SEMICLASSICAL APPROACHES TO THE COUPLINGS BETWEEN NUCLEAR DIPOLE MODES AND SURFACE VIBRATIONS

SEMICLASSICAL APPROACHES TO THE COUPLINGS BETWEEN NUCLEAR DIPOLE MODES AND SURFACE VIBRATIONS VIRGIL BARAN 1, DORIN-GHEORGHE DAVID 1,2, MARIA COLONNA 2, ROXANA ZUS 1 1 Faculty of Physics, University of Bucharest, 45 Atomiştilor, POB MG-11, RO-77125, Bucharest-Măgurele, România 2 Laboratori Nazionali del Sud, INFN, 95123 Catania, Italy E-mail: baran@nipne.ro Received May 1, 216 Within a microscopic transport model based on Vlasov equation we investigate the macroscopic structure of pygmy dipole resonance in terms of the coupled motions of neutrons in excess, protons of the core and neutrons of the core respectively. The resulting picture is consistent with the predictions based on schematic models with generalized separable interactions which extend the Brink model for Giant Dipole Resonance and are able to predict the pygmy dipole modes. Then we explore how the surface vibrations affect the dipole dynamics as a consequence of the coupling between these collective modes. Key words: Giant dipole resonance, pygmy dipole resonance, quadrupole vibrations, strength function. PACS: 5.45.Ac, 5.45.Pq, 5.45.Tp. 1. INTRODUCTION The recent experimental progress at facilities which includes MSU, GANIL, GSI and RIKEN, allows now to investigate in a systematic manner neutron rich systems close to the limits of stability. New researches devoted to the nature of collective motions in such exotic nuclei were stimulated. Several experimental indications suggest the presence of a resonance-like electric dipole (E1) response below the Giant Dipole Resonance (GDR) [1 7]. A recent overview of the present understanding from the experimental point of view can be found in [8, 9]. These controversial modes exhausts few percentages of dipolar EWSR and their nature is a challenge for theory too [1]. PDR was interpreted as a collective motion in phenomenological, hydrodynamic descriptions [11 14], in nonrelativistic [15 17] or relativistic microscopic approaches [18] and in microscopic transport models [19 21]. Other studies however associate the observed strength below GDR to contributions from singleparticle type excitations excluding coherent, collective properties [22]. Also a fragmentation of the E1 response is weakening the collectivity in the low-energy region [23, 24]. One of the essential questions regards the macroscopic picture of nucleon vibrations in the PDR [25]. The GDR is macroscopically described as an out of phase RJP Rom. 61(Nos. Journ. Phys., 5-6), Vol. 875 884 61, Nos. 5-6, (216) P. 875 884, (c) 216 Bucharest, - v.1.3a*216.7.2

876 Virgil Baran et al. 2 oscillation of protons and neutrons. A qualitative picture proposes for PDR an oscillation of neutrons in excess against the inert, more stable core. Our task in the first section is to explore the dynamics of neutrons in excess, of the protons of the core and of the neutron of the core within a transport model based on Vlasov equation aiming to obtain information about the macroscopic motions of protons and neutron both for GDR and PDR. A comparison with the picture obtained within a generalized Brink model will be also presented. 2. THE MACROSCOPIC PICTURE OF PYGMY DIPOLE RESONANCE For the GDR the relevant degree of freedom is the distance between the center of mass of protons and the center of mass of neutrons: X = R p R n (1) from where the dipole moment is expressed as: D = NZ X. Motivated by the role A of the neutrons in excess and the core in the case of PDR we define three subsystems as in the three fluid hydrodynamical models [11] and we introduce two additional degrees of freedom. The distance between the core center of mass and neutrons in excess center of mass Y = N cr n,c + Z crp R N c + Z n,e (2) describes the relative motion core-neutrons in excess while the distance between the core neutrons and the core protons X c = R p R n,c (3) characterizes the core dipole vibration. Then the dipole momentum D can be expressed as the sum: D = N cz X c + N ez A c A Y = D c + D y (4) In this paper we shall study the nuclear dynamics within in a model based on the two coupled Landau-Vlasov kinetic equations for neutrons and protons: f q t + p f q m r U q f q r p = I coll[f n,f p ], (5) which determine the evolution of the one-body distribution functions f q ( r, p,t), with q = n,p [26]. We shall follow the dynamics of various collective modes and we switch-off the collision integral which determine their collisional damping. The neutrons in excess N e = N Z are defined and labelled from the beginning as the most external neutrons in the initial, ground state distribution. The self-consistent nuclear

