THE PYGMY DIPOLE CONTRIBUTION TO POLARIZABILITY: ISOSPIN AND MASS-DEPENDENCE

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1 TH PYGMY DIPOL CONTRIBUTION TO POLARIZABILITY: ISOSPIN AND MASS-DPNDNC V. BARAN 1, A.I. NICOLIN 1,2,, D.G. DAVID 1, M. COLONNA 3, R. ZUS 1 1 Faculty of Physics, University of Bucharest, 405 Atomistilor, POB MG-11, RO , Bucharest-Magurele, Romania 2 Horia Hulubei National Institute for Physics and Nuclear ngineering, Reactorului 30, RO , P.O.B. MG-6, Bucharest-Magurele, Romania 3 Laboratori Nazionali del Sud, INFN, Catania, Italy -mail : alexandru.nicolin@nipne.ro Received December 5, 2016 Abstract. Based on the dipole response of neutron-rich nuclear systems obtained within a transport model which relies on two coupled Vlasov equations for protons and neutrons, respectively, we determine the contribution to polarizability of the pygmy dipole mode. We employ two different density parameterizations of the symmetry energy and investigate the effect of isospin and mass on the pygmy contribution to polarizability. Key words: Pygmy Dipole Resonances, Vlasov equation. 1. INTRODUCTION In the last decade a growing interest to investigate the dipolar dynamics of atomic nuclei was related to the experimental observations which concern an enhanced 1 response below the well known Giant Dipole Resonance (GDR) and ascribed to the Pygmy Dipole Resonance (PDR) (see Refs. [1 3] for recent overviews of experimental information). An appropriate understanding of the features of this mode is important in connection to the density dependence of the symmetry energy and the properties of exotic nuclei, including the neutron skin or the development of new collective motions, and has direct implications in astrophysics. Despite the theoretical and computational efforts, however, there is no consensus on the nature of this resonance, some of the models supporting a collective mode, while other indicating a non-collective structure which is related to the neutron excitation within the outer skin of the core. A systematic experimental investigation on the properties of the pygmy dipole resonance on a wide set of nuclear masses and/or isospin is therefore strongly demanding. The xtreme Light Infrastructure Nuclear Physics (LI-NP, see Ref. [4 6]), currently built within the premises of the Horia Hulubei National Institute for Physics and Nuclear ngineering (IFIN-HH) [7], is a possible infrastructure where such systematic studies can be carried out. The first proposals for experiments are Romanian Journal of Physics 62, 301 (2017) v.2.0* #1fef39e6

2 Article no. 301 V. Baran et al. 2 summarized in the LI-NP White Book [8], while the Day 1 experiments can be found in Ref. [9]. In this article we report preliminary results on the contribution to polarizability of the pygmy dipole mode using a transport model based on two coupled Vlasov equations [10], one for protons and one for neutrons, and two different density parameterizations of the symmetry energy. In Section 2 we discuss the relation between the static polarizability and the symmetry energy in a simple approach, while in Section 3 we present the numerical results concerning the dipolar response, in particular pygmy dipole resonance and the associated contribution to polarizability, derived within a microscopic transport model. In the last section we gather our concluding remarks. 2. STATIC POLARIZABILITY AND TH ROL OF SYMMTRY NRGY The static polarizability of the nucleus, as a response function, is obtained as the ratio of the induced dipole moment to the electric field intensity that induces the polarization. Let ˆD be the dipole operator, 0 the energy of the ground state 0, and n the energies of the excited states n, all being the eigenstates of the unperturbed nuclear Hamiltonian. Under the static perturbation V D = e ˆD determined by the weak external uniform field, the perturbed ground state becomes: Ψ = 0 + e n ˆD 0 n. (1) n>0 n 0 Then, to the first order approximation, the polarizability is given by: α D = e Ψ ˆD Ψ = 2e 2 n>0 n ˆD 0 2 n 0 (2) Let us notice that since the response of a nucleus under an external 1 perturbation is contained in the excitation strength function S() = n>0 n ˆD 0 2 δ( ( n 0 )), (3) the dipole polarizability can be expressed as: α D = 2e 2 0 S() d. (4) This is a particularly useful quantity to additionally constrain the properties of the symmetry energy, specifically its nuclear density dependence, i.e., ε sym = sym A = F (ρ) + C(ρ) ρ (5) 3 2 ρ 0

3 3 The pygmy dipole contribution to polarizability: isospin and mass-dependence Article no. 301 with ρ = ρ n + ρ p. This quantity defines the isospin dependence of the total energy per particle, i.e., A (ρ,i) = A (ρ,i = 0) + sym A (ρ)i2. (6) Already in a simple approach, based on two fluids picture, Migdal has shown [11, 12] that the polarizability depends on the value of symmetry energy at saturation. Indeed, if we consider a constant electric field applied along the z-axis we can express the induced dipole moment as D(t) = NZ NZ X(t) = A A ( 1 Z zρ p d 3 r 1 N ) zρ n d 3 r. (7) Assuming that the total density remains constant one can easily show that the contribution to the energy due to the variations in the density of protons and neutrons, in the presence of the applied electric field, is Fig. 1 The strength function S() for 132 Sn. ( ) 1 δ 1 = 2 e zδρ 3 d 3 r, (8) where ρ 3 = ρ p ρ n and δρ 3 = δρ p δρ n. For this variation of isovector density δρ 3 we obtain an additional contribution to the total energy from the symmetry energy, namely δ 2 = ε sym (δρ 3 ) 2 d 3 r. (9) ρ 0 Minimizing the total energy, i.e., δ tot = δ 1 + δ 2, with respect to δρ 3, i.e., (δ tot )/ (δρ sym ) = 0, we get a linear dependence of δρ 3 on z, namely δρ 3 =

