Accurate Description of the Spinand Parity-Dependent Nuclear Level Densities

Transcription

1 Accurate Description of the Spinand Parity-Dependent Nuclear Level Densities Mihai Horoi Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA

2 Nuclear Level Densities (NLD) Hauser and Feshbach, Phys. Rev 87, 366 (1952), Rauscher at al, Phys. Rev. C 56, 1613 (1997) E x T(E,J,) = E max E min T(E,J,;E x,j x, x )#(E x,j x, x ) de x E r (E x,j,#) The Fermi Gas (FG) Model: To Back-shift or to For-shift? 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) A X + ( p,n) HF+BCS -> ρ HF+BCS (U) - Goriely Nucl. Phys. A605, (1996) Demetriou and Goriely, Nucl. Phys. A695 (2001) J 2 2 A +1 Y J 1 1 J 0 0

3 The Back-Shifted Fermi Gas Model for Nuclear Level Density =!( Z, N)

4 The Spin Cutoff Parameter 28 Si, USD Horoi, Ghita, Zelevinsky, M. Horoi Nucl. CMU Phys. A 758, 142 (2005)

5 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

6 0h 1h (0 + 2)h (1+ 3)h Nuclear Configuration Interaction H'= H + (H CoM # 3/2h) (J ) = [# CoM (NL)# (J') int ] (J ) (J ) # CoM (00)# int N max Center-of-mass spurious states pf d = 2 (2 j + 1) H = # K a + k a k + 1 k 2 # V kl;mn a + k a + l a n a m +L k l m n sd >= # C i i(jt) > i >= # C i i(jt z ) > i >= # C i i(mt z ) > i Lanczos algorithm L 1 = M # C 1 C N % ' ' & JT-scheme: OXBASH, NuShell H L 1 = 1 L 1 + # 1 L 2 L 1 L 2 HL 2 = # 1 L L 2 + # 2 L 3 L 1 L 2 L 3 HL 3 = # 2 L L 3 + # 3 L 4 L 1 L 2 L 3 L 4 L J-scheme: NATHAN, NuShellX M-scheme: Oslo-code, Antoine, MFDN, MSHELL, CMichSM, # < i H j > C j = E C i j % 1 # 1 0 L 0 ( ' * '# 1 2 # 2 0L 0 * N H k L = ' 0 # 2 3 # 3 0 * ' * M M ' & 0 L 0 # N1 * Nk )

7 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,

8 Renormalized Hamiltonians for Large N hω Excitations Model Spaces Renormalization methods: - G-matrix: Physics Reports 261, 125 (1995) - Lee-Suzuki (NCSM): PRC 61, (2000) QP = 0 QQ - V low k : PRC 65, (R) (2002) - Unitary Correlation Operator: PRC 72, (2004) Nucleon-Nucleon Potentials: - Argonne V18: PRC 56, 1720 (1997) - CD-Bonn 2000: PRC 63, (2000) - N 3 LO: PRC 68, (2003) - INOY: PRC 69, (2004) PP PQ = 0 H H = T + V i j + V i j k +L i< j i< j<k Ψ H > Ψ P =PΨ H H = e -S H e S = H 2 + H 3 + H 4 + O > e -S O e S

9 QP = 0 pf sd pf Effective Hamiltonians for One or Two Major Shells QQ PQ = 0 3-body > two-body r# % f G p 1/ 2 f 5 / 2 p 3 / 2 f 7 / 2 H = H m (monopole) + H M (multipole) 56 Ni closed shell(cs) : H m contribution H core polarization: Phys.Rep. 261, 125 (195) Brown & Richter, PRC 74, (2006)

10 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 #

11 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

12 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 ( ) (E x,j,#) = D c (J,#)G FR E,E c (J), c (J) c %conf E c (J), 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.

13 Fixed J Configura tion Centroids and Widths C. Jacquemin, Z. Phys. A 303, 135 (1981)

14 NLD Comparison: CI, Moments, HF+BCS

15 NuShellX: coupled-j code Code sources available at: Can calculate a large number of non-yrast states: ideal for level density Hardware: 3.2 GHz, dual-quad Intel, 16 GB RAM, 700 GB SATA II HD Maximum 8 threads HPC equivalent: computes five J=0 states in 56 Ni (m-scheme dim = 10 9 ) in 4 hours!!!

