The evolution of the low-lying E1 strength in proton-rich nuclei is analyzed
in the framework of the self-consistent relativistic Hartree-Bogoliubov (RHB)
model and the relativistic quasiparticle random-phase approximation (RQRPA).
Model calculations are performed for a series of N=20 isotones and Z=18
isotopes. For nuclei close to the proton drip-line, the occurrence of
pronounced dipole peaks is predicted in the low-energy region below 10 MeV
excitation energy. From the analysis of the proton and neutron transition
densities and the structure of the RQRPA amplitudes, it is shown that these
states correspond to the proton pygmy dipole resonance.Comment: 7 pages, 4 figures, to be published in Phys. Rev. Let

Paar, N.

Ring, P.

Vretenar, D.

English

arXiv

N. Paar

D. Vretenar

P. Ring

Crossref

Proton Electric Pygmy Dipole Resonance

The evolution of the low-lying E1 strength in proton-rich nuclei is analyzed in the framework of the self-consistent relativistic Hartree-Bogoliubov model and the relativistic quasiparticle random-phase approximation (RQRPA). Model calculations are performed for a series of N=20 isotones and Z=18 isotopes. For nuclei close to the proton drip line, the occurrence of pronounced dipole peaks is predicted in the low-energy region below 10 MeV excitation energy. From the analysis of the proton and neutron transition densities and the structure of the RQRPA amplitudes, it is shown that these states correspond to the proton pygmy dipole resonance