3 Semiclassical approaches to couplings between nuclear dipole modes and surface vibrations 877 mean-fields U q appearing in Eq. (5) are derived from the Energy Density Functional (EDF) which results from a Skyrme-like (SKM ) effective interaction [27]. The potential energy density is: E pot (ρ,ρ i ) = A 2 ρ 2 ρ + B ρ σ+1 σ + 1 ρ σ + C(ρ) ρ 2 i, (6) 2 ρ from where total energy of the system is calculated by adding the kinetic contribution: E = d 3 r (E kin (r) + E pot (r)). Here ρ = ρ = ρ n + ρ p is the total density, = ρ i = ρ n ρ p ) is the isovector density while ρ p ρ n are the local proton and neutron densities. The functional derivative of E pot with respect to the proton (neutron) density U q (ρ) = δe pot (r)/δρ q (r) leads to: U p = A ρ ( ρ ) σ ρ i + B C(ρ) + 1 ρ ρ ρ 2 U n = A ρ ( ρ ) σ ρ i + B + C(ρ) + 1 ρ ρ ρ 2 dc(ρ) ρ 2 i, (7) dρ ρ dc(ρ) ρ 2 i. (8) dρ ρ In the EDF formalism these coefficients encode more physics than those provided within a usual HF theory since their values A = 356 MeV, B = 33 MeV, σ = 7/6A, B and σ are fixed from the requirements to reproduce the saturation properties of symmetric nuclear matter with ρ =.16fm 3, E/A = 16 MeV and a compressibility modulus K = 2 MeV. In the isovector sector, while keeping the value of symmetry energy at saturation almost the same, we shall allow for three different dependences with density away from equilibrium which can be distinguished by de sym /A the value of slope parameter around saturation point, L = 3ρ ρ=ρ. In dρ the case of asystiff EOS the coefficient C(ρ) is constant (i.e. C(ρ) = constant 32MeV ), the slope parameter being L = 72.6MeV. For the asysoft EOS we imply a Skyrme-like, SKM*, parametrisation with C(ρ) = (482 1638ρ)MeV fm 3 ρ which leads to a slope parameter small L = 14.4MeV. For an asysuperstiff EOS, C(ρ) = 32 2ρ, the symmetry term increases rapidly around saturation density ρ ρ (ρ + ρ ) being characterized by a value of slope parameter L = 96.6MeV. From the one-body distribution functions we calculate the local densities as integrals over the momentum space and determine the time evolution of the position of the center of mass of the core protons, core neutrons and of the neutrons in excess. The integration of the transport equations is based on the test-particle (t.p.) method [28], with a number of 6 t.p. per nucleon in this case, ensuring in this way a very good spanning of the phase-space. The E1 response is studied by considering a GDR-like initial condition [2], induced by an instantaneous excitation

