G. Colò, V. Kolomietz, S. Shlomo

1. The nuclear matter incompressibility K from isoscalar compression modes WCI-3 workshop, College Station, TX February 12-16, 2005 G. Colò, V. Kolomietz, S. Shlomo

2. Outline • Introduction • Definitions: nuclear matter incompressibility coefficient K • Background: isoscalar giant monopole resonance, • isoscalar giant dipole resonance • Hadron excitation of giant resonances • Theoretical approaches for giant resonances • Hartree-Fock plus Random Phase Approximation (RPA) • Comments: self-consistency ? • Relativistic mean field (RMF) plus RPA • Discussion • ISGMR viz. ISGDR • Non-relativistic viz. Relativistic • Symmetry energy (Colò) • Memory effects (Kolomietz)

3. Introduction The nuclear matter (N = Z and no Coulomb interaction) incompressibility coefficient, K∞ , is a very 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], E/A [MeV] ρ = 0.16 fm-3 ρ [fm-3] E/A = -16 MeV

4. 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].

5. Ψf χf Nucleus VαN α 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.

6. DWBA-Folding model description

8. In fully self-consistent calculations: • Assume a form for the Skyrme parametrization (δ-type). • Carry out HF calculations for ground states and determine the Skyrme parameters by a fit to binding energies and radii. • Determine the residual p-h interaction, • Carry out RPA calculations of strength function, transition density etc. • The RPA Green’s function G is obtained from the free Green’s function G0: • For the scattering operator Hartree-Fock (HF) - Random Phase Approximation (RPA)

9. Are mean-field RPA calculations fully self-consistent ? • NO ! In practice, one makes 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.

10. 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

11. 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.

12. K∞ from the ISGMR in 208Pb: Skyrme calc. Non fully s.c.: ~210 MeV Fully s.c.: ~235 MeV G. Colò and N. Van Giai, NPA 731 (2004) 15.

13. Relativistic RPA Values of K∞ of the order of 250-270 MeV were extracted. T. Nikšić et al., PRC 66 (2002) 064302.

14. A. Kolomiets, O. Pochivalov, and S. Shlomo, PRC 61 (2000) 034312 ISGMR f=r2Y00 SL1 interaction, K∞=230 MeV Eα = 240 MeV

15. ISGDR S. Shlomo and A.I. Sanzhur, Phys. Rev. C 65, 044310 (2002) SL1 interaction, K∞=230 MeV Eα = 240 MeV

16. Nuclear matter properties calculated from RMF theory with NL3 parameters and from the non-relativistic HF calculations ISGMR centroid energy (in MeV) obtained by integrating over the energy range ω1-ω2 with the strength function smeared by using Γ/2 = 1 MeV.

17. CONCLUSION Fully self-consistent calculations of the ISGMR using Skyrme forces lead to K∞~ 230-240 MeV. ISGDR: At high excitation energy, the maximum cross section for the ISGDR drops below the experimental sensitivity. There remain some problems in the experimental analysis. It is possible to build bona fide Skyrme forces so that the incompressibility is close to the relativistic value. Recent relativistic mean field (RMF) plus RPA: lower limit for K∞ equal to 250 MeV. → K∞ = 240 ± 20 MeV. sensitivity to symmetry energy.