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Taming the Pairing Interaction in Nuclei

Taming the Pairing Interaction in Nuclei. Carlos Bertulani, Texas A&M University-Commerce. Physics – TAMU-Commerce. Tsukuba, June 26, 2008. UNEDF collaboration http://unedf.org. Towards the Universal Nuclear Energy Density Functional. isoscalar (T=0) density. isovector (T=1) density.

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Taming the Pairing Interaction in Nuclei

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  1. Taming the Pairing Interaction in Nuclei Carlos Bertulani, Texas A&M University-Commerce Physics – TAMU-Commerce • Tsukuba, June 26, 2008

  2. UNEDF collaboration http://unedf.org

  3. Towards the Universal Nuclear Energy Density Functional isoscalar (T=0) density isovector (T=1) density isoscalar spin density isovector spin density current density spin-current tensor density kinetic density kinetic spin density Example: Skyrme Functional Total ground-state DFT energy

  4. Collaboration on Mass Tables (with ODD-A) Bertsch (University of Washington, Seattle) Nazarewicz (University of Tennessee) Schunck (University of Tennessee) Stoitsov (University of Tennessee) Bertulani (Texas A&M – Commerce)

  5. Efforts with UNEDF collaboration HF + BCS HFB Typically: vNNeff = Skyrme + pairing force Zero-range pairing force:R. R. Chasman, PRC 14 (1976)1935 G. F. Bertsch and H. Esbensen, Ann. Phys. 209 (1991)327

  6. HFODD (J. Dobaczewski)http://www.fuw.edu.pl/~dobaczew/hfodd/hfodd.html 1 – Code has ~ 60,000 lines. 2 - HFB, h.o. expansion, breaks all self-consistent symmetries, and t-symmetry. 3 - Includes t-odd terms in Skyrme functional. 4 – Large continuum space included 5 - Blocking and Lipkin-Nogami (LN) implemented recently by J. Dobaczewski. 6 – People involved: Nazarewicz, Stoitsov, Schunck (ORNL), Dobaczewski (Poland), More, Sarich (Argonne), Bertulani (Tamu-Commerce)

  7. Cost of mass tables with HFODD 1 - Benchmarking in 2007 version, HFODD vs. 2.24 - 2.27 2 – several isotopic chains, different parameters 3 – Tests done at INT/Washington workstations and Jaguar/ORNL. ~ 3 GHz clock speed each processor. 4 – On Jaguar (5 GFlops/s): HFODD = 1.25 GFlops/s = 25% capacity 236U 7

  8. Improvements by Jason Sarich, incorporated in code by Jacek Dobaczewski 1 – Loops of matrix multiplications replaced by BLAS and LAPACK routines 2 – Similarly for eingevalue problem 3 – Presently: routines incorporated in a standalone code (portable), or links to BLAS and LAPACK (faster, platform optimized) 8

  9. More on HFODD • 1 - Present version 2.31 (08/2007) include optimizations (Sarich+Dobaczewski), • blocking and LN treatment of pairing (Dobaczewski+Stoitsov) • 2 – Benchmarks with version 2.31 show similar improvements in speed. • 3 – Schunck + Sarich + Stoitsov: N-Z walk on Jaguar • 4 – UT/ORNL DFT-group + Texas (Bertulani): data analysis (chi-square’s) and • graphics interface. • 5 – Goals using 3+4: • Analysis of equilibrium deformations and excitation energies • for odd-nuclei: • e.g., improving upon • Nazarewicz, Riley, Garret, • NPA 512 (1990) 61 • Preliminary mass tables for • odd-nuclei: • e.g., improving upon • Dobaczewski, Stoitsov, • Nazarewicz, nucl-th/0404077 9

  10. CPU time • Use of Jaguar (ORNL) (has 40,000 processors) • (expensive) • With HFBTHO (Mario Stoitsov): • 2737 even-even bound nuclear states (1527 g.s.) between drip lines. • 2032 converge for 500 iterations. • 404 converge for 1000 iterations • 123 converge for 2000 iterations • 152 converge for 6000 iterations. • 26 nuclei never converge. Self-consistency reached if (m=iterations, V=potentials): usually diverges linear mixing

  11. Broyden method: Broyden, Math. Comput. 19, 577 (1963). Johnson, PRC 38, 12808 (2003). Implemented by Stoitsov in HFBTHO

  12. Second tool of choice: EV8-ODD EV8(P. Bonche, H. Flocard and P.H. Heenen,CPC 171 (2005) 49) 1 – code has only 6,500 lines 2 - HF+BCS, 3-d coordinate mesh, axial symmetry, small continuum space EV8 ODD: 3 – Blocking implemented recently by Bertsch (odd N) and Bertulani (odd Z). 4 - N-Z walker by Bertsch and Bertulani. 5- Calculations so far performed at clusters in Seattle.

