Novel High Performance Computational Aspects of the Shell Model Approach for Medium Nuclei

Transcription

1 Novel High Performance Computational Aspects of the Shell Model Approach for Medium Nuclei Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA Collaborators: R. Senkov, M. Scott, V. Zelevinsky, B.A. Brown Support by NSF grant PHY , and DOE grant DF-FC02-09ER41584 UNEDF

2 Overview New shell model ideas for the low-lying states Novel developments with shell model nuclear level densities Potential impact of the shell model nuclear level densities for nuclear astrophysics predictions

3 0 ω 1 ω (0 + 2) ω (1+ 3) ω Nuclear Configuration Interaction H'= H + β(h CoM 3/2 ω) Ψ (J ) = [Φ CoM (NL)Φ (J') int ] (J ) (J ) Φ CoM (00)Φ int N max Center-of-mass spurious states pf d = 2 (2 j + 1) H = ε K a + k a k + 1 k 2 V kl;mn a + k a + l a n a m + k l m n sd α α >= C i i(jt) > i α α >= C i i(jt z ) > i α α >= C i i(mt z ) > i JT-scheme: OXBASH, NuShell J-scheme: NATHAN, NuShellX M-scheme: Oslo-code, Antoine, MFDN, MSHELL, CMichSM, j α α < i H j > C j = E α C i Lanczos algorithm: provides few lower energies, especially the M-scheme codes.

4 p s sd s pf s pf 5/2 g 9/ g 7/2 sdh 11/2-5x Example: 76 Sr pf 5/2 g 9/2 dimension Current limit: > np valence s.p. states Extensions: Truncations, Exponential Convergence Method, Coupled Clusters, Projected CI,

5 Effective Hamiltonians QP = 0 pf sd pf QQ PQ = 0 H Δ 3-body > two-body core polarization: Phys.Rep. 261, 125 (1995) PRC 74, (2006), 78, (2008) 531 excited states RMS: sd, N3LO ME 2 RMS st order 2nd order 2nd + A-dep 2nd + d3s1 USDA USDB

6

7

8 GXPF1A Effective Interaction: f 7/2 p 3/2 p 1/2 f 5/2 Renormalized G-matrix GXPF1 Phys.Rev. C 69, (2004) 699 energies, 87 nuclei, rms=168 kev GXPF1 GXPF1A M. Honma et al, ENAM04 5 matrix elements adjusted for N=34 56 Ni states J #

9 Koeln Low Spin States in 56 Ni: complete spectroscopy Configuration Interaction (CI) M. Horoi et al., Phys.Rev. C 73, (R) (2006) J 6 Exp. Koeln Theo: CI GXPF1A LBNL high spin D.Rudolf et al.,prl 88,1999 M. Horoi et al., in preparation p e eff n =1.5 e eff = 0.5

10 Angular Momentum Projected CI (PCI) Deformed Slater Determinant Deformed Nilsson Single Particle operator Spherical H.O. Nuclear State Wave-function: Shell Model Hamiltonian Angular Momentum Projection

11 Extension of PCI: Gao, Horoi PRC 79, (2009); PRC 80, (2009). Interaction taken from: K. Kaneko, M. Hasegawa, and T. Mizusaki, Phys. Rev. C 70, (R) (2004). n=20, E cut =1keV Conclusions: 1, The physics of shape coexistence in 68 Se can be clearly seen from the PCI basis. 2, With new method of basis selection the PCI energies are very close to those of full CI for both positive parity and negative parity states.