878 Virgil Baran et al. 4 V ext = ηδ(t t ) ˆD at t = t = 3fm/c after the ground state preparation [29]. In this case, a boost of all neutrons against all protons is induced while keeping the Center of Mass of the nucleus at rest. After the perturbation the excited state becomes Φ(t ) = e iη ˆD Φ, where is the ˆD is the dipole operator and Φ is the state before perturbation. The value of η can be related to the initial expectation value of the collective dipole momentum ˆΠ, Φ(t ) ˆΠ Φ(t ) = η NZ A. Here ˆΠ is canonically conjugated to the collective coordinate operator ˆX, i.e. [ ˆX, ˆΠ] = i [3]. In Figure 1 we plot the time evolution of the collective dipolar coordinates D(t),D y (t) and D c (t). The strength function S(E) = n> n ˆD 2 δ(e (E n D(t)(fm) D y (t)(fm) D c (t)(fm) 2 1-1 -2 1-1 -2 1-1 (a) (b) (c) -2 2 4 6 8 1 t(fm/c) Fig. 1 (Color online) The time evolution of the total dipole moment D(t) (a), of the dipole moments D y(t) (b) and D c(t) (c) with an asystiff EOS following the instantaneous excitation at t = 3fm/c in 148 Sn. E )), which characterizes the response of the system and is directly related to the excitation probability in unit time, is calculated from the imaginary part of the Fourier transform of the time-dependent expectation value of the dipole momentum D(t) = NZ A X(t) = Φ(t) ˆD Φ(t) as S(E) = Im(D(ω)) πη = A Im(D(ω)) NZ π Φ(t ) ˆΠ Φ(t ), (9)

5 Semiclassical approaches to couplings between nuclear dipole modes and surface vibrations 879 Here D(ω) = tmax t D(t)e iωt dt, E n is the excitation energy of the state n while E is the energy of the ground state = Φ. We follow the dynamics of the system until t max = 183fm/c for an initial perturbation along the z-axis. At t = t = 3fm/c we determine the collective momentum which appears in Eq. (9). The artifacts resulting from a finite time domain analysis of the signal were eliminated by adopting a filtering procedure, as described in [31]. A smooth cut-off function was introduced such that D(t) D(t)cos 2 πt ( ). With an asystiff EOS the E1 2t max strength function for 148 Sn obtained as described above is represented in Fig. 2. The PDR response is associated to the peak situated at E P DR = 7.2MeV. The EWSR 25 2 S(E) (fm 2 /MeV) 15 1 5 2 4 6 8 1 12 14 16 18 2 E(MeV) Fig. 2 (color online) The strength function for 148 Sn with asystiff EOS. fraction exhausted by the this mode is obtained as the ratio m 1,y /m 1 with: m 1,y = ES(E)dE (1) P DR and is around 7.% for this isotope and asy-eos. From the time evolution of D y (t) and D c (t) we also calculated the corresponding strength functions associate with the two degrees of freedom and plotted them in Fig. 3. We notice that such an analysis provides the macroscopic structure of normal modes associate with the dipole dynamics, PDR and GDR respectively, in terms of the two degrees of freedom Y and X c. We conclude that the GDR is determined by in phase oscillations of these quantities. With other words the neutrons in excess and neutrons of the core move in the same direction, opposite to the direction of motion of the protons. This is consistent with the qualitative picture expected for

88 Virgil Baran et al. 6 S(E) (fm 2 /MeV) 25 2 15 1 5-5 -1 2 4 6 8 1 12 14 16 18 2 E(MeV) Fig. 3 (color online) The strength function associated to the total dipole response (black solid line), with the D y (red dashed line) and with the D c (blue dot-dashed line). Giant Dipole Resonance. For PDR mode instead, the two quantities Y and X c are out of phase and therefore neutrons in excess are moving in the same direction with the protons of the core as is illustrated in Fig. 4. It is interesting that a similar picture was obtained within a schematic model with a generalized separable interaction [32]. Fig. 4 (Color online) The macroscopic structure of nucleon vibrations for GDR and PDR in terms of the collective degrees of freedom Y and X c.