  13. Details: Binding energies • Example: • VNNeff = Skyrme = SLy4 • Surface pairing: gn = gp = 1000 MeV fm3 • = 0.16 fm-3 EF 5 MeV 13

  14. Goal: reduce rms by refitting interactions rms = 3.3 MeV rms = 1.7 MeV 579 even-even nuclei Bertsch, Sabbey, Uusnakki, PRC 71, 054311 (2005) 14

  15. Bertsch, Pack Forest meeting, 2007

  16. Blocking the levels Starting N=even, Z=even wavefunctions from http://gene.phys.washington.edu/ev8 Blocking added to BCS and Lipkin-Nogami  “ev8-odd” v2 →½, u → 0for one orbit and its time-conjugate partner EF See http://gene.phys.washington.edu/ ~bertsch/pedlist.html for justification.

  17. Dependence on # iterations We’ve used 50 iterations for the mass tables, corresponding to 100 keV error. Using 100-150 iterations, we will expect To reduce this even further to 50 keV.

  18. Pairing Conundrum

  19. Pairing Measure or ? Nilsson, Prior, Mat. Fys. Medd. K. dan. Vidensk.Selsk. 32, 16 (1961) • From experiment: • Δ(3) larger for (-1)N = +1 • Δ(3) smaller for (-1)N = -1 • Δ(4) reflects averageof Δ(3) (N) and Δ(3) (N-1) • (Δ(4) no additional information!)

  20. Pairing vs Level Properties Fermi energy (λ= ∂B/ ∂N) s.p. level density (g(e)= dN/ ∂e) N degenerate shell λ λ does not vary with N  Δ(3) = 0 en+1 measure of gap in single-particlespectrum valence shell full en N=2n λ Jahn-Teller mechanism: spherical symmetry spontaneously broken (2j+1)  double-degenerate orbits Δ(3) alternates for (-1)N = + and - (-1)N = +1, dλ/dN ≈ en+1 – en Δ(3) = (en+1 – en)/2

  21. Pairing vs Level Properties Macroscopic-microscopic model: Satula, Dobaczewski, Nazarewicz, PRL 81,3599 (1998). Rutz, Bender, Reinhard, Maruhn, PL B 468, 1 (1999).

  22. Experiment Summary Can we do better than LDM? experiment Theory Best probe: Δ(3), Δ(4), Δ(5) ….? (not clear) We use Δ(3)(odd) Table by Bertsch

  23. Best of HFB mass tables (e.g. Goriely et al.) Accuracy rrms (2135 nuc) HFB-1 masses: M*s=1.05, volume pairing806 keV HFB-2 masses: M*s=1.04, improved pairing descr.660 keV HFB-3 masses: M*s=1.12, vol+surf pairing (h=a=0.5) 639 keV HFB-4 masses: M*s=0.92, vol. pairing 661 keV HFB-5 masses: M*s=0.92, vol+surf pairing 655 keV HFB-6 masses: M*s=0.80, vol. pairing 666 keV HFB-7 masses: M*s=0.80, vol+surf pairing658 keV HFB-8 masses: M*s=0.80, vol. pairing,PLN (part.nbr proj.)641 keV But Δ(3) > 0.6 MeV i.e. LDM is better!

  24. Odd-even staggering S(N,Z) = M(N-1,Z) + Mn - M(N, Z) Sn isotopes

  25. Separation energy residuals S(N,Z) = M(N-1,Z) + Mn - M(N, Z)

  26. Dependence on pairing strength Skyrme = SLy4Vpairing = g . (r12) . [1- (r)/]  = 0.16 fm-3 search for gn = gp From even-even to even-odd and odd-odd (absolute value): rms residual for LDM with known exp. data = 0.3 MeV

  27. Odd-even staggering with 967 masses Visually, HF-BCS looks like it does better! True?

  28. Odd-even staggering: close view Sn isotopes  = ½ . (-1)N [S(N,Z) – S(N-1, Z)]

  29. Theory ▀Exp. o HFBTHO + Lipkin-Nogami BCS (with ev8)

  30. Wrapping up: Two extreme cases. Energies are in MeV. Note: 59Ni with a starting deformation point is better. But not substantial. RMS residuals of Δ(3)o . There are 443 in the data set. Energies are in MeV. • Fit to even-Z (to avoid np pairing) • Require N > Z+1 to avoid Wigner energy • Surface peaked pairing almost as good as LD • Volume pairing poorer

  31. To be clarified • Extension to isotones • Why can’t we do much better than LDM for pairing gap residuals? • Is pairing interaction too simplified for the fitting purpose? • Can pairing correlation properties be isolated from the full functional?

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