4 Article no. 301 V. Baran et al. 4 eρ 0 /4ε sym z. Then the polarizability is given by e 2 zδρ3 d 3 e r 2 z eρ 0 4ε α = = sym zd 3 r, (10) which, together with R = 1.2A 1/3, provides an A 5/3 mass dependence α = e2 A 8ε sym z 2 = e2 A 24ε sym r 2 = e2 A 3 24ε sym 5 R2 = e 2 (1.2) 2 A ( 5/3 ). (11) F (ρ) + C(ρ) ρ ρ0 The transport model described briefly in the next section provides the expected mass dependence of the polarizability and also shows that the polarizability does not depend only on the value of the symmetry energy at saturation, but is also influenced by the density dependence of the symmetry energy [13]. Here by considering the chain of tin isotopes 108,116,124,132,140,148 Sn we focus on the isospin dependence of α D as well as on the pygmy dipolar resonance contribution to the polarizability defined as α P DR = 2e 2 S() d. (12) P DR We expect that the ratio f α = α P DR /α D will manifest a greater sensitivity to the features of the pygmy dipole resonance than the fraction f m = m 1y /m associated with the energy-weighted-sum-rule m 1y = P DR S()d due to the appearance of the energy in the denominator in equation (12). 3. NUMRICAL RSULTS In order to estimate semi-classically the strength function, we employ a method described in Ref. [14] which provides the dipole response by integrating the Vlasov equations given by f τ (r,p,t) t + p m rf τ (r,p,t) r U(r) p f τ (r,p,t) = 0, (13) where τ = {p,n}, after the instantaneous perturbation V ext = ηδ(t t 0 ) ˆD, with p indicating protons and n neutrons. In these equations the self-consistent mean-field contains an isoscalar part of the form U is (ρ) = Aρ/ρ 0 + Bρ γ /ρ 0 and an isovector part of the form U iv (ρ) = C(ρ) ρ n ρ p τ q + 1 C (ρ n ρ p ) 2. (14) ρ 0 2 ρ ρ 0 Its numerical implementation, including the OpenMP parallelization, is described in detail in Ref. [10]. The strength function derived in this approach for 132 Sn is shown in Fig. 3. The coefficients in the mean-field are taken to reproduce the known features

5 5 The pygmy dipole contribution to polarizability: isospin and mass-dependence Article no α D (fm 3 ) I 2 Fig. 2 (Color online) The dipole polarizability of the Sn isotopes mentioned in the text, as a function of the square of the isospin for asysoft (the green circles) and asysuperstiff (the blue squares) OSs. In each case the dashed line corresponds to the linear fit. of nuclear matter at saturation [15]. In Fig. 2 we report the isospin dependence of the total dipole polarizability by considering two asy-os, asysoft and asysuperstiff [16], while in Fig. 3 we plot the contribution to polarlizability from the pygmy dipole region. We notice in both cases a linear dependence to the square of the isospin which is an indication of the role of the neutron skin on this quantity. Moreover, the pygmy contribution seems to go towards zero when the number of neutrons approaches the number of protons α PDR (fm 3 ) I 2 Fig. 3 (Color online) The contribution to dipole polarizability of the Sn isotopes mentioned in the text due to the pygmy dipole resonance. The green circles correpond to the asysoft case while the blue squares to the asysuperstiff. In both cases the dashed lines represents the linear fit.

6 Article no. 301 V. Baran et al CONCLUSIONS In this paper we discussed the isospin dependence of dipole polarizability and the contribution of the pygmy dipole resonance to the polarizability considering a semi-classical transport approach for two different density dependencies of the symmetry energy. We observe a linear dependence of the total dipole polarizability on the square of the isospin. A similar trend is observed for the contribution of the pygmy dipole resonance to this quantity. Since the neutron skin also depends quadratically on isospin in our approach we conclude that there an interplay between the dynamics of pygmy dipole resonance and the features of the neutron skin. The slope of α P DR exhibits a clear dependence on the asy OS which requires a more detailed investigation. Acknowledgements. For this work V. Baran, A.I. Nicolin, and R. Zus has been supported by the project 29 LI RO financed by the Institute of Atomic Physics. A.I. Nicolin has also been supported by PN /2016 financed by The National Authority for Scientific Research and Innovation. RFRNCS 1. O. Wieland and A. Bracco, Prog. Part. Nucl. Phys. 66, 374 (2011) 2. D. Savran et al., Prog. Part. Nucl. Phys. 70, 210 (2013). 3. D.M. Rossi et al., Phys. Rev. Lett. 111, (2013). 4. N.V. Zamfir, Rom. Rep. Phys. 68, S3 (2016). 5. S. Gales, Rom. Rep. Phys. 68, S5 (2016). 6. See the website of the infrastructure at 7. See the website of the institute at 8. The LI-NP White Book is availble at 9. F. Camera et al., Rom. Rep. Phys. 68, S539 (2016). 10. R. Tabacu et al., Rom. Journ. Phys. 60, 1441 (2015). 11. A. Migdal, J. xptl. Theoret. Phys. U.S.S.R. 15, 81 (1945). 12. J.S. Levinger, Phys. Rev. 107, 554 (1957). 13. V. Baran et al., AIP Conference Proceedings 1645, 267 (2015). 14. V. Baran et al., Phys. Rev. C 88, (2013). 15. V. Baran et al., Phys. Rev. C 85, (2012). 16. V. Baran et al., Phys. Rep. 410, 335 (2005).