Alex Brown PREX Aug Neutron Radii and the Neutron Equation of State

Alex Brown PREX Aug Skx-s20(5) Skyrme energy density functional Skx-s15 Skx-s20 Skx-s fm for the 208 Pb neutron skin Neutron skin = S = Δ R np = R n – R p where R n is the rms radius for neutrons and R p is the rms radius for protons

Alex Brown PREX Aug 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, (2001). Charge Densities with the Skyrme Hartree-Fock Method, W. A. Richter and BAB, Phys. Rev. C67, (2003). Tensor interaction contributions to single-particle energies, BAB, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, , (2006). Neutron Skin Deduced from Antiprotonic Atom Data, BAB, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C76, (2007). Skx family of Skyrme functionals Skx, Skx-ce Skx-csb Skx-ta, Skx-tb Skx-s15, Skx-s20, Skx-s25

Alex Brown PREX Aug Skyrme interaction (σ = α)

Alex Brown PREX Aug 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

Alex Brown PREX Aug

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

Alex Brown PREX Aug Skyrme potential

Alex Brown PREX Aug Focus on properties of spherical nuclei in a spherical potential model – fast but limited to properties of a few key nuclei 208 Pb 132 Sn 100 Sn

Alex Brown PREX Aug

Skx-s15

Alex Brown PREX Aug Skx-s20

Alex Brown PREX Aug Skx-s25

Alex Brown PREX Aug Data for Skx BE for 16 O, 24 O, 34 Si, 40 Ca, 48 Ca, 48 Ni, 68 Ni, 88 Sr, 100 Sn, 132 Sn and 208 Pb with “errors” ranging from 1.0 MeV for 16 O to 0.5 MeV for 208 Pb rms charge radii for 16 O, 40 Ca, 48 Ca, 88 Sr and 208 Pb with “errors” ranging from 0.03 fm for 16 O to 0.01 fm for 208 Pb About 50 Single particle energies with “errors” ranging from 2.0 MeV for 16 O to 0.5 MeV for 208 Pb.

Alex Brown PREX Aug Skx - fit to these data Fitted parameters: t 0 t 1 t 2 t 3 x 0 x 1 x 2 x 3 W (spin orbit term) t 0s (isospin symmetry breaking) Vary α (power of the density dependence) by hand minimum at α = 0.5 (K=270 nuclear matter incompressibility) t 0 t 0s t 1 t 2 t 3 x 0 and W well determined from exp data x 3 depends on neutron EOS x 1 and x 2 not determined

Alex Brown PREX Aug 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)

Alex Brown PREX Aug Skx Skyrme single-particle energies - implies that (m*/m)=1.00

Alex Brown PREX Aug Skx Skyrme single-particle energies

Alex Brown PREX Aug Neutron EOS and neutron skin -- x 3 How can we constrain the neutron equation of state? Friedman-Pandharipanda neutron EOS - Phys. Rev. C33, 335 (1986)

Alex Brown PREX Aug Nuclear charge densities

Alex Brown PREX Aug

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

Alex Brown PREX Aug

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

Alex Brown PREX Aug S (fm) Phys. Rev. C 76, (2007). Ratios of neutron densities (Skm*)

Alex Brown PREX Aug S (fm) K=200 MeV for nuclear matter incompressibility α = 1/6 Phys. Rev. C 76, (2007). Skx for charge density diffuseness and neutron skin

Alex Brown PREX Aug Assumption about neutron matter effective mass (m*/m)=1.00 used as a fit constraint

Alex Brown PREX Aug to to 1.0

Alex Brown PREX Aug

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

Alex Brown PREX Aug BE( 132 Sn)-BE( 100 Sn) (MeV) Exp = 278(1)

Alex Brown PREX Aug 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

Alex Brown PREX Aug

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

Alex Brown PREX Aug

122 Zr S BE (fm) (MeV) – –934.2

Alex Brown PREX Aug S (fm) =

Alex Brown PREX Aug Participating Institutions and Co-Investigators: Ames National Laboratory - Sosonkina ANL - Pieper, Wiringa, Lusk, Moré, Norris Lawrence Berkeley National Laboratory - Ng, Yang LLNL - Escher, Navratil, Ormand, Thompson Los Alamos National Laboratory - Carlson, Kawano ORNL - Arbanas, Dean, Nazarewicz, Fann, Roche, Shelton Central Michigan University - Horoi Iowa State University - Vary Michigan State University - Brown, Bogner University of North Carolina at Chapel Hill - Engel Ohio State University - Furnstahl San Diego State University - Johnson University of Tennessee - Bertulani, Papenbrock University of Washington - Bertsch, Bulgac Funding Partners: Office of Science, Advanced Scientific Computing Research, and National Nuclear Security Agency SciDAC -Building a universal energy density functional

Alex Brown PREX Aug Nuclear Structure Theory - Confrontation and Convergence (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 ….. 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 E x, 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 Each of these has its own computational challenges

Alex Brown PREX Aug Mihai Horoi Thomas Duguet Werner Richter Taka Otsuka D. Abe T. Suzuki Funding from the NSF Collaborations

Alex Brown PREX Aug Skx Skyrme Interaction

Alex Brown PREX Aug Displacement energy requires a new parameter

Alex Brown PREX Aug 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 – iterations. 10 nuclei, 8 parameters, so each fit requires 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

Alex Brown PREX Aug Rms charge radii

Alex Brown PREX Aug

114 Sn to 115 Sb proton spectroscopic factors

Alex Brown PREX Aug For Skxtb α t = -118, β t = 110 For Skxta α t = 60, β t = 110 For Skx α t = 0, β t = 0

Alex Brown PREX Aug Skx – fit to single-particle energies

Alex Brown PREX Aug Skx with G matrix tensor CHI jumps up from 0.6 to 1.5 due to spe

Alex Brown PREX Aug normal spin-orbit tensor terms

Alex Brown PREX Aug A α β δ γ ε η θ κ λ μ ν π ρ σ τ υ φ χ ψ ω A Ώ Γ Δ Λ Π Σ Φ Ψ Ω