Shalom Shlomo Texas A&M University

1. Shalom Shlomo Texas A&M University Determining Modern Energy Functional for Nuclei And The Status of The Equation of State of Nuclear Matter

2. Outline • Introduction. Collective States, Equation of State, 2. Energy Density Functional. Hartree-Fock Equations (HF), Skyrme Interaction Simulated Annealing Method, Data and Constraint • Results and Discussion. • HF-based Random-Phase-Approximation (RPA). Fully Self Consitent HF-RPA, Hadron Excitation of Giant Resonances, Compression Modes and the NM EOS, Symmetry Energy Density 5. Results and Discussion. 6. Conclusions.

3. Introduction • Important task: Develop a modern Energy Density Functional (EDF), E = E[ρ], with enhanced predictive power for properties of rare nuclei. • We start from EDF obtained from the Skyrme N-N interaction. • The effective Skyrme interaction has been used in mean-field models for several decades. Many different parameterizations of the interaction have been realized to better reproduce nuclear masses, radii, and various other data. Today, there is more experimental data of nuclei far from the stability line. It is time to improve the parameters of Skyrme interactions. We fit our mean-field results to an extensive set of experimental data and obtain the parameters of the Skyrme type effective interaction for nuclei at and far from the stability line.

4. Map of the existing nuclei. The black squares in the central zone are stable nuclei, the broken inner lines show the status of known unstable nuclei as of 1986 and the outer lines are the assessed proton and neutron drip lines (Hansen 1991).

5. Equation of state and nuclear matter compressibility The symmetric nuclear matter (N=Z and no Coulomb) incompressibility coefficient, K, is a important physical quantity in the study of nuclei, supernova collapse, neutron stars, and heavy-ion collisions, since it is directly related to the curvature of the nuclear matter (NM) equation of state (EOS), E = E(ρ). ρ = 0.16 fm-3 E/A [MeV] E/A = -16 MeV ρ [fm-3]

6. Macroscopic picture of giant resonance L = 0 L = 2 L = 1

7. A Classical Picture of the Breathing Mode In the classical description of the breathing mode, the nucleus is modeled after a drop of liquid that oscillates by expanding and contracting about its spherical shape. We consider the isoscalar breathing mode in which the neutrons and protons move in phase (∆T=0, ∆S=0).

8. In the scaling model, we have the matter density oscillates as We consider small oscillations, so є is very close to zero (≤ 0.1). Performing a Taylor expansion of density we obtain,,

9. We have, Where is equal to This nicely agrees with the transition density obtained from microscopic (HF based RPA) calculations

10. Modern Energy Density Functional Within the HF approximation: the ground state wave function In spherical case HF equations: minimize

11. Skyrme interaction For the nucleon-nucleon interaction we adopt the standard Skyrme type interaction are 10 Skyrme parameters.

12. The total Hamiltonian of the nucleus where The total energy

13. The total energy where

15. Now we apply the variation principle to derive the Hartree-Fock equations. We minimize the Energy E, given in terms of the energy density functional .

16. where

17. After carrying out the minimization of energy, we obtain the HF equations: where , , and are the effective mass, the potential and the spin orbit potential. They are given in terms of the Skyrme parameters and the nuclear densities.

19. With an initial guess of the single-particle wave functions (example; harmonic oscillator wave functions), we can determine m*, U(r), and W(r) and solve the HF equation to get a set of new single-particle wave functions; then one can proceed in this way until reaching convergence. NOTES: One should start close to the solution. Accuracy and convergence in three dimension 3 Convergence of HFB equations in three dimension?

20. Approximations Coulomb energy Note that here the direct term and exchange Coulomb terms each include the spurious self-interaction term. Interaction: KDE0, neglect exchange term KDE, include exchange term KDEX, include contributions of g.s. correlations

21. Center of mass correction a). Correction to the total binding energy: We use the harmonic oscillator approximation. The CM energy is taken as but Is determined by using the mass mean-square radii b). Correction to the charge rms radii The charge mean-square radius to be fitted to the experimental data is obtained as

22. Simulated Annealing Method (SAM) The SAM is a method for optimization problems of large scale, in particular, where a desired global extremum is hidden among many local extrema. We use the SAM to determine the values of the Skyrme parameters by searching the global minimum for the chi-square function Ndis the number of experimental data points. Npis the number of parameters to be fitted. andare the experimental and the corresponding theoretical values of the physical quantities. is the adopted uncertainty. is the adopted uncertainty.