7 Semiclassical approaches to couplings between nuclear dipole modes and surface vibrations 881 3. THE COUPLING OF DIPOLE MODES TO QUADRUPOLE VIBRATIONS In this section we briefly present first results related to the possible nonlinear couplings between dipolar modes and isoscalar quadrupole vibrations. Already in a Goldhaber-Teller model is expected that the shift of the proton and neutron spheres will determine a shape deformation of the nuclear system which will generate latter the quadrupole oscillations. Some features of the dipole and quadrupole dynamics induced by such couplings due to residual interaction were studied in a time-dependent Hartree-Fock theory [33]. The early time evolution of these coupled motions was investigated for different initial conditions and different intensities of the perturbations. Here, within a Vlasov approach, we shall study the quadrupole dynamics induced by a dipole boost along z-axis described in the previous section over much larger time scales. At each time step we determined the quadrupole moment in real space, QK(t) 2 15 1 5-5 -1-15 -2 2 4 6 8 1 12 14 16 18 2 2 1 QK(t) -1-2 2 4 6 8 1 12 14 16 18 2 t(fm/c) Fig. 5 (Color online) The quadrupole oscillations in momentum space for the initial condition corresponding to a dipolar boost in the case of asysoft EOS (red dashed lines) and asysuperstiff EOS (black solid line). and in momentum space: Q = QK = A (2zi 2 x 2 i yi 2 ) (11) i=1 A i=1 (2p z 2 i p x 2 i p y 2 i ) (12)

882 Virgil Baran et al. 8 In Fig. 5 for 14 Sn we show the time evolution of QK after a dipole boost in momentum space at t = 3fm/c in the case of asysoft EOS (red dashed line) and asysuperstiff EOS (black solid line). We notice that the energy transfer between the two modes 4e+5 3e+5 QK(E) 2e+5 1e+5 1.5 3 4.5 6 7.5 9 1.5 12 13.5 15 4e+5 3e+5 QK(E) 2e+5 1e+5 1.5 3 4.5 6 7.5 9 1.5 12 13.5 15 E(MeV) Fig. 6 (Color online) The Fourier spectrum corresponding to quadrupole oscillations in momentum space for the dipolar initial condition in the case of asysoft EOS (red dashed lines) and asysuperstiff EOS (black solid line). manifests some differences: the quadrupole oscillations appears faster for asysoft EOS but larger amplitudes are maintained for asysuperstiff EOS. Both features can be related to the differences in the neutron skin properties determined by the two asy-eos. The Fourier spectrum obtained in the two cases, see Fig. 6 shows a mode at an energy around 11.5 MeV. In order to clarify the nature of the quadrupole vibration induced by the dipole boost we studied the evolution of the quadrupole moment after a quadrupole boost in momentum space. In this case we expect an excitation of Giant Quadrupole Resonance. The results are reported in the Fig. 7 for the same asy-eos. Being an isoscalar mode the frequency of this vibration is not influenced by the asy-eos, as expected. The position of the peak is the same as for the previous initial condition. Therefore we conclude that the excited mode following the dipolar boost is the isoscalar Giant Quadrupole Resonance.

9 Semiclassical approaches to couplings between nuclear dipole modes and surface vibrations 883 5 QK(t) -5 2e+7 5 1 15 2 t(fm/c) 1.5e+7 QK(E) 1e+7 5e+6 1.5 3 4.5 6 7.5 9 1.5 12 13.5 15 E(MeV) Fig. 7 (Color online) Top: The quadrupole oscillations in momentum space for the dipolar initial condition in the case of asysoft EOS (red dashed lines) and asysuperstiff EOS (black solid line). Bottom: The Fourier spectrum corresponding to quadrupole oscillations in momentum space for the quadrupolar initial boost in the case of asysoft EOS (red dashed lines) and asysuperstiff EOS (black solid line) 4. CONCLUSIONS Summarizing, we addressed some open questions raised recently concerning the structure of PDR within a microscopic semi-classical transport approach which is based on Landau-Vlasov equations. Our analysis do not confirm the oversimplified picture of PDR as corresponding to the oscillations of excess neutrons against the inert isospin symmetric core. Within the transport model the dynamical simulations show a more complex structure which includes an isovector excitation of the core. Consequently, the PDR mode appears from our calculations as an out of phase superposition of oscillations associated with the collective coordinates Y and X c which describe the motion of excess neutrons against the core center of mass and the relative motion of core neutrons against core protons respectively. In the second part we presented first results related to the nonlinear couplings between the dipole and quadrupole modes in a transport approach based on Vlasov equation. Acknowledgements. This work for V. Baran has been supported by the project from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE- 211-3-972.

