=1,n c,d=n+1,N Solution of coupled-cluster equation Basic numerical operation: • System of non-linear coupled algebraic equations  solve by iteration • n = number of neutrons and protons • N = number of basis states • Solution tensor memory • (N-n)**2*n**2 • Interaction tensor memory • N**4 • Operations count scaling • O(n**2*N**4) • O(n**4*N**4) with 3-body • O(n**3*N**5) at CCSDT • Many such terms exist. • Cast into a matrix-matrix multiply algorithm. • Parallel issue: block sizes of V and t.

  12. t2(ab,ij) = V(kl,cd)tij tkl cd ab kl <f cd >f Code parallelism Memory distribution across processors Partial sum t2 reside on each processor V(ab, c1, d1) t2 partial sum V(ab, c1, d2) t2 partial sum V(ab, c2, d1) t2 partial sum V(a, b, c2, d2) t2 partial sum  Global reduce (sum) t2, distribute V(ab, c1, d1) t2 partial sum V(ab, c1, d2) t2 partial sum V(ab, c2, d1) t2 partial sum V(ab, c2, d2) t2 partial sum 

  13. Future direction • Current algorithm scales to 1K processors with about 20% efficiency. Attacking problems in mass 40 region is doable with current code. • Develop algorithm that spreads both the 2-body matrix elements and the CC amplitudes (in collaboration with Ken Roche)  Enables nuclei in the mass 100 region and should scale to 100K processors (under way). • Designing further parallel algorithms that calculate nuclear properties to calculate densities and electromagnetic transition amplitudes. • Eventual time-dependent CC for fission dynamics.

  14. Contact David J. Dean Physical Sciences Directorate Nuclear Theory (865) 576-5229 deanj@ornl.gov References: Dean and Hjorth-Jensen, PRC 69, 054320 (2004); Kowalski, Dean, Hjorth-Jensen, Papenbrock, Piecuch,PRL 92, 132501 (2004); Wloch, Dean, Gour, Hjorth-Jensen, Papenbrock, Piecuch, PRL 94, 21501 (2005);Gour, Piecuch, Wloc, Hjorth-Jensen, Dean, PRC (2006); Hagen, Dean, Hjorth-Jensen, Papenbrock, PLB (2007); Hagen, Dean, Hjorth-Jensen, Papenbrock, Schwenk, PRC 76, 044305 (2007); Hagen, Papenbrock, Dean, Schwenk, Nogga, Wloch, Piecuch, PRC 76, 034302(2007); Dean, Physics Today (November 2007) 14 Dean_NCCT_SC07

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