Paar, Nils

Vretenar, Dario

Ring, Peter

University of Zagreb Repository

PRL 94, 182501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending13 MAY 2005Proton Electric Pygmy Dipole ResonanceN. PaarInstitut fu¨r Kernphysik, Technische Universita¨t Darmstadt, Schlossgartenstrasse 9, D-64289 Darmstadt, GermanyD. VretenarPhysics Department, Faculty of Science, University of Zagreb, CroatiaP. RingPhysik-Department der Technischen Universita¨t Mu¨nchen, D-85748 Garching, Germany(Received 8 February 2005; published 10 May 2005)0031-9007=The evolution of the low-lying E1 strength in proton-rich nuclei is analyzed in the framework of theself-consistent relativistic Hartree-Bogoliubov model and the relativistic quasiparticle random-phaseapproximation (RQRPA). Model calculations are performed for a series of N  20 isotones and Z 18 isotopes. For nuclei close to the proton drip line, the occurrence of pronounced dipole peaks ispredicted in the low-energy region below 10 MeV excitation energy. From the analysis of the proton andneutron transition densities and the structure of the RQRPA amplitudes, it is shown that these statescorrespond to the proton pygmy dipole resonance.DOI: 10.1103/PhysRevLett.94.182501 PACS numbers: 21.10.Gv, 21.30.Fe, 21.60.Jz, 24.30.CzA number of experimental and theoretical studies of themultipole response of exotic nuclei far from stability havebeen reported in recent years. On the neutron-rich side, inparticular, the possible occurrence of the pygmy dipoleresonance (PDR), i.e., the resonant oscillation of theweakly bound neutron skin against the isospin-saturatedproton-neutron core, has been investigated. The onset oflow-lying E1 strength has been observed not only in exoticnuclei with a large neutron excess, e.g., for neutron-richoxygen isotopes [1], but also in stable nuclei with moderateproton-neutron asymmetry, such as 44;48Ca and 208Pb [2–4]. On the theory side, various models have been employedin the investigation of the nature of the low-lying dipolestrength. Recent studies include the application of theSkyrme Hartree-Fock plus quasiparticle random-phase ap-proximation (RPA) with phonon coupling [5,6], the time-dependent density-matrix theory [7], the continuum linearresponse in the coordinate-space Hartree-Fock Bogoliubovformalism [8], the quasiparticle phonon model [2,3,9,10],the relativistic RPA [11,12], and the relativistic quasipar-ticle RPA (RQRPA) [13,14]. The PDR is interesting notonly as a new and exotic mode of nuclear excitations, but italso plays an important role in predictions of neutroncapture rates in the r-process nucleosynthesis, and conse-quently in the calculated elemental abundance distribution.Even though the E1 strength of the PDR is small comparedto the total dipole strength, the occurrence of the PDRsignificantly enhances the radiative neutron capture crosssection on neutron-rich nuclei, as shown in recent large-scale QRPA calculations of the E1 strength for the wholenuclear chart [15,16].In the analysis based on the relativistic (Q)RPA [11–14],it has been shown that in neutron-rich nuclei the electricdipole response is characterized by the fragmentation ofthe strength distribution and its spreading into the low-05=94(18)=182501(4)$23.00 18250energy region. In light nuclei, the low-lying dipole strengthis not collective and originates from nonresonant single-neutron excitations. In medium-heavy and heavy neutron-rich nuclei, on the other hand, some of the low-lying dipolestates display a more distributed structure of the RQRPAamplitudes. The study of the corresponding transition den-sities and velocity distributions revealed the dynamics ofthe neutron pygmy dipole resonance.In this work, we employ the relativistic QRPA in thestudy of the evolution of low-lying dipole strength inproton-rich nuclei. Because the proton drip line is muchcloser to the line of  stability than to the neutron drip line,bound nuclei with an excess of protons over neutrons canbe found only in the region of light Z  20 and mediummass 20< Z  50 elements. For Z > 50, nuclei in the re-gion of the proton drip line are neutron deficient rather thanproton rich. In addition, in contrast to the evolution of theneutron skin in neutron-rich systems, because of the pres-ence of the Coulomb barrier, nuclei close to the proton dripline generally do not exhibit a pronounced proton skin, ex-cept for very light elements. Since in light nuclei themultipole response is generally less collective, all these ef-fects seem to preclude the formation of the pygmy dipolestates in nuclei close to the proton drip line. Nevertheless,the present analysis will show that proton pygmy dipolestates can develop in light and medium mass proton-richnuclei.The relativistic QRPA [13] is formulated in the canoni-cal single-nucleon basis of the relativistic Hartree-Bogoliubov (RHB) model and is fully self-consistent. Forthe interaction in the particle-hole