Effects of self-consistence violations in HF based RPA calculations for giant resonances Shalom Shlomo Texas A&M University

Outline 1.Introduction Definitions: nuclear matter incompressibility coefficient K  Background:isoscalar giant monopole resonance, isoscalar giant dipole resonance 2.Theoretical approaches for giant resonances Hartree-Fock plus Random Phase Approximation (RPA) Comments: self-consistency ? Relativistic mean field (RMF) plus RPA

ρ [fm -3 ] ρ = 0.16 fm -3 E/A [MeV] E/A = -16 MeV The EOS is an important ingredient in the study of properties of nuclei, heavy- ion collisions, and in astrophysics (neutron stars, supernova). The nuclear matter equation of state (EOS)

Hartree-Fock (HF) Within the HF approximation: the ground state wave function In spherical case HF equations:minimize

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

The total energy Where

. Now we apply the variation principle to derive the Hartree-Fock equations. We minimize

where

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

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.

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.

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

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

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: R max = 90 fm Δr = 0.1 fm (continuum RPA) E ph max ~ 500 MeV ω 1 – ω 2 ≡ Experimental range

Self-consistent calculation within constrained HF

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 m σ = MeV, m ω = MeV, m ρ = MeV, g σ =10.217, g ω =12.868, g ρ =4.474, g 2 = fm -1 and g 3 = (in this case there is no self-interaction for the ω meson). NL3: K ∞ = MeV, G.A.Lalazissis et al., PRC 55 (1997) 540. RMF-RPA: J. Piekarewicz PRC 62 (2000) ; Z.Y. Ma et al., NPA 686 (2001) 173. Relativistic Mean Field + Random Phase Approximation

E ph max E ss E0E0 E1E1 E2E ∞ Dependence of the energy E ss of the spurious state (T=0, L=1) and the centroid energies E L of the isoscalar multipole giant resonances (L=0, 1, and 2), in MeV, on the value of E ph max (in MeV) adopted in HF-discretized RPA calculation for 80 Zr using a Skyrme interaction. The corresponding HF-Continuum RPA results are placed in the last row

Strength function for the spurious state and ISGDR calculated using a smearing parameter Г/2 = 1 MeV in CRPA. The transition strength S 1, S 3 and S η correspond to the scattering operators f 1, f 3 and f η, respectively. The SSM caused due to long tail of spurious state is projected out using the operator f η

Isoscalar strength functions of 208 Pb for L=0-3 multi-polarities are displayed. SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (dotted line) represent the calculations without the ph spin-orbit and the Coulomb interactions in the RPA, respectively. The Skyrme interaction SGII was used.

Isovector strength functions of 208 Pb for L=0-3 multi-polarities are displayed. SC (full line) corresponds to the fully self-consistent calculation where LS (dashed line) and CO (dotted line) represent the calculations without the ph spin-orbit and the Coulomb interactions in the RPA, respectively. The Skyrme interaction SGII was used.

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

Fully self-consistent HF-RPA results for ISGDR centroid energy (in MeV) with the Skyrme interaction 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, RCNP Osaka. Nucleusω1-ω2ω1-ω2 Expt.NL3SGIIKDE0 90 Zr ± ± ± Sn ± ± ± Sm ± ± Pb ± ± ±0.2 K (MeV) J (MeV)

Nucleusω1-ω2ω1-ω2 Expt.NL3SK255SGIIKDE0 90 Zr ± Sn ± Sm ± Pb ± K (MeV) J (MeV) 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.

CONCLUSION Fully self-consistent calculations of the ISGMR using Skyrme forces lead to K ∞ ~ 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.