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Electric and Vorticity Strengths in Heavier Nuclei J. Kvasil 1), V.O. Nesterenko 3), W. Kleinig 2), P.-G. Reinhard 4), P. Vesely 1) 1) Institute of Particle and Nuclear Physics, Charles University, CZ-18000 Praha 8, Czech Republic 2) Technical Universiy of Dresden, Institute for Analysis, D-01062, Dresden, Germany 3) Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia 4) Institute of Theoretical Physics II, University of Erlangen, D-91058, Erlangen, Germany

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1.Motivation and brief formulation of the Separable RPA approach 2. Different Skyrme parametrizations are analyzed from the point of view of the photoabsorption cross section 3. Photoabsorption cross section in the Pigmy region is discussed 4. M1 resonance is dicussed from the point of view of different Skyrme parametrizations 4. Vorticity multipole operator strength function is introduced as a measure of the irrotationality of a nuclear matter 5. Some preliminary results of the vorticity strength is presented

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Effective n-n interactions (Skyrme, Gogny, relativistic mean field) are widely used for the description of the static characteristics of spherical and deformed nuclei Dynamics of small amplitude vibrations is mainly described by the RPA. However, for heavy nuclei the standard RPA method requires the construction and diagonalization of huge matrices. RPA problem becomes simpler if the residual two-body interaction in the nuclear Hamiltonian is factorized as a product of two s.p. operators ( see e.g. P.Ring, P.Schuck, The Nuclear Many-Body Problem, Springer N.Y. (1980) ). where and are two-quasiparticle parts of s.p. operators

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We developed a general self-consistent separable RPA (SRPA) approach applicable to any density- and current- dependent functional - see e.g. sperical nuclei: deformed nuclei: V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, Phys.Rev. C66, 044307 (2002) V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard Phys.Rev. C74, 064306 (2006) for the determination of operators and by fully self-consistent method starting from the general energy functional with some are time-even and some are time-odd Basic idea of the SRPA method: nucleus is excited by external s.p. fields :

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where strength constant matrixes are Using TDHFB with the linear response theory we obtain : for electric type excitation for magnetic type excitation

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RPA equations: gives energies, forward and backward amplitudes of phonon operator RPA equations with the separable residual interactions can be transferred into the homogeneous system of algebraic equations. Dimension of the matrix of this system is given by the number of s.p. operators and in the residual interaction. Detailed description of our SRPA method can be found in the papers: W.Kleinig, V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, P.Vesely, PRC78, 044315 (2008) V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC74, 064306 (2006) Knowing the structure of phonons we can calculate el.mg. reduced probability from the RPA ground state to one-phonon state with the energy transition multipolaritytransition multipole operator

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Then the energy weighted strength function is: This quantity can be determined even without the solving the RPA equations for each individual phonon state using the Cauchy theorem and the substitution see V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC74, 064306 (2006)

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or in more detail: where Knowing the reduced probability or strength function we can determine the photoabsorption cross section:

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We use the Skyrme energy density for the energy functional - see e.g. J.Dobaczewski, J.Dudek, Phys.Rev. C52, 1827 (1995): with interaction parameters gauge invariance

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The dependence of the energy density on goes through the following densities and currents: density spin-orbit current kinetic energy density current spin-current kinetic energy – spin current pairing density

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Comparison of experimental photoabsorption cross-section with calculated values for different Skyrme parametrizations photoabsorption cross section gives possibility to test different parametrizations exp. taken from P.Carlos et al. NPA 172, 437(1971) similar results and similar agreement were obtained also for Mo, Sm, Sn izotopes

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W. Kleinig, V.O. Nesterenko, J. Kvasil, P.-G. Reinhard and P. Vesely, Phys. Rev. C, 78, 044313 (2008) - Z=102, 114, 120; isotopic chains -deformations: good agreement with Lublin-Strasbourg drop model; - energy trend: Skyrme-RPA description of E1(T=1) GR in rare-earth, actinide and superheavy nuclei --- --x-- SLy6

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Cumulative integral photoabsorption cross section in the low-energy (Pigmy) region (4 - 13 MeV) for bigger deformation steeper increase of the cumulative cross section starting from some energy (see 100 Mo) this starting energy depends on deformation splitting of strength funtion - see next pages 92-100 Mo

