Excitations in light deformed nuclei investigated by self consistent quasiparticle random phase approximation

Excitations in light deformed nuclei investigated by self consistent quasiparticle random phase approximation C. Losa, A. Pastore, T. Døssing, E. Vigezzi, R. A. Broglia SISSA, Trieste Trento, December 2011 C. Losa (SISSA) Deformed QRPA Trento, December 2011 1 / 22

Outline Phenomenological introduction about vibrational modes in nuclei Theoretical description of collective vibrational modes QRPA theory Details about our work: approach, type of interaction used Results: linear response in 20 O, 24 26 Mg, 34 Mg Comparison with other approaches Conclusions and perspectives C. Losa (SISSA) Deformed QRPA Trento, December 2011 2 / 22

Isoscalar monopole vibrational modes J = 0 C. Losa (SISSA) Deformed QRPA Trento, December 2011 3 / 22

Isoscalar monopole vibrational modes J = 0 C. Losa (SISSA) Deformed QRPA Trento, December 2011 4 / 22

Isovector dipole vibrational modes J = 1 C. Losa (SISSA) Deformed QRPA Trento, December 2011 5 / 22

Isovector dipole vibrational modes J = 1 C. Losa (SISSA) Deformed QRPA Trento, December 2011 6 / 22

Isoscalar quadrupole vibrational modes J = 2 C. Losa (SISSA) Deformed QRPA Trento, December 2011 7 / 22

Isoscalar quadrupole vibrational modes J = 2 C. Losa (SISSA) Deformed QRPA Trento, December 2011 8 / 22

Mean Field Theory H = A i=1 2 2m i + A v(i, j) i<j The description of the many body vibrational systems is given at a first order by a mean field H 0 = A i=1 ) ( 2 2m i + V (i) = H = H 0 + V res H 0 V res = i<j v(i, j) i V (i) C. Losa (SISSA) Deformed QRPA Trento, December 2011 9 / 22

Linear Response Theory λ = Q λ 0 Q λ : phonon creation operator RPA Q λ = ph ( ) Xph λ a pa h Yph λ a h a p QRPA Q λ = L<L (X λ LL α L α L Y λ LL α L α L ) 0 HF 0 HFB qp states α K = u K a K v K a K, α K = u K a K v K a K C. Losa (SISSA) Deformed QRPA Trento, December 2011 10 / 22

Linear Response Theory λ = Q λ 0 Q λ : phonon creation operator RPA Q λ = ph ( ) Xph λ a pa h Yph λ a h a p QRPA Q λ = L<L (X λ LL α L α L Y λ LL α L α L ) 0 HF 0 HFB qp states α K = u K a K v K a K, α K = u K a K v K a K Spherical symmetry Axial symmetry {Q } are obtained for each block J π {Q } are obtained for each block Ω π J π C. Losa (SISSA) Deformed QRPA Trento, December 2011 10 / 22

Approach as Mean Field we consider HF(B) theory as Linear Response we consider (Q)RPA theory we impose Axial Symmetry using Cylindrical Coordinates r, z, φ we work in canonical HO and THO basis wave functions 10 0 10 5 10 10 10 15 40 Ca s 1/2 R s1/2 (r) Y 0,0 ψ s1/2 (0,z) / (2π) 1/2 ψ s1/2 THO (0,z) / (2π) 1/2 THO JT HO 10 20 0 5 10 15 20 r ( fm ) f (R) r r R, z z f (R) R r 2 R = b 2 + z2 bz 2 JT J. Terasaki et al. Phys. Rev. C71, 2005 Skyrme (SKM ) force is used both in HFB and QRPA C. Losa (SISSA) Deformed QRPA Trento, December 2011 11 / 22

Density Dependent Delta Interaction V pair ( r, r ) = 1 P σ 2 [ V 0 + V ] 1 6 ργ 00 (r) δ ( r r ) R. R. Chasman et al. Phys. Rev. C14, 1976 C. Losa (SISSA) Deformed QRPA Trento, December 2011 12 / 22

Density Dependent Delta Interaction V pair ( r, r ) = 1 P σ 2 [ V 0 + V ] 1 6 ργ 00 (r) δ ( r r ) R. R. Chasman et al. Phys. Rev. C14, 1976 The shape of the pairing field (r) depends on the choice of the parameters η, V 1. DDDI - parameters type V 1 γ Surface -37.5V 0 1 Volume 0 1 Mixed -18.75V 0 1 J. Dobaczewski et al., EPJ A15, 2002 C. Losa (SISSA) Deformed QRPA Trento, December 2011 12 / 22

(Q)RPA equations A B B A X Y = ω λ X Y, λ = Q λ 0 C. Losa (SISSA) Deformed QRPA Trento, December 2011 13 / 22

