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

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

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

The extension of the continuum QRPA with 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 Velocity dependent terms Simple delta-type int. Spin dependent terms are dropped Previous continuum QRPA Simple delta-type int. Landau-Migdal force Important for the conservation of the EWSR Current conservation => ISEWSR Enhancement factor => IVEWSR

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

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

Numerical calculation p-h channel Skyrme interaction: SkM* for 20 O, 54 Ca p-p interaction V 0 = 280[MeV fm -3 ] for 20 O, 285[MeV fm -3 ] for 54 Ca ρ 0 = 0.32 fm -3 R Max =15fm l cut =7,8 E cut =60[MeV] Smoothing constant ε ＝ 1.0[MeV] we compare the previous continuum QRPA with Landau-Migdal force (LM) and the new continuum QRPA (Full) in the neutron rich nuclei. -- Parameters Calculations -- HFB ground state (Spherical) Continuum QRPA cal. Isovector dipole and Isoscalar quadrupole in 20 O and 54 Ca

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

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. Energy weighted sum rule Running energy weighted sum EWSR (κ=0.32) EWSR

Interaction dependence of the strength function & the effect of the velocity dependent terms E x [MeV] S(E x )[e 2 fm 2 /MeV] The effect of the velocity dependent terms depends on the interaction. 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.

Transition densities p-h transition density p-pair transition density h-pair transition density 54 Ca SkM* IV 1 - Low -lying IVGDR 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. Previous cQRPANew cQRPA

Transition densities p-h transition density p-pair transition density h-pair transition density 54 Ca SkM* IS 2 + Low -lying ISGQR Previous cQRPA New cQRPA 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.

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.)

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

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

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