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Nuclear vorticity and general treatment of vortical, toroidal, and compression modes J. Kvasil 1), V.O. Nesterenko 2), W. Kleinig 2,3), P.-G. Reinhard 4), and N.Lo Iudice 5) 1) Institute of Particle and Nuclear Physics, Charles University, CZ-18000 Praha 8, Czech Republic 2) Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, 141980, Russia 3) Technical University of Dresden, Institute for Analysis, D-01062, Dresden, Germany 4) Institute of Theoretical Physics II, University of Erlangen, D-91058, Erlangen, Germany 5) Dipartimento di Scienze Fisiche, Universita di Napoli “Federico II”, Monte S. Angelo, Via Cinzia I – 80126 Napoli, Italy

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Motivation Toroidal and vortical character of the nucleon motion in the nucleus is the subject of investigation for many years – e.g.: G.T.Berstch, Nucl.Phys.A246, 253 (1975). P.E.Seer, T.S.Dumitrescu, T.Suzuki, C.H.Dasso, Nucl.Phys. A404, 350 (1985). D.C.Raventhall, J.Wambach, Nucl.Phys. A475, 468 (1987). E.C.Caparelli, E.J.V.de Passos, J.Phys.G.: Nucl.Part.Phys. 25, 537 (1999). V.M.Dubovik, A.A.Cheskov, Sov.J.Part.Nucl. 5, 318 (1975). S.F.Semenko, Sov.J.Nucl.Phys. 34, 356 (1981). H.L.Clark Y.W.Lui, D.H.Youngblood, Phys.Rev. C63, 031301(R) (2001). K.Kvasil, N.Lo Iudice, Ch.Stoyanov, P.Alexa, J.Phys.G.: Nucl.Part.Phys. 29, 753 (2003). or the review: N.Paar, D.Vretenar, E.Khan, G.Colo, Rep.Prog.Phys. 70, 691 (2007). Vorticity and compression modes can be excited as a second order modes above the standard electric multipole modes (corresponding transition operators appear as a second order terms in the long-wave decomposition of the standard electric multipole operators). However, the recent experimental techniques enable to detect compression dipole modes (or isoscalar E1 modes) - see e.g.: H.L.Clark Y.W.Lui, D.H.Youngblood, Phys.Rev. C63, 031301(R) (2001). J.Endres, et al., Phys.Rev.Letters 105, 212503 (2010). B.A.Davis, et al., Phys.Rev.Letters 79, 609 (1997). reaction

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Theoretical background Vorticity, toroidal and compression multipole operators Given transition is or is not connected with the vortical motion if: vortical motion (with whirls) irrotational motion (without whirls) where is the velocity field operator It is necessary to connect the velocity field with some measurable quantity which involves. Usually it is the nuclear current are s.p. creation and anihilation operators

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with where is effective charges and is effective gyromagnetic ratios ( ), are the s.p. spinors.) In the literature there are two ways how to connect the velocity field with the nuclear current: hydrodynamic approach ( see e.g. S.Misicu, PRC 73, 024301 (2006). ) Raventhall-Wambach approach (see D.C.Raventhall, J.Wambach, Nucl.Phys. A475, 468 (1987). )

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Hydrodynamic approach. velocity field is defined as: if given transition is vortical and vice versa If any measurable quantity is in the QM represented by an operator involving then the operator of the new quantity measuring the vorticity can be obtained from the former operator by the substitution: However, this evaluation of the vorticity ( based on ) is approximative because and are bound by charge - current conservation: (and therefore also ) is not independent on (see further discussions) one does not know a priori what part of information about nuclear flow is involved in m.el. of and what in

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Raventhall-Wambach (RW) approach D.C.Raventhall, J.Wambach, Nucl.Phys. A475, 468 (1987). The RW concept of the vorticity starts with the assumption that the nucleus current (generally ) can be written as a sum of two terms: where the component carries an information about a possible nonzero vorticity. By other words the quantity: is a measure of the vorticity. Given transition is connected with the vortical motion (with whirls) if and vice versa. In the RW paper m.el. of and are decomposed into sp. harmonics and vector sp. harmonics : RW paper

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Formally similar decomposition can be written also for : Matrix elements of the density and nuclear current are bound by the charge- density current conservation (here we suppose that the transition represents the excitation of the nucleus, therefore ) substituting above decompositions into charge -current constraint three functions are bound only two of them are really independent Using the charge-current constraint and using so called moment of function technique (see Appendix A for details) in the RW paper it was shown that the vector spherical harmonics decomposition for can be rewritten as (see Appendix A): with

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In the RW paper it was also shown that under this decomposition in the second term in the expr. for can be written as (see Appendix A): If we accept general assumption that nuclear current is connected with velocity field by and also if we accept the approximative correspondence then the second term of the expression for can be identify with the second term in and, really, the matrix elms. of are a measure of the vorticity (see the Appendix A)