(Q)RPA equations A B B A X Y = ω λ X Y, λ = Q λ 0 Residual Interaction V res V ph KK LL = δ2 E[ρ, κ, κ ] = KK eff eff V Skyrme + V Coul δρ LK δρ LL L K V pp KK LL = δ2 E[ρ, κ, κ ] δκ LK δκ = KK V pair LL eff L K E [ρ, κ, κ ] = E Skyrme [ρ] + E Coul [ρ p ] + E pair [ρ, κ, κ ] C. Losa (SISSA) Deformed QRPA Trento, December 2011 13 / 22

Ground state in 20 O and 24 Mg Potential Energy Surfaces Pairing energies C. Losa (SISSA) Deformed QRPA Trento, December 2011 14 / 22

Ground state in 26 Mg and 34 Mg Potential Energy Surfaces Pairing energies C. Losa (SISSA) Deformed QRPA Trento, December 2011 15 / 22

20 O: without spin orbit and Coulomb interaction in IS 2 + Present work K. Yoshida et al. Phys. Rev. C78, 2008 peaks below 10 MeV: shifted up 200 kev with heights lowered by 20% ISGQR: peak shifted down 200 300 kev ISGQR: peak heights increase 30%(JT), 15%(HO), 25%(THO) C. Losa (SISSA) Deformed QRPA Trento, December 2011 16 / 22

24 Mg: fractions of EWSR in IS 2 + Present work compared to D. H. Youngblood et al. Phys. Rev. C60, 1999 S. Péru et al. Phys. Rev. C77, 2008 D1S SKM HO SKM THO SLY4 HO SLY4 THO Exp. 20.54 19.40 19.33 20.02 19.94 16.9 ± 0.6 C. Losa (SISSA) Deformed QRPA Trento, December 2011 17 / 22

24 26 Mg: strength functions in IV 1 24 Mg 26 Mg S 1 ( e 2 fm 2 /MeV ) 1 0.5 0 1 0.5 HO THO Ω π =0 Ω π =1 IV 1 24 Mg 0 0 10 20 30 40 50 E ( MeV ) S 1 ( e 2 fm 2 /MeV ) 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 0.8 0.6 0.4 0.2 HO THO Ω π =0 Ω π =1 IV 1 26 Mg 0 0 10 20 30 40 50 E ( MeV ) C. Losa (SISSA) Deformed QRPA Trento, December 2011 18 / 22

34 Mg: strength functions in IS 2 + S. Ebata Canonical basis time dependent Hartree Fock Bogoliubov (Cb TDHFB) approach K. Yoshida QRPA approach in quasi particle basis Present work QRPA approach in THO basis C. Losa (SISSA) Deformed QRPA Trento, December 2011 19 / 22

FAM: Finite Amplitude Method T. Nakatsukasa, T. Inakura, and K. Yabana Phys. Rev. C76 024318, 2007 P. Avogadro and T. Nakatsukasa Phys. Rev. C84 014314, 2011 C. Losa (SISSA) Deformed QRPA Trento, December 2011 20 / 22

FAM: Finite Amplitude Method T. Nakatsukasa, T. Inakura, and K. Yabana Phys. Rev. C76 024318, 2007 P. Avogadro and T. Nakatsukasa Phys. Rev. C84 014314, 2011 QRPA A B B A X Y = ω λ X Y FAM (E µ + E ν )X νµ + δh 20 νµ(ω) (E µ + E ν )Y νµ + δh 20 νµ(ω) = ω λ X νµ Y νµ δh 20 δh (δh, δ ) δh = H(ρ η, κ η ) H(ρ, κ) η ρ η = (V + ηu X )(V + ηu Y ) T κ η = (U + ηv Y )(V + ηu X ) C. Losa (SISSA) Deformed QRPA Trento, December 2011 20 / 22

FAM: 24 Mg for 0 + modes M. Stoitsov, M. Kortelainen, T. Nakatsukasa, C. Losa, and W. Nazarewicz Phys. Rev. C84 041305(R), 2011 QRPA (Present work) cutoff in ɛ crit = 200 MeV v crit = 10 4 MeV QRPA matrices 32000 32000 requiring 16 GB memory FAM requires 0.5 GB memory C. Losa (SISSA) Deformed QRPA Trento, December 2011 21 / 22

Conclusions and perspectives software for self consistent HFB+QRPA calculations in canonical HO and THO basis 20 O test of our calculations with those of J. Terasaki comparison with results of K. Yoshida role of self consistency 24 26 Mg role of intrinsic deformation in the splitting between projections comparison with experimental data and S. Péru 34 Mg agreement with the result of K. Yoshida, S. Ebata comparison with FAM agreement between the IS monopole strength functions future comparisons for higher multipolarities C. Losa (SISSA) Deformed QRPA Trento, December 2011 22 / 22