23. Implementing the SAM to search the global minimum of function: 1. are written in term of 2. Define 3. Calculate for a given set of experimental data and the corresponding HF results (using an initial guess for Skyrme parameters). . 4. Determine a new set of Skyrme parameters by the following steps: + Use a random number to select a component of vector + Use another random number to get a new value of + Use this modified vector to generate a new set of Skyrme parameters.

24. 5. Go back to HF and calculate 6. The new set of Skyrme parameters is accepted only if 7. Starting with an initial value of , we repeat steps 4 - 6 for a large number of loops. 8. Reduce the parameter T as and repeat steps 1 – 7. 9. Repeat this until hopefully reaching global minimum of

25. Fitted data - The binding energies for 14 nuclei ranging from normal to the exotic (proton or neutron) ones: 16O, 24O, 34Si, 40Ca, 48Ca, 48Ni, 56Ni, 68Ni, 78Ni, 88Sr, 90Zr, 100Sn, 132Sn, and 208Pb. - Charge rms radii for 7 nuclei: 16O, 40Ca, 48Ca, 56Ni, 88Sr, 90Zr, 208Pb. • The spin-orbit splittings for 2p proton and neutron orbits for 56Ni (2p1/2) - (2p3/2) = 1.88 MeV (neutron) (2p1/2) - (2p3/2) = 1.83 MeV (proton). - Rms radii for the valence neutron: in the 1d5/2 orbit for 17O in the 1f7/2 orbit for 41Ca - The breathing mode energy for 4 nuclei: 90Zr (17.81 MeV), 116Sn (15.9 MeV), 144Sm (15.25 MeV), and 208Pb (14.18 MeV).

26. Constraints 1. The critical density Landau stability condition: Example: 2. The Landau parameter should be positive at must be positive for densities up to 3. The quantity 4. The IVGDR enhancement factor

27. v v0 v1 d B/A (MeV) 16.0 17.0 15.0 0.4 Knm (MeV) 230.0 200.0 300.0 20.0 ρnm (fm-3) 0.160 0.150 0.170 0.005 m*/m 0.70 0.60 0.90 0.04 Es (MeV) 18.0 17.0 19.0 0.3 J (MeV) 32.0 25.0 40.0 4.0 L (MeV) 47.0 20.0 80.0 10.0 Kappa 0.25 0.1 0.5 0.1 G’0 0.08 0.00 0.40 0.10 W0 (MeV fm5) 120.0 100.0 150.0 5.0

28. Variation of the average value of as a function of the inverse of the control parameter T for the KDE0 interaction for the two different choices of the starting parameter.

29. Parameter KDE0 KDE SLy7 t0 (MeV fm3) -2526.51 (140.63) -2532.88 (115.32) -2482.41 t1 (MeV fm5) 430.94 (16.67) 403.73 (27.63) 457.97 t2 (MeV fm5) -398.38 (27.31) -394.56 (14.26) -419.85 t3 (MeV fm3(1+α)) 14235.5 (680.73) 14575.0 (641.99) 13677.0 x0 0.7583 (.0.0655) 0.7707 (0.0579) 0.8460 x1 -0.3087 (0.0165) -0.5229 (0.0298) -0.5110 x2 -0.9495 (0.0179) -0.8956 (0.0270) -1.000 x3 1.1445 (0.0882) 1.1716 (0.0767) 1.3910 W0 (MeV fm5) 128.96 (3.33) 128.06 (4.39) 126.00 α 0.1676 (0.0163) 0.1690 (0.0144) 0.1667 The values of the Skyrme parameters

30. Parameter KDE0 KDE SLy7 B/A (MeV) 16.11 15.99 15.92 K (MeV) 228.82 223.89 229.7 ρnm 0.161 0.164 0.158 m*/m 0.72 0.76 0.69 Es (MeV) 17.91 17.98 17.89 J (MeV) 33.00 31.97 31.99 L (MeV) 45.22 41.43 47.21 қ 0.30 0.16 0.25 G’0 0.05 0.03 0.04 χ2min 1.3 2.2 Nuclear matter properties

