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Giant resonances in rotating superdeformed nuclei

Giant resonances in rotating superdeformed nuclei. J. Kvasil 1) , N. Lo Iudice 2) , F. Andreozzi 2) , A. Porrino 2) , F. Knapp 1). 1) Institute of Particle and Nuclear Physics, Charles University, Prague, Czech Republic

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Giant resonances in rotating superdeformed nuclei

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  1. Giant resonances in rotatingsuperdeformed nuclei J. Kvasil1), N. Lo Iudice2), F. Andreozzi2), A. Porrino2), F. Knapp1) 1) Institute of Particle and Nuclear Physics, Charles University, Prague, Czech Republic 2) Dipartamento di Scienze Fisiche, Università di Napoli “ Federico II ”, Napoli, Italy

  2. Motivation motivation • SD bands – the first observation: 1986 – P.J.Twin et al., PRL 57, 811 (1986) • since then: ~200 SD bands observed in A~150 and A~190 • (plus others in A~80,130) • nature of SD bands: • - SD bands = rotational bands built on the intrinsic states with • single-quasiparticle structure (odd-A nuclei) or two-quasiparticle structure • (even-A nuclei) where single-quasiparticle scheme is connected with large • quadrupole deformation ( ) • - large octupole collectivity in some SD bands also expected • see: * octupole softness – J.Dudek et al. Phys.Lett. B248, 235 (1990) • * RPA calculations for 152Dy, 190-194Hg : • Nakatsukasa et al. PRC 53, 2213 (1996) experimental evidence of such a nature: 1) existence of superdeformation – kinematical moment of inertia

  3. motivation 2) existence of octupole collectivity a) dynamical moment of inertia with Harris formule Using and (from Harris exp.) we have information about which can be calculated theoretically. This was 152Dy done in Nakatsukasa et al, PRC 53, 2213 (1996) J.Kvasil, N.Lu Iudice, F.Andreozzi, F.Knapp, A.Porrino, PRC 73, 034302 (2006) and it was found that most of SD bands observed have an octupole character.

  4. motivation b) B(E1) values next evidence of octupole collectivity in most observed SD bands are strong E1 transitions from these SD bands into the SD yrast band – see in 152Dy: see T. Lauritsen et al. PRL 89, 282501 (2002) PRL 88, 042501 (2002) and similarly for 194Hg in: G. Hackman et al. PRL 79, 4100 (1997) for 152Dy

  5. motivation in this presentation: • we studied the collective properties of SD bands in 152Dy and 190, 192, 194Hg • adopting a cranked Nilsson model + QRPA similar to work of • Nakatsukasa et al, PRC 53, 2213 (1996) but a little bit more selfconsistent • in our case – see J.Kvasil, N.Lu Iudice, F.Andreozzi, F.Knapp, A.Porrino, • PRC 73, 034302 (2006), PRC 75, 034306 (2007) • we studied an evolution of deformation with the increasing rotational • frequency (transition from the normal deformation to SD) • we studied an evolution of E0, E1, E2 and M1 resonances in dependence • on (and consequently on the deformation ). Namely, we made • an analysis of the orbital (scissor) and spin parts of M1 resonance.

  6. model Hamiltonian Brief outline of the model and results Hamiltonian Cranking Hamiltonian (in the rotating frame – axis 1 = rotation axis) where one-body mean field is taken in the sum of Nilsson part and of Galilean invariance restoring term ( ): where with the volume conservation constraint:

  7. model Hamiltonian residual interactions – multipole multipole separable form: we introduce multipole and spin-multipole operators with good signature ( ) where From the deformation point of view it is better to express multipoles in residual interactions in the double stretched coordinates:

  8. model Hamiltonian deformed mean field looks like spherical in double stretched coordinates with strength constants: (see A.Bohr, B.Mottelson, Nuclear Structure II, Benjamin, 1975) from mass differences However, str. constants were in the RPA varied in order to obtain spurious modes in right place. see B.Castel, I.Hamamoto, PRC 65, 27 (1976)

  9. HFB scheme HFB equations Bogoliubov transformation: with the constraint: HFB equation: where yrast state for given Since the numerical instability in the crossing region of (backbending area) HFB eqs. are not solved fully selfconsistently, instead: R.Wyss et al., NP A511, 324 (1990) pairing gap taken in simple dependence: • equlibrium deformation found • from the condition of minimum of the • total energy for for

  10. HFB results HFB results maps of total energy in dependence on deformation for 152Dy total angular moment in the depenedence on rotational frequency for 152Dy

  11. RPA scheme – cranked quasiparticle RPA cranked QRPA Using Bogoliubov transformation we express cranking Hamiltonian: in terms of quasiparticle operators . Then we use RPA eqs. for the determination of the RPA eigenstates where RPA coordinate and linear momenta operators th phonon creation operator of two-quasiparticle creation operator we can solve the RPA equations independently for and (indepndently for each signature) since

  12. cranked QRPA cranking Hamiltonian symmetry conditions: where comparing the RPA symmetry conditions with the RPA equation: • among the RPA solutions of we have two zero-energy Goldstone • modes (one connected with and the second with ) • among the RPA solutions of we have one RPA solution with the • energy connected with operators or Symmetry conditions above are in the framework of the RPA fulfilled automatically when the residual interactions in the Hamiltonian are fully consistent with the mean field .

  13. cranked QRPA However, our phenomenological separable residual interactions: do not restore symmetries above violated by our phenomenological Nilsson like mean field. we slightly varied parameters for each value of comparing to: restoration of symmetries of total Hamiltonian in rotating nuclei for each rotational frequency in order to get right position of Goldstone modes discussed above and in such a way Hamiltonian symmetries above are restored in the framework of the RPA After the solution of the RPA equations with the Hamiltonian involving separable residual interactions with such a way modified strength we can easily extract spurious and Goldstone redundant modes and obtain the energies and structure , of all intrinsic one-phonon states ( it can be done in the case of triaxial deformations independently for each parity and signature ).

