1. NEUTRON SKIN AND GIANT RESONANCES Shalom Shlomo Cyclotron Institute Texas A&M University

2. Outline 1.Introduction Isovector giant dipole resonance, Giant resonances (GR) and bulk properties of nuclei 2.Experimental and theoretical approaches for GR Hadron excitation of giant resonances Hartree-Fock plus Random Phase Approximation (RPA) 3.Density dependence of symmetry energy and neutron skin A study within the Energy Density Functional Approach (EDF) 4. Giant resonances and symmetry energy density ISGMR—Incompressibility and Symmetry energy IVGDR and ISGMR in Ca isotopes 5. Nuclear + Coulomb excitations of GR and neutron skin 6. Conclusions

3. The total photoabsorption cross-section for 197 Au, illustrating the absorption of photons on a giant resonating electric dipole state. The solid curve show a Breit-Wigner shape. (Bohr and Mottelson, Nuclear Structure, vol. 2, 1975). The isovector giant dipole resonance

4. Macroscopic picture of giant resonances L = 0 L = 1 L = 2

5. Theorists: calculate transition strength S(E) within HF-RPA using a simple scattering operator F ~ r L Y LM : Experimentalists: calculate cross sections within Distorted Wave Born Approximation (DWBA): or using folding model. Hadron excitation of giant resonances Nucleus α χiχi χfχf ΨiΨi ΨfΨf VαNVαN

6. DWBA-Folding model description

7. we adopt the standard Skyrme type interaction For the nucleon-nucleon interaction Hartree-Fock with Skyrme interaction are 10 Skyrme parameters.

8. Carry out the minimization of energy, we obtain the HF equations:

10. Hartree-Fock (HF) - Random Phase Approximation (RPA) 1) Assume a form of Skyrme interaction (  - 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 particle-hole interaction, 4) Carry out RPA calculations of the strength function, transition density, etc. In fully self-consistent calculations :

11. Giant Resonance In the Green’s Function formulation of RPA, one starts with the RPA- Green’s function which is given by where V ph is the particle-hole interaction and the free particle-hole Green’s function is defined as where φ i is the single-particle wave function, є i is the single-particle energy, and h o is the single-particle Hamiltonian.

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

13. The energy density functional is decomposed as Where ρ n and ρ p are the density distributions of neutrons and protons respectively, and E.Friedman and S. Shlomo, Z. Phyzik, A283, 67 (1977) Density dependence of symmetry energy and Neutron skin within EDF

14. For the Coulomb energy density, ε c, one usually uses the form where the first term is the direct Coulomb term with V c (r) given by

15. For the symmetry energy density, ε sym, we assume the form The interaction V 1 (r) is taken to be of the form where ρ m (r) is the nuclear matter density distribution, ρ 0 =0.165 fm -3. In accordance with the semiemperical mass formula we impose the constraint The terms with a 2, a 3, and a 4 have been used previously in nuclear matter calculations and in applications of the EDF to finite nuclei.

16. Considering now the constraint We introduce a Lagrange multiplier λ and minimize using δρ m =δρ p +δρ n =0.

17. We obtain with where V c (r) and V 1 (r) are given by previous equations.

18. The EDF is not known for low density. Thus the variational equation for ρ 1 (r) must be used only in an internal region r R M the resulting ρ n (r) and ρ p (r) should be positive and decay exponentially with r. Taking R M =R, then for the internal region, r < R, we have where,

19. For the external region, r > R, we choose where the coefficients C and γ are determined by imposing (i) the continuity of the densities and (ii) the total normalizations A surface enhancement parameter y is defined by

20. Values of r n -r p Parameterization calculations have been made for 48 Ca and 208 Pb using a parabolic Fermi for the proton distribution, with c=3.74 fm, a=0.53 fm and ω=-0.03, leading to r p = 3.482 fm for 48 Ca, and c=6.66 fm, a=0.50 fm and ω=0 leading to r p = 5.483 fm for 208 Pb

24. Giant Resonances and Symmetry Energy ISGMR --Incompressibility and symmetry energy ISGMR in Ca isotopes IVGDR in Ca isotopes and symmetry energy

26. Nucleusω1-ω2ω1-ω2 Expt.NL3SK255SGIIKDE0 90 Zr0-6018.718.917.918.0 10-3517.81±0.3018.917.918.0 116 Sn0-6017.117.316.416.6 10-3515.85±0.2017.316.416.6 144 Sm0-6016.116.215.315.5 10-3515.40±0.4016.215.215.5 208 Pb0-6014.214.313.613.8 10-3513.96±0.3014.413.613.8 K (MeV)272255215229 J (MeV)37.4 26.833.0 Fully self-consistent HF-RPA results for ISGMR centroid energy (in MeV) with the Skyrme interaction SK255, SGII and KDE0 and compared with the RRPA results using the NL3 interaction. Note the coressponding values of the nuclear matter incompressibility, K, and the symmetry energy, J, coefficients. ω 1 -ω 2 is the range of excitation energy. The experimental data are from TAMU.

32. Nuclear and Coulomb Excitations of Giant Resonances Neutron skin and nuclear excitation of IVGDR by alpha (T=0) scattering Interference between Nuclear and Coulomb excitations of GR and neutron skin

33. Definitions: Assuming uniform density distributions For:

34. For Isovector Dipole (T=1, L=1) oscillations; CoM: g = -Z/A for a neutron and N/A for a proton. Transition density and transition potential are:

35. For a proton projectile the transition potential is: With Note: Un and Up are of different geometry

36. Expanding the ground state densities: Where, And

37. We obtain for a proton projrectile

38. For excitaion of IVGDR by a proton: With

39. For excitaion of IVGDR by an alpha particle (T=0), adding the contributions of the two neutron and two protons, we have Note that;

40. For excitaion of ISGMR by an alpha particle;

47. CONCLUSIONS 1.Fully self-consistent HF-based RPA calculations of the ISGMR lead to K = 210-250 MeV with uncertainty due to the uncertaint in the symetry energy density. 2.The neutron skin depends strongly on the density dependence of the symmetry energy. 3.The dependence of the centroid energy of the Isovector giant dipole resonance is clouded by the effects of (i) momentum dependence of the interaction (ii) the spin-orbit interaction. 4.Interference between Nuclear and Coulomb excitations of GR can be used to determine the depependence of neutron skin on N-Z. 5.Accurate determination of the magnitude of the neutron skin in neutron rich nuclei is very much need.