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Cumulative integral photoabsorption crossection in the low-energy (Pigmy) region (4 - 13 MeV) -see also D.P.Arteaga, E.Khan, P.Ring, PRC 79, 034311 (2009) for bigger deformation steeper increas of the cumulative cross section starting from some energy this starting energy depends on deformation splitting of strength funtion - see next pages

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E1 excitation strength function exciting operators: one can see splitting and broadening of E1 resonance with increasing

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HFB total energy for neutron rich Sn isotopes In the paper D.P.Arteaga, E.Khan, P.Ring, PRC 79, 034311 (2009) neutron very rich Sn isotopes were analysed in the framework of RMF approach - we tried to compare with different Skyrme parametrizations 144-160 Sn nuclei are deformed but soft

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Comparison of equilibrium neutron, proton and total deformations for different Skyrme parametrizations with the RMF results (see D.P.Arteaga, E.Khan, P.Ring, PRC 79, 034311 (2009) )

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Energy weighted E1 strength function for selected neutron rich Sn isotopes for Pygmy energy interval with the increasing number of neutrons E1 strength goes down in the Pygmy region

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Strength function for 142-152 Nd isotopes orbital (scissor) part contributes only for deformed systems neutrons protons time-odd densities and currents should be involved in the Skyrme functional so called high-energy M1 mode (induced by E2) - see: I.Hamamoto, W.Nazarewicz Phys.Lett. B297, 25 (1992) J.Kvasil, N.Lo Iudice, F.Andreozzi, F.Knapp, A.Porrino, Phys.Rev. C73, 034502 (2006) excited by operators:

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Comparison of M1 strength functions calculated with SkI3, SkM*, SkT6, SLy6 Skyrme parametrizations. relatively big differencies of different parametrizations possibility to test parametrizations by a comparison with experimental data on M1 strength excited by operators:

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Comparison of M1 strength functions calculated with different Skyrme parametrizations with experimental values. experimental values from: P.Sarriguren, et al.,PRC 54, 690 (1996) none of used parametrizations describes M1 strength for all investigated nuclei exciting operators: deformed nuclei - two peaks structure of the spin-flip resonance

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Comparison of M1 strength functions calculated with different Skyrme parametrizations with experimental values. experimental values from: Ca - S.K.Nanda, et al., PRC 29, 660 (1984)) Pb - R.M.Laszewski, et al., PRL 61, 1710 (1988) none of used parametrizations describes M1 strength for all investigated nuclei exciting operators: sperical nuclei - one peak structure of the spin-flip resonance

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RPA shifts and spin-orbital neutron-proton splitting the shape (one-peak or two-peaks structure) of the spin-flip resonance is a result of the interplay between the RPA energy shift and spin-orbital splitting - energy shift of the centroid of the spin-flip resonance caused by the switching on the residual interaction - centroids of proton and neutron spin-orbital splittings

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- M1 vs M1(T=1) - impact of tensor interaction Essential difference between M1 and M1(T=1) strength: tensor term in Skyrme functional G. Colo, H. Sagawa, S. Fracasso, and P.F. Bortignon, Phys. Lett. B, v.646, 227 (2007) no tensor with tensor SV-bas - strike effect of tensor interaction! - principle possibility to get both 1- and 2-bump structure - importance of refitting of Skyrme parameters

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Vorticity One of the basic questions of all hydrodynamical nuclear models: irrotationality of nuclear matter (with or without whirls?) irrotationality: velocity field operator : condition does not guarantee because: One can expect that if the nuclear matter is irrotational then: The question: how to investigate the irrotationality of nuclear matter in practice? But this is a problem because and are coupled by charge-current conservation:

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In papers: D.G.Raventhall, J.Wambach, NPA 475, 468 (1987). E.C.Caparelli, E.J.V.de Passos, J.Phys.G 25, 537 (1999). N.Ryezayeva, T.Hartmann, Y.Kalmykov, H.Lenske, P.von Neumann-Cosel, V.Yu.Ponomarev, A.Richter, A.Shevchenko, S.Volz, J.Wambach, PRL 89, 272502 (2002). so called transitional vorticity strength is defined – the idea is folllowing: vorticity operator: or: if nuclear matter is irrotational then all matrix elements of the vorticity operator are zero for the first sight (see in the previous slide) it seems: but it is not so because of the charge-current conservation gives uncertainty what is and what is