31. Nuclei Bexp ∆B = Bexp -Bth KDE0 KDE SLy7 16O 127.620 0.394 1.011 -0.93 24O 168.384 -0.581 0.370 34Si 283.427 -0.656 0.060 40Ca 342.050 0.005 0.252 -2.85 48Ca 415.99 0.188 1.165 0.11 48Ni 347.136 -1.437 -3.67 56Ni 483.991 1.091 1.016 1.71 68Ni 590.408 0.169 0.539 1.06 78Ni 641.940 -0.252 0.763 88Sr 768.468 0.826 1.132 90Zr 783.892 -0.127 -0.200 100Sn 824.800 -3.664 -4.928 -4.83 132Sn 1102.850 -0.422 -0.314 0.08 208Pb 1636.430 0.945 -0.338 -0.33 Results for the total binding-energy B G. Audi et al, Nucl. Phys. A729, 337 (2003)

32. Nuclei Expt. KDE0 KDE 16O 2.730 2.771 2.769 24O 2.778 2.771 34Si 3.220 3.208 40Ca 3.490 3.490 3.479 48Ca 3.480 3.501 3.488 48Ni 3.795 3.777 56Ni 3.750 3.768 3.750 68Ni 3.910 3.893 78Ni 3.969 3.950 88Sr 4.219 4.221 4.200 90Zr 4.258 4.266 4.1005 100Sn 4.480 4.457 132Sn 4.709 4.710 4.685 208Pb 5.500 5.489 5.459 Results for the charge rms radii E. W. Otten, in Treatise onn Heavy-Ion Science, Vol 8 (1989). H. D. Vries et al, At. Data Nucl. Tables 36, 495 (1987). F. Le Blanc et al, Phys. Rev. C 72, 034305 (2005).

33. GIANT RESONANCES • Hadron Scattering • HF-Based RPA • Results

34. Ψf χf VαN α Nucleus χi Ψi Hadron excitation of giant resonances Theorists: calculate transition strength S(E) within HF-RPA using a simple scattering operator F ~ rLYLM: Experimentalists: calculate cross sections within Distorted Wave Born Approximation (DWBA): or using folding model.

35. momentum transfer The Scattering Operator can be simplified in certain cases to have a simple form A) Born Approximation: α-wave functions are plane wave: B) C) Coulomb excitation by a classical particle

36. (Electric 0+, 1-, 2-; Magnetic 0-, 1+, 2-) (Isoscalar, Isovector) Thus: the cross section is directly related to the matrix element Onedefines: S(E)≡ strength function, response function, structure function. Example: inelastic electron scattering in Born approximation General transition operator F: Energy weighted sum rule (EWSR) Giant resonance: contributes 30 - 100% to EWSR.

37. DWBA-Folding model description

38. EWSR = energy weight sum rule

39. Elastic angular distributions for 240 MeV alpha particle. Filled squares represent the experimental data. Solid lines are fit to the experimental data using the folding model DWBA with nucleon-alpha interaction.

40. History ISOSCALAR GIANT MONOPOLE RESONANCE (ISGMR): 1977 – DISCOVERY OF THE CENTROID ENERGY OF THE ISGMR IN 208Pb E0~ 13.5 MeV (TAMU) • This led to modification of commonly used effective nucleon-nucleon interactions. Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations, with effective interactions (Skyrme and others) which reproduce data on masses, radii and the ISGMR energies have: • K∞ = 210 ± 20 MeV (J.P. BLAIZOT, 1980). • ISOSCALAR GIANT DIPOLE RESONANCE (ISGDR): • 1980 – EXPERIMENTAL CENTROID ENERGY IN 208Pb AT • E1 ~ 21.3 MeV (Jülich), PRL 45 (1980) 337; ~ 19 MeV, PRC 63 (2001) 031301 • HF-RPA with interactions reproducing E0 predicted E1 ~ 25 MeV. • K∞ ~ 170 MeV from ISGDR ? • T.S. Dimitrescu and F.E. Serr [PRC 27 (1983) 211] pointed out “If further measurement confirm the value of 21.3 MeV for this mode, the discrepancy may be significant”. • → Relativistic mean field (RMF) plus RPA with NL3 interaction predict K∞=270 MeV from the ISGMR [N. Van Giai et al., NPA 687 (2001) 449].