16 NLD of 56 Fe: CI, Moments, HF+BCS Ohio data: PRC 74, (2006)

17 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

18 NLD for the rp-process

19 0h 1h (0 + 2)h (1+ 3)h Nuclear Configuration Interaction H'= H + (H CoM # 3/2h) (J ) = [# CoM (NL)# (J') int ] (J ) (J ) # CoM (00)# int N max Center-of-mass spurious states pf d = 2 (2 j + 1) H = # K a + k a k + 1 k 2 # V kl;mn a + k a + l a n a m +L k l m n sd >= # C i i(jt) > i >= # C i i(jt z ) > i >= # C i i(mt z ) > i Lanczos algorithm L 1 = M # C 1 C N % ' ' & JT-scheme: OXBASH, NuShell H L 1 = 1 L 1 + # 1 L 2 L 1 L 2 HL 2 = # 1 L L 2 + # 2 L 3 L 1 L 2 L 3 HL 3 = # 2 L L 3 + # 3 L 4 L 1 L 2 L 3 L 4 L J-scheme: NATHAN, NuShellX M-scheme: Oslo-code, Antoine, MFDN, MSHELL, CMichSM, # < i H j > C j = E C i j % 1 # 1 0 L 0 ( ' * '# 1 2 # 2 0L 0 * N H k L = ' 0 # 2 3 # 3 0 * ' * M M ' & 0 L 0 # N1 * Nk )

20 Fixed J Restricted Configura tion Widths C. Jacquemin, Z. Phys. A 303, 135 (1981) i j i j r D, D, D i j r s

21 Removal of Spurious Center-of-Mass Excitations H'= H + {(H CM # 3/2h) } 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 nsp (E,J = 2, 0 + 2) = (E,2,0 + 2) # 2 2+J K nsp (E,J',0) # nsp (E,J',1) J K = 0 step 2 J '= 2#J K 10 B: 10 particles in s-p-sd-pf shell model space 3 J'=1 Horoi and Zelevinsky, PRL 98, (2007)

22 Nonspurious Level Density: General 3 nsp [E,J,1+ 3] = [E,J,1+ 3] # nsp [E,J',(1# K) + (3# K)] K=1J K = J K min step K=1J K = J K min step 2 nsp [E,J,0 + 2] = [E,J,0 + 2] # nsp [E,J',(0 # K) + (2 # K)] 2 2 K=1J K = 0 step 2 J '= J #J K J +J K J +J K if (1# K) < 0 % nsp [E,J',(1# K) + (3# K)] = nsp [E,J',(3 # K)] 3 J +J K J '= J #J K = [E,J,0 + 2] # nsp [E,J',(2 # K)] J '= J #J K nsp [E,J,(n) + (n + 2) +K+ (n + 2m)] = [E,J,(n) + (n + 2) +K+ (n + 2m)] n +2m K=1 n +2m J +J K # nsp [E,J',(n # K) + (n + 2 # K) +K+ (n + 2m # K)] J K = J K min step 2 J '= J #J K Horoi and Zelevinsky, PRL 98, (2007)

23 Summary and Outlook CI NLD look very promising, at least up to the particle emission threshold. More comparison with experimental data necessary. J-dependent CI NLD are very accurately described by a sum of finite range Gaussians with fixed-j centroids and widths, if one knows with good precision the energy of g.s. and yrast states. We derived explicit expression to calculate fixed-j centroids and widths for restricted class of configurations, such Nhω configurations. We found recursive formulae for calculating exactly the nonspurious level density when one knows the level densities for a restricted class of Nhω configurations.

24 Summary and Outlook - We plan to investigate level densities for the r-process path. Calculate the relevant reaction rates. - We plan to improve the efficiency of the algorithm to scale on large parallel computers. - Restrict the class of intermediate configuration for second moments σ c (J); useful for removal of center-ofmass spurious components (PRL 98, (2007)).

25 Horoi, Volya, Zelevinsky Exponential Convergence Method ( E, J ) = d ( FRG) p P p p ( J )( FRG) p ( E, J ) ( E, J ) = ( FRG)( E + E0 # E p ( J ),! p ( J )) E n = Aexp(! 2 n / b) + C

26 Exponential Convergence Method for fp-nuclei Central Michigan Shell Model (CMichSM) code E( n) = A + Be! n Exact: MeV