12 Nuclear Level Densities (NLD) E x E r ρ(e x,j,π) (n,γ), (n,xn), (n, ʹ n ), (n, p), (n, f ), A X + ( p,n) Hauser and Feshbach, Phys. Rev 87, 366 (1952) J 2 π 2 A +1 Y J 1 π 1 J 0 π 0 T m (E,J,π) = E max E min T(E,J,π;E x,j x,π x )ρ(e x,j x,π x ) de x

13 Where are NLD Needed: Nuclear Astrophysics Binary stars XRB: Sirius o SN 1987 A o

14 The Back-Shifted Fermi Gas Model for Rauscher at al, Phys. Rev. C 56, 1613 (1997) Nuclear Level Density δ = Δ( Z, N)

15 The Spin Cutoff Parameter 28 Si, USD T. von Egidy & D. Bucurescu, Phys.Rev. C 80, (2009) Horoi, Ghita, Zelevinsky, Nucl. Phys. A 758, 142 (2005)

16 Other Models of Nuclear Level Densities ρ(e x, J, π) = (1/2) F(U, J) ρ FG (U) U = E x - Δ HF+BCS -> ρ HF+BCS (U) - Goriely Nucl. Phys. A605, (1996) Demetriou and Goriely, Nucl. Phys. A695 (2001) HFB+Combinatorial: ρ(e x, J, π) - S. Hilaire, J.P. Delaroche and A.J. Koning, Nucl. Phys. A632, 417 (1998). - S. Hilaire, J.P. Delaroche and M. Girod, Eur. Phys. J. A12 (2001) S. Hilaire and S. Goriely, Nucl. Phys. A779 (2006)

17 Experimental Data Sc 50 V, 51 V 56 Fe, 57 Fe 93 Mo, 94 Mo, 95 Mo, 96 Mo, 97 Mo, 98 Mo 148 Sm, 149 Sm 160 Dy, 161 Dy, 162 Dy 167 Er, 168 Er 47 Ti 56 Fe, 57 Fe 60 Ni, 60 Co PRC 74, (2006) 56 Fe 170 Yb, 171 Yb, 172 Yb

18 Experimental Data: Neutron Resonances ρ(b n ) = 1000 D 0 f rms ~ 2 Hilaire and Goriely, NPA 799, 63 (2006)

19 Accurate Nuclear Level Densities Comparison of: 1. CI, 2. HF+BCS www-astro.ulb.ac.be/html/nld.html 3. experimental data 28 Si Complete spectroscopy: sdshell nuclei Conclusions: - HF+BCS overestimates the data 26 Al - CI accurately describes the data

20 M. Horoi et al. : PRC 67, (2003), PRC 69, (R) (2004), NPA 785, 142 (2005). PRL 98, (2007) NLD and Statistical Spectroscopy Configurations: e.g. 4 particles in sd d3 d5 s preserve rotational invariance and parity ( ) ρ(e x,j,π) = D c (J,π)G FR E,E c (J),σ c (J) c conf E c (J),σ c (J) Tr SDc < M H q M > SDc E x = E E g.s. E c (J), σ c (J): computational intensive Configurations can be calculated in parallel 28 Si π = + staircase: CI, USD E g.s. from CI, PCI, Exponential Convergence Method (PRL 82, 2064 (1999)), CC, etc.

21 Fixed J Configura tion Centroids and Widths C. Jacquemin, Z. Phys. A 303, 135 (1981) D k (n m M) = ( 1) t k M. Horoi D((n CMU t)( k t n t 1 + +t k =t m t 1 U 1 t k U k )(M t 1 m k1 t k m kk ))

22 ( ) ρ(e x,j,π) = D c (J,π)G FR E,E c (J),σ c (J) c conf Study of Errors E c ησ c E c + ησ c E c ε rms = exp 1 N N i ln ρ MOM i CI ρ i 2 1 error relative to CI η = 2.6

23 NLD Comparison: CI, Moments, HF+BCS

24 NLD of 56 Fe: CI, Moments, HF+BCS Ohio data: PRC 74, (2006) Theory: pf model space, GXPF1A interaction

25 Ratio of unnatural to natural NLD of different parities at low energies ρ(e x, J, π) = (1/2) F(U, J) ρ FG (U) U = E x - Δ Equal contribution to both parities Remedy by Alhassid, Bertsch, Liu, Nakada, PRL 84, 4313 (2000) + Basel group (Rauscher) HF+BCS moments CI (NuShellX) Configurations: e.g. 4 particles in fpg f5 p3 p1 g9 π preserve rotational invariance and parity