channel, effectiveLagrangians with nonlinear meson self-interactions ordensity-dependent meson-nucleon couplings are used,and pairing correlations are described by the pairing partof the finite-range Gogny interaction. In both the particle-1-1  2005 The American Physical SocietyPRL 94, 182501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending13 MAY 2005hole and pairing channels, the same interactions are used inthe RHB equations that determine the canonical quasipar-ticle basis and in the matrix equations of the RQRPA. Thisfeature is essential for the decoupling of the zero-energymode which corresponds to the spurious center-of-massmotion. In addition to configurations built from two-quasiparticle states of positive energy, the RQRPA con-figuration space contains quasiparticle excitations formedfrom the ground-state configurations of fully or partially0 10 20 30E[MeV]00.511.52R[e2fm2 /MeV]0 10 20 30E[MeV]0 10 20 30E[MeV]0 2 4 6 8r[fm]-0.2-0.10.00.10.2r2δρ[fm-1 ]neutronsprotons0 10 20 30E [MeV]0 2 4 6 8r [fm]0 2 4 6 8 10r[fm]40Ca44Cr 46Fe42Ti44Cr9.44 MeV10.15 MeV46Fe46Fe 18.78 MeVFIG. 1. The RHB RQRPA isovector dipole strength distri-butions in the N  20 isotones, calculated with the DD-ME1effective interaction (upper panels). For 44Cr and 46Fe the protonand neutron transition densities for the main peak in the low-energy region are displayed in the lower panels and, for 46Fe, thetransition densities for the main GDR peak.18250occupied states of positive energy and the empty negative-energy states from the Dirac sea. In the present analysisof the dipole response of proton-rich nuclei in the fullyself-consistent RHB RQRPA framework, the density-dependent effective interaction DD-ME1 [17] is employedin the ph channel, and the finite range Gogny interactionD1S [18] in the pp channel.In Fig. 1 we display the RQRPA dipole strength distri-butions in the N  20 isotones: 40Ca, 42Ti, 44Cr, and 46Fe.The electric dipole responseBTEJ;!  12Ji  1X0fX;J00 hkQ^TJk0i  1jj0JY;J00 h0kQ^TJkiguv0  1Jvu0 2(1)is calculated for the isovector dipole operatorQ^ T11 NN  ZXZp1rpY1  ZN  ZXNn1rnY1: (2)For each RQRPA energy !, X and Y denote the corre-sponding forward- and backward-going two-quasiparticleamplitudes, respectively. v and u are the occupationnumbers of the single-particle levels in the canonical basis.The discrete spectra are averaged with the LorentziandistributionRE XiBE1; 1i ! 0f =2E Ei2  2=4; (3)with   1 MeV as an arbitrary choice for the width of theLorentzian.The dipole strength distributions in Fig. 1 are domi-nated by the isovector giant dipole resonance (GDR) at20 MeV excitation energy. In these relatively light sys-tems, the GDR still exhibits pronounced fragmentation.With the increase of the number of protons, low-lying di-pole strength appears in the region below the GDR, and, for44Cr and 46Fe, a pronounced low-energy peak is found at10 MeV excitation energy. What is the nature of thislow-lying dipole state? In the lower panels of Fig. 1 we plotthe proton and neutron transition densities for the peaks at10.15 MeV in 44Cr and 9.44 MeV in 46Fe, and comparethem with the transition densities of the GDR state at18.78 MeV in 46Fe. Obviously, the dynamics of the twolow-energy peaks are very different from that of the iso-vector GDR: the proton and neutron transition densities arein phase in the nuclear interior, and there is almost nocontribution from the neutrons in the surface region. Thelow-lying state does not belong to statistical E1 excitationssitting on the tail of the GDR, but represents a fundamentalmode of excitation: the proton electric PDR.In Fig. 2 we analyze the RQRPA structure of the dipoleresponse in 46Fe. This nucleus is located at the proton dripline (see Ref. [19] for a recent review on the limits ofnuclear stability), and, very recently, for the first timeevidence for ground-state two-proton radioactivity wasreported in the decay of 45Fe [20,21]. The discrete unper-turbed Hartree-Bogoliubov and full RQRPA E1 spectra of46Fe are shown in the left panel. Since the interaction isrepulsive in the isovector channel, one expects that theinclusion of the residual interaction will result in the shiftof the full RQRPA spectrum to higher energies with respectto the unperturbed spectrum. This is clearly the case for thestates in the GDR region. The lowest group of states at10 MeV, however, gains strength and is shifted to lowerenergy. Obviously, the nature of these states is not isovec-tor. The QRPA structure of these states is shown in the rightpanel, where for the three low-lying states at 9.02, 9.44, and10.26 MeV, as well as for the strongest state in the GDRregion at 18.78 MeV, we plot the corresponding QRPAamplitude of proton and neutron 2qp configurations 2qp  jX2qpj2  jY2qpj2; (4)with the normalization conditionX2qp 2qp  1: (5)For each of the four dipole states, in addition to theexcitation energy, we have also included the correspondingBE1 value. The