Neutron Radii and the Neutron Equation of State

1. Neutron Radii and the Neutron Equation of State

2. Skx-s20(5) Skyrme energy density functional Skx-s15 Skx-s20 Skx-s25 0.15 0.20 0.25 fm for the 208Pb neutron skin Neutron skin = S = ΔRnp = Rn – Rp where Rn is the rms radius for neutrons and Rp is the rms radius for protons

3. 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, Skx, Skxc BAB, Phys. Rev. C58, 220 (1998). Displacement Energies with the Skyrme Hartree-Fock Method, BAB, W. A. Richter and R. Lindsay, Phys. Lett. B483, 49 (2000). Neutron Radii in Nuclei and the Neutron Equation of State, BAB, Phys. Rev. Lett. 85, 5296 (2000). S. Typel and BAB, Phys. Rev. C64, 027302 (2001). Charge Densities with the Skyrme Hartree-Fock Method, W. A. Richter and BAB, Phys. Rev. C67, 034317 (2003). Tensor interaction contributions to single-particle energies, BAB, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, 061303, (2006). Neutron Skin Deduced from Antiprotonic Atom Data, BAB, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C76, 034305 (2007). Skx family of Skyrme functionals Skx, Skx-ce Skx-csb Skx-ta, Skx-tb Skx-s15, Skx-s20, Skx-s25

4. Skyrme interaction (σ = α)

5. Skyrme energy density functional Nuclear matter is this without the surface terms1 Nuclear matter (N=Z) depends on the t’s Symmetry energy and neutron matter also depends on the x’s

7. Skyrme single-particle wave equation Effective mass m*(r)/m

8. Skyrme potential

9. Focus on properties of spherical nuclei in a spherical potential model – fast but limited to properties of a few key nuclei 208Pb 100Sn 132Sn

11. Skx-s15

12. Skx-s20

13. Skx-s25

14. 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.

15. 1998 - Skx - fit to these data Fitted parameters: t0 t1 t2 t3 x0 x1 x2 x3 W (spin orbit term) t0s (isospin symmetry breaking) Vary α (power of the density dependence) by hand minimum at α = 0.5 (K=270 nuclear matter incompressibility) t0 t0s t1 t2 t3 x0 and W well determined from exp data x3 depends on neutron EOS x1 and x2 not determined

16. Skx – single-particle energies 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 (Dobascewski, Duguet)

17. Skx Skyrme single-particle energies - implies that (m*/m)=1.00

18. Skx Skyrme single-particle energies

19. 1998 - Neutron EOS and neutron skin -- x3 How can we constrain the neutron equation of state? • Friedman-Pandharipanda neutron EOS - Phys. Rev. C33, 335 (1986)

20. Nuclear charge densities

23. Neutron density for 208Pb Shows the shell layers (Dashed line is the proton density)

25. Ratios of charge densities (Skm*) Diffuseness of the charge density is correlated with nuclear matter incompressibility Best fit to charge density requires K=200-230 MeV Skx-s20(5) takes α = 1/6 Which gives K=200 Phys. Rev. C 76, 034305 (2007).

26. Ratios of neutron densities (Skm*) S (fm) 0.25 0.20 0.15 Phys. Rev. C 76, 034305 (2007).

27. Skx for charge density diffuseness and neutron skin S (fm) 0.25 0.20 0.15 K=200 MeV for nuclear matter incompressibility α = 1/6 Phys. Rev. C 76, 034305 (2007).

28. Assumption about neutron matter effective mass (m*/m)=1.00 used as a fit constraint

29. -0.5 to 0.5 -1.0 to 1.0

31. Ab-initio low-density value A. Gezerlis and J. Carlson, PRC77, 032801 (2008) also important to get low-density part right (Andrew Steiner…)

32. BE(132Sn)-BE(100Sn) (MeV) 277 278 282 283 284 291 296 299 Exp = 278(1)

33. K=270 α =1/2 S=0.25 K=200 α = 1/6 S=0.20 So next step would be to introduce two α values One for nuclear matter and another for the symmetry potential

35. 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.

38. 122Zr S BE (fm) (MeV) 0.15 -928.6 0.20 –931.3 0.25 –934.2

39. S (fm) = 0.12 0.16

40. SciDAC -Building a universal energy density functional • Participating Institutions and Co-Investigators:Ames National Laboratory - SosonkinaANL - Pieper, Wiringa, Lusk, Moré, NorrisLawrence Berkeley National Laboratory - Ng, Yang LLNL - Escher, Navratil, Ormand, ThompsonLos Alamos National Laboratory - Carlson, KawanoORNL - Arbanas, Dean, Nazarewicz, Fann, Roche, SheltonCentral Michigan University - HoroiIowa State University - VaryMichigan State University - Brown, BognerUniversity of North Carolina at Chapel Hill - EngelOhio State University - FurnstahlSan Diego State University - JohnsonUniversity of Tennessee - Bertulani, PapenbrockUniversity of Washington - Bertsch, Bulgac • Funding Partners: Office of Science, Advanced Scientific Computing Research, and National Nuclear Security Agency

41. Nuclear Structure Theory - Confrontation and Convergence • Good – most “fundamental” • Bad – only for light nuclei, need NNN parameters, “complicated wf” • Good – applicable to more nuclei, 150 keV rms, “good wf” • Bad – limited to specific mass regions and Ex, need effective spe and tbme for good results • Good – applicable to all nuclei • Bad – limited mainly to gs and yrast, 600 keV rms mass, need interaction parameters • Good – simple understanding of special situations • Bad – certain classes of states, need effective hamiltonian • (AI) Ab initio methods with NN and NNN • (CI) Shell model configuration interactions with effective single-particle and two-body matrix elements • (DFT) Density functionals plus GCM… My examples with Skyrme Hartree-Fock (Skx) • Cluster models, group theoretical models ….. Each of these has its own computational challenges

42. Collaborations • Mihai Horoi • Thomas Duguet • Werner Richter • Taka Otsuka • D. Abe • T. Suzuki • Funding from the NSF

43. Skx Skyrme Interaction

44. Displacement energy requires a new parameter

45. 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

46. Rms charge radii

49. 114Sn to 115Sb proton spectroscopic factors

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