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Toroidal, compression, vorticity multipole operators (see J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, P.Vesely, PRC 84, 034303 (2011)) Starting point in the derivation of the tor., com. and vor. multipole operators is the standard electric multipole operator: Spherical Bessel function can be decomposed in (long-wave decomposition): charge-density constraint where long-wave limit of the standard multipole operator long-wave limit of the toroidal multipole operator (see e.g. N.Paar, D.Vretenar, E.Khan, G.Colo, Rep.Prog.Phys. 70, 691 (2007) ) toroidal multipole operator appears as a second order term in long-wave decomposition of standard operator

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Vortical multipole operator can be obtained from the standard multipole operator by the substitution So: using sph. Bessel function decomposition and the expression where is given by and with and using per partes integration and vector spherical harmonics identities familiar compression multipole operator the lowest term in the long-wave decomposition of the vorticity multipole operator is linear in k

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in this work only transitions in even-even nuclei starting from their ground state are discussed in case the spurious modes connected with CoM motion should be removed correction terms in corresponding multipole operators appear: there is no CoM correction in the vorticity dipole operator – see Appendix B in: J.Kvasil, V.O.Nesterenko,W.Kleinig, P.-G.Reinhard, P.Vesely, PRC 84, 034303 (2011)

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Separable RPA approach In this work the SRPA approach is used for the description of nuclear states - V.O.Nesterenko, J.Kvasil, P.-G.Reinhard, PRC 66, 044307 (2002). V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC 74, 064306 (2006) The SRPA starts with the Skyrme energy functional (see Appendix B for details): - nucleon density - kinetic-energy density - spin-orbit density - nucleon current - spin current - vector kinetic-energy current - total kinetic energy - total Skyrme energy - total pairing energy - total Coulomb energy Basic idea of the SRPA: nucleus is excited by external s.p. fields: time even time odd

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The optimal set of generators was developed: V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC 74, 064306 (2006) P.Vesely, J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, V.Yu.Ponomarev, PRC. 80, 031302(R) (2009) J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, P.Vesely, PRC 84, 034303 (2011) E1 T=1 modes M1 modes Using linear response theory corresponding Hamiltonian is: HFB mean field separable residual interaction self-consistently obtained from mean field with one-body operators: where with inverse strength matrices:

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The self-consistent separable for, of the residual interaction enables to transform the RPA equations of motion into the system of homogeneous algebraic for forward and backward amplitudes of the RPA phonon creation operators with corresponding energy one-phonon states: quasiparticle creation and anihilation operators The rang of the RPA algebraic system of equations is given by. Usually is sufficient. Knowing the structure and energies of one-phonon states one can determine the strength function of given transition operator to simulate escape width and coupling to complex configurations were is the Lorentz weight function: Using the Cauchy theorem it is possible to determine even without knowledge of Individual solutions of the RPA equations – other advantage of the separability of residual interaction: V.O.Nesterenko, W.Kleinig, J.Kvasil, P.Vesely, P.-G.Reinhard, PRC 74, 064306 (2006)

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However, the actual smoothing effects depend on excitation energies. The escape width appear above the particle emission threshold and grow with energy due to widening of the emission phase space for and corrected strength function is given by double folding: where is the energy of the first emission threshold and is a minimal width ( in this work ). Photoabsorbtion cross section is practically given only by transitions (the higher and contributions can be neglected) where is the fine-structure constant.

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In this paper strength functions of standard electric dipole operator, toroidal dipole operator, compression dipole operator, and vortical dipole operator are analyzed for isoscalar (T=0), isovector (T=1) and electromagnetic transitions. The type of transition is given by effective charge and gyromagnetic ratios in corresponding transition operator: (T=0) transitions: (T=1) transitions: el.mg. transitions: - gyromagnetic ratios for free nucleons quenching factor connected with the mesonic degres of freedom in the nucleus ( see R.Alacron, R.M.Laszewski, D.S.Dale, PRC 40, R1097 (1989) )

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Results and discussions Photoabsorbtion cross section in 124 Sn in dependence on excitation energy Skt6 (m* = 1), SV-bas (m* = 0.9), SkM * (m* =0.79), SLy6 (m* =0.69), SkI3 (m* = 0.58) overall agreement with experimet for the most of Skyrme paremter. discrepancies can be seen in the particle-emission threshold and Pygmy regions experimental values taken form: V.V.Varlamov, M.E.Stepanov, V.V. Chesnakov, Izv.Ross.Ak.Nauk, ser.fiz. 67, 656 (2003)

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Reduced isovector E1 probabilities from the ground state to one-phonon states in Pygmy energy region (exp. taken from J.Enders, et al., PRL 105, 212503 (2010) ) Compression dipole strength function for the Pygmy energy region (exp. taken from J.Enders, et al., PRL 105, 212503 (2010) ) In order to obtain an agreement of calculated E1 and compression dipole strengths with experiment it is necessary to go beyond the RPA and to take into account also more complex configurations than only one-phonon ones (see e.g. J.Enders, et al., PRL 105, 212503 (2010) or E.Ltvinova, P.Ring, V.Tselyaev, PRC 78, 014312 (2008)).