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In the paper D.G.Raventhall, J.Wambach, NPA 475, 468 (1987) decomposition into the spherical vectors is done: and it was shown (using the charge-current conservation) that with all information about the transitional vorticity is given by the radial transitional component of the nuclear charge current it was also shown that

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In papers: D.G.Raventhall, J.Wambach, NPA 475, 468 (1987). E.C.Caparelli, E.J.V.de Passos, J.Phys.G 25, 537 (1999). N.Ryezayeva, T.Hartmann, Y.Kalmykov, H.Lenske, P.von Neumann-Cosel, V.Yu.Ponomarev, A.Richter, A.Shevchenko, S.Volz, J.Wambach, PRL 89, 272502 (2002). so called transition vorticity strength : was introduced as a measure of the irrotationality of the nuclear matter (usually ). It was shown that the vorticity strength is significant for the transitions from the ground state to states in the Pigmy region and that these states have a toroidal character (for some lighter spherical nuclei).

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we introduced another quantity as a measure of the irrotationality - vorticity multipole operator vorticity multipole operator is directly connected with the long-wave decomposition of the standard electric multipole operator: using Bessel function decomposition where is the transition energy and toroidal multipole operator see e.g. D.Vretenar, N.Paar, P.Ring, T.Niksic, PRC 65, 021301 (2002) S.F.Semenko, Yad.Fiz. 34, 639 (1981) (nonstandard normalization of el.mg. multipoles)

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Vorticity multipole operator is obtained from (see exp. ) by the following substitution: Then Bessel function decomposition with long-wave limit of the vorticity multipole operator:

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The nonzero value of all matrix elements of all vorticity multipole operators can serve as a measure of the irrotationality of the nuclear matter. We restrict ourselves for and we calculate the strength function of : Dipole vorticity strength function can be compared with the dipole toroidal strength function: where dipole toroidal operator (involving the corrections to the C.o M. motion) is:

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Dipole vorticity strength function can be also compared with the squeezed dipole electric (or isoscalar E1) strength function: with the squeezed dipole E1 transition operator: Connection between squeezed dipole E1 operator and the dipole toroidal operator is discussed in J.Kvasil, N.Lo Iudice, Ch.Stoyanov, P.Alexa, J.Phys G 29, 753 (2003)

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Nuclear charge density operator: Nuclear charge current operator consists from convectional and magnetization parts: where effective charges and gyromagnetic ratios depend on the process of excitation ( see M.N.Harakeh, A.van der Woude, Giant Resonances, Clarendon 2001 ) el.mag.isoscalar T=0isovector T=1

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isoscalar vorticity strength is mainly formed by convective part of the nuclear charge current isovector vorticity strength is mainly formed by magnetization part of the nuclear charge current

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vorticity – exc. operators: toroidal– exc. operators: squeezed E1– exc. operators: similarity of the basic structure of the vorticity, toroidal and sqeezed dipole resonance

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Velocity field for the RPA state definition of the velocity field: with the density and current operator: in the figure the velocity projection onto the plane is plotted ( )

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Conclusions SRPA – effective method for the investigation of excited states in heavy nuclei different Skyrme parametrisations (SkI3, SkM*, SkT6, SLy6) give very similar and good agreement with experimental photoabsorption cross section (not so for M1 giant resonance) for bigger deformation steeper increase of the cumulative integral photoabsorption cross section with the increasing excitation energy is observed for energies above the particle emission threshold. Below this threshold this increase is not so conclusive significant vorticity dipole strength is observed in the excitation energy intervals (for 208 Pb): in these energy intervals one can expect a significant irrotationality of the nuclear matter in positive parity excited states isoscalar dipole vorticity strength is mainly formed by the convective charge current while the isovector dipole vorticity strength (low energy part) is mainly caused by the magnetization charge current

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