Continuum quasiparticle linear response theory using the Skyrme functional for exotic nuclei

1. First FIDIPRO-JSPS Workshop on Energy Density Functionals in Nuclei October 25-27, 2007, Filand. Continuum quasiparticle linear response theory using the Skyrme functional for exotic nuclei University of Jyväskylä Kazuhito Mizuyama, Niigata University, Japan Masayuki Matsuo, Yasuyoshi Serizawa

2. Collective excitations in Unstable Nuclei and Quasiparticle Random Phase Approximation Neutron-rich unstable nuclei Many weakly bound neutrons Realistic description D.F.T. + Effective int. Self-consistency Continuum Pairing Continuum RPA S. Shlomo and G. Bertsch. NPA243(1975). I. Hamamoto, H. Sagawa, and X. Z. Zhang. PRC 57, R1064 (1998) Self-consistent Skyrme QRPA J. Terasaki and J. Engel. PRC 74, 044301, (2006). Relativistic QRPA P. Ring, N. Paar, T. Niksic, and D. Vretenar. NPA722, (2003). Continuum QRPA M. Matsuo. NPA696, 371, (2001). E. Khan, et al. PRC66, 024309, (2002).

3. Continuum quasiparticle linear response theory(Continuum QRPA) Continuum QRPA in coordinate space HFB formalism M.Matsuo Nucl.Phys.A696(2001)371 M. Matsuo, K. Mizuyama, and Y. Serizawa. PRC 71, 064326, (2005). Response function • Shallow Fermi level • Continuum • Pairing correlation HFB Green’s function Pairing and Exact boundary condition of out-going wave for continuum states : correct asymptotic of out going wave : regular solution of q.p. wave function at r=0.

4. The extension of the continuum QRPAwith the Skyrme energy density functional • The purpose of this study is to formulate the continuum QRPA with the Skyrme functional keeping the velocity dependent terms. -- Skyrme-Hartree-Fock-Bogoliubov functional -- For the purpose of this study, the linear response equation for various densities fluctuations is needed Spin dependent terms are dropped Velocity dependent terms Simple delta-type int. Previous continuum QRPA Important for the conservation of the EWSR • Current conservation => ISEWSR • Enhancement factor => IVEWSR • Simple delta-type int. • Landau-Migdal force

5. From Previous continuum QRPA to the new continuum QRPA -- Previous continuum QRPA equation -- -- New continuum quasi linear response equation -- -- Response function -- -- Induced field --

6. The treatment of singular terms in the response function -- Response function -- The treatment of delta-type singular terms in the response function (cf. K.F.LIU, N.V.GIAI, Phys.Lett.Vol.65,23(1976)) Regular terms Singular terms

7. Numerical calculation -- Parameters -- Smoothing constant ε＝1.0[MeV] p-h channel Skyrme interaction: SkM* for 20O, 54Ca RMax=15fm lcut=7,8 Ecut=60[MeV] p-p interaction V0= 280[MeV fm-3] for 20O , 285[MeV fm-3] for 54Ca ρ0 = 0.32 fm-3 Isovector dipole and Isoscalar quadrupole in 20O and 54Ca -- Calculations -- • HFB ground state (Spherical) • Continuum QRPA cal. we compare the previous continuum QRPA with Landau-Migdal force (LM) and the new continuum QRPA (Full) in the neutron rich nuclei.

8. Strength function The structure is not changed between two calculations in the IS E2 strength IS 54Ca SkM* IS & IV 2+ The structure is changed between two calculations in the E1 strength IV Centroid energy is different about 2-3MeV in the IV mode.

9. Energy weighted sum rule EWSR (κ=0.32) Running energy weighted sum EWSR Continuum QRPA with the Skyrme functional satisfies the energy weighted sum rule about 95-99%. On the other hand, Lndau-Migdal (LM) approx. underestimated the sum rule about 15% in the E1 excitation, overestimated about 10% in the IS quqdrupole excitation.

10. S(Ex)[e2fm2/MeV] Ex[MeV] Interaction dependence of the strength function & the effect of the velocity dependent terms In the “t0+t3” approximation, the velocity dependent terms are completely ignored in the residual interaction. (Only the density dependent terms(t0,t3) are used.) The calculation with SLy4 also satisfies EWSR by 96% until Ex=55 MeV. The effect of the velocity dependent terms depends on the interaction.

11. p-h transition density p-pair transition density h-pair transition density Transition densities 54Ca SkM* IV 1- Low -lying • The basic structures are same in the low-lying state and the GDR between two calculations. • Low-lying state • p-pair transition density is enhanced. • IVGDR • Transition densities are almost equivalent between two calculations, note that the peak energy is different. New cQRPA Previous cQRPA IVGDR

12. p-h transition density p-pair transition density h-pair transition density Transition densities 54Ca SkM* IS 2+ Low -lying • The basic structures are same in the low-lying state and the GQR between two calculations. • Low-lying state • h-pair transition density is enhanced. • ISGQR • Transition densities are almost equivalent between two calculations, note that the peak energy is also same. Previous cQRPA New cQRPA ISGQR

13. Summary • We formulated the continuum QRPA based on the Skyrme energy functional • with keeping the velocity dependent terms. • We applied the Skyrme continuum QRPA to the isovector dipole • and the isoscalar quadrupole responces in neutron-rich O and Ca isotopes. • The Skyrme continuum QRPA satisfies the EWSR. • There are quantitative improvements, • compared with the previous cQRPA (Landau-Migdal approx.)

14. Translational symmetry and spurious mode Improvement of the self-consistency Violation of the self-consistency ⇒ Spurious mode at E≠0 Renormalization factorfR (residual force) ⇒ fR× (residual force) The renormalization factor is one of the indicators of the self-consistency fR=１：Self-consistent fR≠１：Self-consistent is violated

16. C Δ→ 0 C’ Continuum RPA ・Shlomo-Bertsch Nucl.Phys.A243(1975) ・Hamamoto-Sagawa-Zhang Phys.Rev.C53 (1996) Etc…

18. Time-odd velocity-dependent term’s effect in the strength function 20O SkM* IV 1- Time-odd terms are most essential role to restore the conservation of the EWSR.