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Toroidal dipole strength function for isoscalar T=0 and isovector T=1 transition operators with total nuclear current (red line), only convection part of the current (green line) and magnetization part of the current (blue line). Toroidal str. function for the isoscalar T=0 case is formed dominantly by the concvection part of the current while this strength for isovector T=1 case is built mainly by the magnetization part of the current Overall structure of the toroidal dipole giant resonance is similar for all Skyrme paramet- rizations

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Vorticity strength function for isoscalar T=0 and isovector T=1 transition operators with total nuclear current (red line), only convection part of current (green line) and only magnetization part of current (blue line). Vorticity str. function for the isoscalar T=0 case is formed dominantly by the concvection part of the current while this strength for isovector T=1 case is built mainly by the magnetization part of the current Overall structure of the vorticity dipole giant resonance is similar for all Skyrme paramet- rizations

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Compression dipole strength function for isoscalar T=0 and isovector T=1 transition operator The isoscalar T=0 compression dipole strengths for different Skyrme parameterizations are similar each other while the isovector T=1 compression dipole strengths are different The higher energy (E~25-35 MeV component of the isoscalar T=0 dipole strength is comparable with the lower energy component (E<20 MeV) while for isovector T=1 case the lower energy part is dominating

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Toroidal dipole, vorticity dipole and compression dipole strength functions in 124 Sn with Skyrme SLy6 interaction are compared for isoscalar T=0, isovector T=1 and el.mg. transitions. the T1 and V1 strengths for el.mg. transitions are formed mainly by the magnetization part of the current shapes of T1 and V1 str. functions are similar while the shape of C1 strength is diffe- rent ( especially for T=1 and el.mg. transitions )

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RPA values of toroidal dipole, vorticity dipole and compression dipole strength functions for the Skyrme SLy6 parameterization are compared with mean field values for both isoscalar T=0 and isovector T=1 transitions. like in the standard E1 transitions also for T1, V1 and C1 transitions the strength for T=0 case is shifted to lower energy after the residual inter- actions are taken into account while for the T=1 case the RPA cause a small shift to higher energies.

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Conclusion Comparison of calculated photoabsorbtion cross section and compression dipole strength with corresponding experimental data confirm the known fact that the RPA is able to describe only overall characteristics of the E1 giant resonance. In order to describe details of the electric dipole strenths in the Pygmy region one should go beyond the RPA (in accordance with papers: J.Enders, et al., PRL 105, 212503 (2010); E.Litvinova, P.Ring, V.Tselyaev, PRC 78, 014312 (2008); J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, P.Vesely, PRC 84, 034303 (2011) Toroidal and vorticity dipole strength functions for isoscalar T=0 transitions are formed dominantly by the convection part of the nuclear current while for isovector T=1 and electric E1 transitions the strengths are given by the magne- tization part of the nuclear current (see also J.Kvasil, V.O.Nesterenko, W.Kleinig, P.-G.Reinhard, P.Vesely, PRC 84, 034303 (2011) Toroidal dipole and vorticity dipole strength distributions are similar each other while the shape of compression dipole strength is a little bit different excited states by the toroidal transition operator are vortical while compression dipole modes are vortical only partly and only in some energy intervals (see fig.7)

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Appendix A: Raventhall and Wambach concept of vorticity D.G.Raventhall, J.Wambach, NPA 475, 468 (1987) Transition density and current can be decomposed into sphr. harmonics and vector sphr. harmonics: The substitution of these decompositions into the charge-current conservation : gives the condition for radial functions : three functions are bound only two of them are really independent 1

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Further we define the moment of : 2 Using per partes integration and this quantity can be rewritten: 1 3 From and it follows that only are really independent. 1 3 Further curl of will be analyzed. Using standard differential identities for vector sphr. harmonics (see e.g. D.A.Varshalovich, A.N.Moskalev, V.K.Khersonski, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1976 ) this curl can be written as: with Similarly as in we will determine the moment of (using per partes) 2 4 5 6

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From it can be seen that is dependent on (or on ) and therefore still contains a constraint due to the charge- current conservation. In order to remove this constraint we subtract from the following quantity: 6 4 7 and in such a way we define the operator : 8 with (using and ) 15 9 This function has zero moment: quantity is without charge- density constraint the quantity can be considered as free of charge- current constraint it containts a pure information about the current (without the influation from density)

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The quantity is independent on density and contains information only about the nuclear current. Its connection with the vorticity can be seen from the following considerations. Similarly as in the hydrodynamical approach we can suppose that the velocity field can be related to the nuclear current by: 10 Let us compare the matrix element of the second term in with the following quantity: 10 and matrix elements of can serve as a measure of the vorticity. 11 Comparison of with leads to the conclusion that under the expansion in we can write: 11 7 If we approximate then the second term in can be identified with the m.el. of the second term in 7 10

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29 Appendix B: Brief formulation of the SRPA approach (more details)

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32 with where and where enumerates T- even densities enumerates T- odd densities where enumerates T- even densities enumerates T- odd densities where

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34 with

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35,

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36 the matrix of the eq. system for and is symmetric and real this eq. system has nontrivial solution only if the determinat of its matrix is zero, - dispersion equation for

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37 p-h ( two-qp ) part of corresponding operator

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