26 NLD for the rp-process f 5/2 p g 9/2 model space

27 Scaling of the MPI JMOMENTS Code Domain decomposition: many-body configurations Algorithm: Dynamical Load Balancing Machine: Franklin/NERSC Nucleus / model space Isospin configs PN - configs 52Fe/pf Fe/pf Ge/pfg9/ Strong scaling t

28 M. Horoi et al. : PRC 67, (2003), PRC 69, (R) (2004), NPA 785, 142 (2005). PRL 98, (2007) NLD and Statistical Spectroscopy: PN Formalism ( ) ρ(e x,j,π) = D c (J,π)G FR E,E c (J),σ c (J) c conf Configurations: e.g. 4P and 4N in pf model space Pf7 Pp3 Pf5 Pp1 Nf7 Np3 Nf5 Np preserve rotational invariance and parity E c (J),σ c (J) Tr SDc < M H q M > SDc E x = E E g.s. E c (J), σ c (J): computational intensive Many more configurations, which can be more efficiently be calculated in parallel E g.s. from CI, PCI, Exponential Convergence Method (PRL 82, 2064 (1999)), CC, etc, or determined. D k (PN m M) = M P +M N =M D k P (P m M P )D k N (N m M N )

29 JMOMENTS PN Code for Nuclear Level Density Domain decomposition: many-body configurations Algorithm: Dynamical Load Balancing Machine: Franklin/NERSC Strong scaling T calc = 1 min!!! waiting point nuclei

30 Results E gs (pf) = MeV E gs (pf+g9/2) t = MeV E gs (pf+g9/2) s = MeV 2.5 MeV E gs (pf) = MeV E gs (pf+g9/2) s = MeV

31 NLD and Hauser-Feshbach From A. Voinov et al., PRC 76, (2007) θ = He + 58 Fe E 3 He =10 MeV 61 Ni * compund θ =150 0 d + 59 Co E d = 7.5 MeV

32 talys 1.2 : NLD and Hauser-Feshbach 58 Fe( 3 He, α) θ = Co( d, α) θ =150 0

33 NLD: reaction rates talys 1.2 : Rauscher & Thielemann ADNDT 75, 1 (2000) G(T) = ( 2I µ +1) /( 2I 0 +1)e E x µ / kt 2I π +1 µ I,π ( ) / 2I 0 +1 ( )ρ(e x,i,π)e E x / kt 50

34 NLD: rp-process See also PRC 75, (R) (2007) From. P. Shury et al., PRC 75, (2007) Rauscher & Thielemann ADNDT 75, 1 (2000)

35 Removal of Spurious Center-of-Mass Excitations H'= H + β {(H CM 3/2 ω) } ρ(e, J, 0+2): total density in a model space including all 0+2 h.o. excitations ρ nsp (E, J, 0+2): center-of-mass excitations removed ρ nsp (E,J = 2, 0 + 2) = ρ(e,2,0 + 2) 2 2+J K ρ nsp (E,J',0) ρ nsp (E,J',1) J K = 0 step 2 J '= 2 J K 3 J'=1 10 B: 10 particles in s-p-sd-pf shell model space Horoi and Zelevinsky, PRL 98, (2007)

36 Fixed J Restricted Configuration Widths i j i j r D, D, D i j r s

37 Moments method with restricted configurations Si in sd model space: only 4p-4p configurations included J = 0 J =1 J = 2 J = 3 J = 4 J = 5 J = 6 J = 7 J = 8

38 Extreme Computing Goals Reaction DB TENDL-2009: MCNPX: mcnpx.lanl.gov

39 Summary and Outlook Shell model techniques describe and predict a large number of data in medium and heavy nuclei. BSFG and mean-field models describe the existing data within a factor of 10 (average 2), but they can calculate NLD for all nuclei. CI J-dependent NLD seem to be more accurate, but can only be obtained for a limited number of nuclei. J-dependent moments method reproduces very well the CI J-dependent NLD and could provide accurate NLD for a much larger class of nuclei. Work to integrated SM NLD with reaction codes is in progress.