884 Virgil Baran et al. 1 REFERENCES 1. T. Hartmann et al., Phys. Rev. Lett. 85 (2) 274. 2. T. Hartmann et al., Phys. Rev. C 65 (22) 3431. 3. P. Adrich et al., Phys. Rev. Lett. 95, 13251 (25). 4. O. Wieland et al., Phys. Rev. Lett. 12, 9252 (29). 5. A. Klimkiewicz et al., Phys. Rev. C 76, 5163(R) (27). 6. A. Tamii et al., Phys. Rev. Lett. 17, 6252 (211). 7. T. Kondo et al., Phys. Rev. C 86, 14316 (212). 8. T. Aumann and T. Nakamura, Phys. Scr. T 152, 1412 (213). 9. D. Savran, T. Aumann and A. Zilges, Prog. Part. Nucl. Phys. 7, 21 (213). 1. N. Paar, D. Vretenar, E. Kahn, G. Colo, Rep. Prog. Phys. 7, 691 (27). 11. R. Mohan, M. Danos, L.C. Biedenharn, Phys. Rev.3 174 (1971). 12. Y. Suzuki, K. Ikeda, H. Sato, Prog. Theor. Phys.83 18 (199). 13. S.I. Bastrukov et al., Zeit. Phys. A 664 (1992) 258. 14. S.I. Bastrukov et al., Phys. Lett. B 664 (28) 258. 15. N. Tsoneva, H. Lenske, Phys. Rev. C 77 24321 (28). 16. E.G. Lanza, A. Vitturi, M.V. Andres, F.Catara, D.Gambacurta, Phys. Rev. C 84 6462 (211). 17. X. Roca-Maza et al., Phys. Rev. C 85 (212) 2461. 18. D. Vretenar, N. Paar, P. Ring, G.A. Lalazissis, Nucl. Phys. A 692 496 (21); D. Vretenar, T. Niksic, N. Paar, P. Ring, Nucl. Phys. A 731 281 (24). 19. M. Urban, Phys. Rev. C 85 34322 (212). 2. V. Baran et al., Phys. Rev. C 85 5161 (212). 21. C. Tan et al., Phys. Rev. C 87 14621 (213). 22. P.-G. Reinhard et al., Phys. Rev. C 87 14324 (213). 23. D. Sarchi, P.F. Bortignon, G. Colo, Phys. Lett. B 61 27 (24). 24. D. Gambacurta, M. Grasso, F. Catara, Phys. Rev. C 84 3431 (211). 25. N. Paar, J. Phys. G: Nucl. Part. Phys. 37, 6414 (21). 26. V.Baran, M. Colonna, M. Di Toro, V. Greco, Phys. Rep. 41, 335 (25). 27. M. Colonna, M. Di Toro, A. B. Larionov, Phys. Lett. B 428, 1 (1998). 28. C. Gregoire et al.,nucl. Phys. A 465, 77 (1987). 29. F. Calvayrac, P.G. Reinhard, E. Suraud, Ann. Phys. 225, 125 (1997). 3. V. Baran et al., Rom. J. Phys. 57, 36 (212). 31. P.-G. Reinhard, P.D. Stevenson, D. Almehed, J.A. Maruhn, M.R. Strayer, Phys. Rev. E 73, 3679 (26). 32. A. Croitoru et al., Rom. J. Phys. 6, 748 (215). 33. C. Simenel, Ph. Chomaz, Phys. Rev. C 8 6439 (29).