  14. RPA results SD1- SD2- SD3- SD4- SD5- SD6- lowest phonon energies dynamical moment of inertia crossing points:

  15. RPA results dynamical moment of inertia for SD bands in Hg isotopes

  16. RPA results • Right description of fluctuation of dynamical moment of inertia for SD band • as thats of with octupole character represents the indirect evidence of octupole • character of the most of observed rotational SD bands in 152Dy and 190, 192, 194Hg. • direct evidence of octupole collectivity of SD6 band in 152Dy and SD2 band in • 190Hg is given by a good description of strong B(E1) values observed: E1 transitions SD6 SD1 in 152Dy. • 32+ 1676 2.2x10-4 2.86x10-4 • 35- 34+ 1696 3.8x10-4 1.86x10-4 • 37- 36+ 1715 4.5x10-4 1.94x10-4 • 39- 38+ 1734 3.9x10-4 1.74x10-4 • 41- 40+ 1751 4.9x10-4 3.69x10-4 exp. from T.Lauritsen et al., Phys. Rev. Lett. 89, 282501 (2002)

  17. RPA results E1 transitions SD2 SD1 in 190Hg. • 24+ 911 >1.4x10-3a) 2.29x10-3 • 27- 26+ 864 1.2x10-3a) 1.58x10-3 • 3.8x10-3b) • 29- 28+ 812 1.5x10-3a) 0.99x10-3 • 1.5x10-3b) • 31- 30+ 757 < 2.4x10-3a) 1.62x10-3 • 1.6x10-3b) 25- exp. from a) B.Crowell et al., Phys. Rev. C51, R1599 (1995) b) A.Korichi et al., Phys.Rev.Lett. 86, 2746 (2001) E1 transitions SD3 SD1 in 194Hg. • 10+ 824 0.92x10-4 2.29x10-3 • 13- 12+ 832 1.12x10-4 1.58x10-3 • 15- 14+ 839 0.83x10-4 0.99x10-3 • 17- 16+ 845 0.72x10-4 1.62x10-3 • 19- 18+ 849 0.91x10-4 11- experiment: ~10-5 W.u. see: G.Hackman et al., PRL 79, 4199 (1997)

  18. strength function Strength functions giant resonances of type ( X=el. or X=mag.) = peaks in the dependence of strength function of the excitation reduced probability on the excitation energy nonrotating nucleus – low rot. frequency limit with fast rotating nucleus – high rot. frequency limit with where

  19. strength function In the paper: J.Kvasil, N.Lo Iudice, V.O.Nesterenko, M.Kopal, PRC 58,209 (1998) and in other papers the strength function method is described which allows to determine the strength function without the explicite solving the RPA equations for each RPA phonon state . It is based on the replacing the Dirac -function in the definition of by a Lorentzian averaging function:

  20. strength function and then using the Cauchy theorem. After that we obtain the alternative expr. for : where: single-particle part going from the mean field (contains s.p. matrix elements of the transition operator collective part going from the residual interactions (contains s.p. matrix elements of the transition operator and all s.p. operators involved in separable residual interactions) Other quantity used in the comparison with experiments – moments of given strength function: energy unweighted summed strength (SR) energy weighted summed strength (EWSR)

  21. str. function results Isovector E1 resonance for neutrons for protons Relatively narrow peak with centroid of energy ~13 MeV becomes broader with increasing and mainly with increasing for all Hg isotopes (the same for 152Dy – see J.Kvasil, N.Lo Iudice, F.Andreozzi, F.Knapp, A.Porrino, PRC 73,034302 (2006))

  22. Monopole E0 resonance str. function results for neutrons for protons not high dependence on and only a slight dependence on

  23. Isoscalar E2 resonance str. function results for neutrons for protons not high dependence on but with increasing peak at 10 MeV is washed and broad distribution in the interval 12 MeV<E<25 MeV appears

  24. str. function results M1 – resonance (scissor mode in SD muclei ?) neutrons protons spin flip scissor in normal deformation scissor mode was predicted: N.Lo Iudice, F.Palumbo, PRL 41 (1978) and in superdeformation: I.Hamamoto, W.Nazarewicz, PL B297 (1992)

  25. str. function results • For orbital (scissor) part of M1 strength is small (peaked at 2.5-3 MeV) • and most of the total M1 strength is given by spin part (peaked at 5-6 MeV) • With increasing and mainly with increasing the role of the orbital part • increases and for the orbital (scissor) part is dominant and • peaked at 6 MeV. This increase of or is clearly correlated • with the expectations followed from the physical picture of the scissor mode • and from the simple oscillator estimations (see E.Lipparini, S.Stringari, Phys. • Lett. B65, 27 (1976), or N.Lo Iudice, Phys.Rev. C57, 11246 (1998): where is the centroid of the scissor like excitations.

  26. str. function results again the confirmation of the fact that the orbital (scissor) part of M1-strength increases with increasing and .

  27. conclusion Conclusion • Our cranked Nilsson + RPA method: • accounts well for the dynamical moments of inertia all along their deformation • and rotational paths; • reproduces the E1 transition probabilities among SD bands; • predicts a strong enhancement of the M1 strength of the scissor mode in SD • bands. Experimental question is if it possible to measure experimentally the resonances above the yrast line states? Theoretical question: we described SD bands using a simple cranked Nilsson + RPA model – is it possible to do it also with more microscopical approaches like cranked Skyrme filed + RPA model?

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