Alex Brown UNEDF Feb Strategies for extracting optimal effective Hamiltonians for CI and Skyrme EDF applications

Alex Brown UNEDF Feb sd pf

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

Alex Brown UNEDF Feb spe 63 tbme for the sd-shell

Alex Brown UNEDF Feb Starting Hamiltonian Renormalized NN

Alex Brown UNEDF Feb Sigma_th = 100 keV

Alex Brown UNEDF Feb

About 5 iterations needed

Alex Brown UNEDF Feb

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

Alex Brown UNEDF Feb USDA 170 keV rms

Alex Brown UNEDF Feb USDB 137 keV rms

Alex Brown UNEDF Feb USDA ground state energy differences MeV theory underbound oxygen beyond N=16 all unbound

Alex Brown UNEDF Feb USDA 170 keV rms for 608 levels 290 keV rms for tbme (4.1% of largest)

Alex Brown UNEDF Feb USDB 137 keV rms for 608 levels 376 keV rms for tbme

Alex Brown UNEDF Feb USD 150 keV rms for 380 levels 450 keV rms for tbme

Alex Brown UNEDF Feb 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.

Alex Brown UNEDF Feb TBME depend on the target nucleus and model space Comparison of 24 O (with proton p 1/2 ) and 22 C (without p 1/2 )

Alex Brown UNEDF Feb USD G Effective spe for the oxygen isotopes

Alex Brown UNEDF Feb A tour of the sd shell on the web

Alex Brown UNEDF Feb Positive parity states for 26 Al

Alex Brown UNEDF Feb Positive parity states for 26 Mg

Alex Brown UNEDF Feb g s p = g s n = g l p = 1 g l n = 0

Alex Brown UNEDF Feb g s p = g s n = g l p = 1 g l n = 0

Alex Brown UNEDF Feb g s p = g s n = g l p = 1 g l n = 0 g s p = g s n = g l p = 1 g l n = 0

Alex Brown UNEDF Feb g s p = g s n = g l p = g l n =

Alex Brown UNEDF Feb g s p = g s n = g l p = 1 g l n = 0

Alex Brown UNEDF Feb g s p = g s n = g l p = g l n =

Alex Brown UNEDF Feb sd pf

Alex Brown UNEDF Feb jj44 means f 5/2, p 3/2, p 1/2, g 9/2 orbits for protons and neutrons

Alex Brown UNEDF Feb USDA 170 keV rms for 608 levels 290 keV rms for tbme (4.1% of largest)

Alex Brown UNEDF Feb 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

Alex Brown UNEDF Feb 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, (2003). Tensor interaction contributions to single-particle energies, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, , (2006). Neutron Skin Deduced from Antiprotonic Atom Data, B. A. Brown, G. Shen, G. C. Hillhouse, J. Meng and A. Trzcinska, Phys. Rev. C76, (2007).

Alex Brown UNEDF Feb 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. Constraint to FP curve for the neutron EOS

Alex Brown UNEDF Feb Skx - fit to these data Fitted parameters: t 0 t 1 t 2 t 3 x 0 x 1 x 2 x 3 W W x (extra spin orbit term) t 0s (isospin symmetry breaking) Vary α by hand (density dependence) minimum at α = 0.5 (K=270) t 0 t 0s t 1 t 2 t 3 x 0 and W well determined from exp data x 3 constrained from neutron EOS W x x 1 and x 2 poorly determined

Alex Brown UNEDF Feb 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 UNEDF Feb 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)

Alex Brown UNEDF Feb

Displacement energy requires a new parameter

Alex Brown UNEDF Feb Rms charge radii

Alex Brown UNEDF Feb Skx Skyrme Interaction

Alex Brown UNEDF Feb Skx Skyrme Interaction

Alex Brown UNEDF Feb Skx Skyrme Interaction

Alex Brown UNEDF Feb Neutron EOS related to neutron skin -- x 3 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

Alex Brown UNEDF Feb

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 UNEDF Feb Skx – fit to single-particle energies

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

Alex Brown UNEDF Feb normal spin-orbit tensor terms

Alex Brown UNEDF Feb

S (fm) K=200 MeV for nuclear matter incompressibility Phys. Rev. C 76, (2007). Skx for charge density diffuseness and neutron skin

Alex Brown UNEDF Feb Zr S BE (fm) (MeV) – –934.2

Alex Brown UNEDF Feb S (fm) =

Alex Brown UNEDF Feb Neutron matter effective mass can constrain x 1 and x 2

Alex Brown UNEDF Feb Phys. Rev. C 76, (2007).

Alex Brown UNEDF Feb O 34 Si 42 Si 48 Ca 24 O

Alex Brown UNEDF Feb Skxta/b: Skx with the inclusion of the rho+pi tensor in fits to spe, BE and radii, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, (R) (2006).. Z=8

Alex Brown UNEDF Feb Skxta/b: Skx with the inclusion of the rho+pi tensor in fits to spe, BE and radii, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, (R) (2006).. Z=8

Alex Brown UNEDF Feb Skxta/b: Skx with the inclusion of the rho+pi tensor in fits to spe, BE and radii, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, (R) (2006).. N=16

Alex Brown UNEDF Feb In 28 O the d 3/2 is bound by 0.2 MeV

Alex Brown UNEDF Feb Skxta/b: Skx with the inclusion of the rho+pi tensor in fits to spe, BE and radii, B. A. Brown, T. Duguet, T. Otsuka, D. Abe and T. Suzuki, Phys. Rev. C 74, (R) (2006).. N=20

Alex Brown UNEDF Feb N=20

Alex Brown UNEDF Feb N=28

Alex Brown UNEDF Feb N=28

Alex Brown UNEDF Feb

114 Sn to 115 Sb proton spectroscopic factors

Alex Brown UNEDF Feb

32 Cl 33 Ar p 32 Cl(p,gamma) 33 Ar Rp-process path Experiment needed to get energy of states in 33 Ar to 5 keV accuracy. Theory needed to get proton decay widths to ground and excited states of 32 Cl and gamma widths for 33 Ar 32 P 33 P R. R. C. Clement et al., Phys. Rev. Lett. 92, (2004) H. Schatz, et al., Phys. Rev. C 72, (2005) Role of excited state in other nuclei - Janina Grineviciute

Alex Brown UNEDF Feb Full pf space for 56 Ni with GXPF1A Hamiltonian (order of one day computing time) M. Horoi, B. A. Brown, T. Otsuka, M. Honma and T. Mizusaki, Phys. Rev. C 73, (R) (2006).

Alex Brown UNEDF Feb e p =1 e n =0

Alex Brown UNEDF Feb e p =1.37 e n =0.45

Alex Brown UNEDF Feb e p =1.37 e n =0.45e p =1.10 e n =0.68

Alex Brown UNEDF Feb e p =1 e n =0

Alex Brown UNEDF Feb e p =1.37 e n =0.45

Alex Brown UNEDF Feb |g a /g v |=1.26

Alex Brown UNEDF Feb |g a /g v |=0.97

Alex Brown UNEDF Feb 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 UNEDF Feb USDB ground state energy differences MeV theory underbound

Alex Brown UNEDF Feb

PRL98, (2007) RIKEN PRL (2007) NSCL Theory has 10 eV width

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

Alex Brown UNEDF Feb Monopole interactions

Alex Brown UNEDF Feb Monopole interaction changes