41. Hartree-Fock (HF) - Random Phase Approximation (RPA) In fully self-consistent calculations: 1. Assume a form for the Skyrme parametrization (δ-type). 2. Carry out HF calculations for ground states and determine the Skyrme parameters by a fit to binding energies and radii. 3. Determine the residual p-h interaction 4. Carry out RPA calculations of strength function, transition density etc.

42. Green’s Function Formulation of RPA In the Green’s Function formulation of RPA, one starts with the RPA-Green’s function which is given by where Vphis the particle-hole interaction and the free particle-hole Green’s function is defined as where φi is the single-particle wave function, єiis the single-particle energy, and ho is the single-particle Hamiltonian.

43. The continuum effects, such as particle escape width, can be taken into account using where r< and r> are the lesser and greater of r1 and r2 respectively, U and V are the regular and irregular solution of (H0-Z)ψ = 0, with the appropriate boundary conditions, and W is the Wronskian. NOTE the two terms in the free particle-hole greens function

44. We use the scattering operator F to obtain the strength function and the transition density is consistent with the strength in

45. Relativistic Mean Field + Random Phase Approximation The steps involved in the relativistic mean field based RPA calculations are analogous to those for the non-relativistic HF-RPA approach. The nucleon-nucleon interaction is generated through the exchange of various effective mesons. An effective Lagrangian which represents a system of interacting nucleons looks like It contains nucleons (ψ) with mass M; σ, ω, ρ mesons; the electromagnetic field; non linear self-interactions for the σ (and possibly ω) field. Values of the parameters for the most widely used NL3 interaction are mσ=508.194 MeV, mω=782.501 MeV, mρ=763.000 MeV, gσ=10.217, gω=12.868, gρ=4.474, g2=-10.431 fm-1 and g3=-28.885 (in this case there is no self-interaction for the ω meson). NL3: K∞=271.76 MeV, G.A.Lalazissis et al., PRC 55 (1997) 540. RMF-RPA: J. Piekarewicz PRC 62 (2000) 051304; Z.Y. Ma et al., NPA 686 (2001) 173.

46. Are mean-field RPA calculations fully self-consistent ? NO ! In practice, has made approximations. Mean field and Vph determined independently → no information on K∞. In HF-RPA one 1. neglects the Coulomb part in Vph; 2. neglects the two-body spin-orbit; 3. uses limited upper energy for s.p. states (e.g.: Eph(max) = 60 MeV); 4. introduces smearing parameters. Main effects: change in the moments of S(E), of the order of 0.5-1 MeV; note: spurious state mixing in the ISGDR; inaccuracy of transition densities.

47. Commonly used scattering operators: • for ISGMR • for ISGDR • In fully self-consistent HF-RPA calculations the (T=0, L=1) spurious state (associated with the center-of-mass motion) appears at E=0 and no mixing (SSM) in the ISGDR occurs. • In practice SSM takes place and we have to correct for it. • Replace the ISGDR operator with • (prescriptions for η: discussion in the literature) NUMERICS: Rmax = 90 fm Δr = 0.1 fm (continuum RPA) Ephmax ~ 500 MeV ω1 – ω2 ≡ Experimental range

48. Self-consistent calculation within constrained HF

49. Strength function for the spurious state and ISGDR calculated using a smearing parameter Г/2 = 1 MeV in CRPA. The transition strengths S1, S3 and Sη correspond to the scattering operators f1, f3 and fη, respectively. The SSM caused due to the long tail of the spurious state is projected out using the operator fη

50. Isoscalar strength functions of 208Pb for L = 0 - 3 multipolarities are displayed. The SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (open circle) represent the calculations without the ph spin-orbit and Coulomb interaction in the RPA, respectively. The Skyrme interaction SGII [Phys. Lett. B 106, 379 (1981)] was used.