Strategies for extracting optimal effective Hamiltonians for CI and Skyrme EDF applications

1. Strategies for extracting optimal effective Hamiltonians for CI and Skyrme EDF applications

2. pf sd

3. 77 gs BE and 530 excited states, 137 keV rms B. A. Brown and W. A. Richter, Phys. Rev. C 74, 034315 (2006). Number of data for each nucleus

4. 3 spe 63 tbme for the sd-shell

5. Starting Hamiltonian Renormalized NN

6. Sigma_th = 100 keV

8. About 5 iterations needed

13. Linear combinations of two-body matrix elements USDA 30 USDB 56 rms for the tbme rms for the 608 levels

14. USDA 170 keV rms

15. USDB 137 keV rms

16. USDA ground state energy differences MeV theory underbound oxygen beyond N=16 all unbound

17. USDA 170 keV rms for 608 levels 290 keV rms for tbme (4.1% of largest)

18. USDB 137 keV rms for 608 levels 376 keV rms for tbme

19. USD 150 keV rms for 380 levels 450 keV rms for tbme

20. A few notes • Need a realistic model for the starting and background Hamiltonians • What do we use for undetermined linear combinations? starting Hamiltonian or Hamiltonian from previous iteration Not obvious that the same (universal) Hamiltonian should apply to all sd-shell nuclei – probably a special case – we now know that other situations (like O vs C) require an explicit change in the TBME due to changes coming from core-polarization of difference cores.

21. TBME depend on the target nucleus and model space Comparison of 24O (with proton p1/2) and 22C (without p1/2)

22. Effective spe for the oxygen isotopes USD G

23. A tour of the sd shell on the web

24. Positive parity states for 26Al

25. Positive parity states for 26Mg

26. gsp = 5.586 gsn = -3.826 glp = 1 gln = 0

27. gsp = 5.586 gsn = -3.826 glp = 1 gln = 0

28. gsp = 5.586 gsn = -3.826 glp = 1 gln = 0 gsp = 5.586 gsn = -3.826 glp = 1 gln = 0

29. gsp = 5.127 gsn = -3.543 glp = 1.147 gln = -0.090

30. gsp = 5.586 gsn = -3.826 glp = 1 gln = 0

31. gsp = 5.127 gsn = -3.543 glp = 1.147 gln = -0.090

32. pf sd

33. jj44 means f5/2, p3/2, p1/2, g 9/2 orbits for protons and neutrons

34. USDA 170 keV rms for 608 levels 290 keV rms for tbme (4.1% of largest)

35. Why do we need to modify the renormalized G matrix for USD • Is the renormalization adequate • Difference between HO and finite well • Effective three-body terms • Real three-body interactions

36. Skyrme parameters based on fits to experimental data for properties of spherical nuclei, including single-particle energies, and nuclear matter. A New Skyrme Interaction for Normal and Exotic Nuclei, B. A. Brown, Phys. Rev. C58, 220 (1998). Displacement Energies with the Skyrme Hartree-Fock Method, B. A. Brown, W. A. Richter and R. Lindsay, Phys. Lett. B483, 49 (2000). Neutron Radii in Nuclei and the Neutron Equation of State, B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000). Charge Densities with the Skyrme Hartree-Fock Method, W. A. Richter and B. A. Brown, Phys. Rev. C67, 034317 (2003). Tensor interaction contributions to single-particle energies, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, 061303, (2006). Neutron Skin Deduced from Antiprotonic Atom Data, B. A. Brown, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C76, 034305 (2007).

37. Data for Skx • BE for 16O, 24O, 34Si, 40Ca, 48Ca, 48Ni, 68Ni, 88Sr, 100Sn, 132Sn and 208Pb with “errors” ranging from 1.0 MeV for 16O to 0.5 MeV for 208Pb • rms charge radii for 16O, 40Ca, 48Ca, 88Sr and 208Pb with “errors” ranging from 0.03 fm for 16O to 0.01 fm for 208Pb • About 50 Single particle energies with “errors” ranging from 2.0 MeV for 16O to 0.5 MeV for 208Pb. Constraint to FP curve for the neutron EOS

38. Skx - fit to these data Fitted parameters: t0 t1 t2 t3 x0 x1 x2 x3 W Wx (extra spin orbit term) t0s (isospin symmetry breaking) Vary α by hand (density dependence) minimum at α = 0.5 (K=270) t0 t0s t1 t2 t3 x0 and W well determined from exp data x3 constrained from neutron EOS Wx x1 and x2 poorly determined

39. Skx - fit to all of these data Fit done by 2p calculations for the values V and V+epsilon of the p parameters. Then using Bevington’s routine for a “fit to an arbitrary function”. After one fit, iterate until convergence – 20-50 iterations. 10 nuclei, 8 parameters, so each fit requires 2000-5000 spherical calculations. Takes about 30 min on the laptop. Goodness of fit characterized by CHI with best fit obtained for “Skx” with CHI=0.6

40. Skx - fit to all of these data Single-particle states from the Skyrme potential of the close-shell nucleus (A) are associated with experimental values for the differences -[BE(A) - BE(A-1)] or = -[BE(A+1)-BE(A)] based on the HF model The potential spe are typically within 200 keV of those calculated from the theoretical values for -[BE(A) - BE(A-1)] or = -[BE(A+1)-BE(A)] No time-odd type interactions, but time-odd contribution to spe are typically not more than 200 keV (Thomas Duguet)

42. Displacement energy requires a new parameter

43. Rms charge radii

44. Skx Skyrme Interaction

45. Skx Skyrme Interaction

46. Skx Skyrme Interaction

47. Neutron EOS related to neutron skin -- x3 How can we constrain the neutron equation of state? • We know the proton density from electron scattering • The neutron skin is S = R_p – R_n where R are the rms radii

49. For Skx α t = 0, β t = 0 For Skxta α t = 60, β t = 110 For Skxtb α t = -118, β t = 110

50. Skx – fit to single-particle energies