QRPA amplitudes are shown in a loga-rithmic plot as functions of the unperturbed energy of therespective 2qp configurations. Only amplitudes whichcontribute more than 0.01% are shown, and we also differ-entiate between proton and neutron configurations. We1-267891011E[MeV]0 10 20 30 40E[MeV]00.20.40.60.81R[e2fm2 /MeV]28 30 32 34 36A00.51B(E1)[e2 fm2 ]32ArZ=18FIG. 3. The RHB RQRPA isovector dipole strength distri-bution in 32Ar (left panel). For the argon isotopes, the massdependence of the centroid energy of the pygmy peak and thecorresponding values of the integrated BE1 strength below10 MeV excitation energy are shown in the right panel.0.010.1110100protonsneutrons0.010.11100.010.1110X2 -Y2 [%]0 20 40 60 80 100E2qp0.010.11100 10 20 30E[MeV]00.511.522.533.54R[e2fm2 /MeV]RQRPA(full)RHB response9.02 MeV, 0.11 e2fm246Fe9.44 MeV, 0.26 e2fm210.26 MeV, 0.06 e2fm218.78 MeV, 3.09 e2fm2FIG. 2. The discrete Hartree-Bogoliubov and full RQRPA E1 spectrain 46Fe (left panel). For the main peaks inthe low-energy region and in the regionof the isovector GDR, the distributionsof the RQRPA amplitudes are shown asfunctions of the unperturbed energy ofthe respective 2qp configurations (rightpanel).PRL 94, 182501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending13 MAY 2005note that, rather than a single proton 2qp excitation, thelow-lying states are characterized by a superposition of anumber of 2qp configurations. Obviously, the pygmystates display a degree of collectivity that can be directlycompared with the QRPA structure of the GDR state at18.78 MeV. In addition, proton 2qp configurations accountfor 99% of the QRPA amplitude of the pygmy states,whereas the ratio of the proton to neutron contribution tothe GDR state is 2. For the GDR states, the 2qp con-figurations predominantly correspond to excitations fromthe sd shell to the fp shell. The structure of the pygmystates, on the other hand, is dominated by transitions fromthe 1f7=2 proton state at 0:17 MeV and from the 2p3=2proton state at 3.60 MeV (this state is bound only becauseof the Coulomb barrier). The energy weighted sum of thestrength below 11 MeV excitation energy corresponds to2.5% of the Thomas-Reiche-Kuhn (TRK) sum rule.Another example of particularly pronounced protonPDR is the proton-rich isotopes of Ar. In the left panel ofFig. 3 we display the RHB RQRPA electric dipolestrength distribution in 32Ar. In addition to the ratherfragmented GDR structure at  20 MeV, prominent pro-ton PDR peaks are obtained just below 10 MeV. For thefour states at 8.30, 8.94, 9.36, and 9.60 MeV, which con-stitute the pygmy structure, in Fig. 4 we plot the proton andneutron transition densities and the distributions of theRQRPA amplitudes, in comparison with the GDR state at18.01 MeV. In contrast to the isovector GDR, the protonand neutron transition densities of the pygmy states are inphase. The RQRPA amplitudes of the low-lying statespresent superpositions of many proton 2qp configura-tions, with the neutron contributions at the level of 1%.The dominant configurations correspond to transitionsfrom the proton states 1d3=2 at 2:09 MeV and 2s1=2 at4:07 MeV. The corresponding BE1 values are includedin Fig. 4. The energy weighted sum of the strength below10 MeV takes 5.6% of the TRK sum rule. In the right panel18250of Fig. 3 we display the mass dependence of the centroidenergy of the pygmy peaks and the corresponding values ofthe integrated BE1 strength below 10 MeV excitationenergy. In contrast to the case of medium-heavy and heavyneutron-rich isotopes, in which both the PDR and GDR arelowered in energy with the increase of the neutron number[14], in proton-rich isotopes the mass dependence of thePDR excitation energy and BE1 strength is opposite tothat of the GDR. The proton PDR decreases in energy withthe growth of the proton excess. This mass dependence isintuitively expected since, as we have shown, the protonPDR mode is dominated by transitions from the weaklybound proton orbitals. As the proton drip line is ap-proached, either by increasing the number of protons orby decreasing the number of neutrons, due to the weaker1-30.010.1110100protonsneutrons0.010.11100.010.1110X2 -Y2 [%]0.010.11100 20 40 60 80 100E2qp0.010.1110-0.100.1-0.100.1-0.100.1r2δρ[fm-1 ]-0.100.10 2 4 6 8 10r[fm]-0.100.18.30 MeV, 0.19 e2fm232Ar8.94 MeV, 0.26 e2fm29.36 MeV, 0.14 e2fm29.60 MeV, 0.13 e2fm218.01 MeV, 0.64 e2fm2FIG. 4. Transition densities andRQRPA amplitudes for selected 1states in 32Ar. For the main peaks inthe low-energy region and in the regionof the isovector GDR, the proton andneutron transition densities are shownin the left panel. The corresponding dis-tributions of the RQRPA amplitudes aredisplayed as functions of the unperturbedenergy of the respective 2qp configura-tions (right panel).PRL 94, 182501 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending13 MAY 2005binding of higher proton orbitals, one expects more inertoscillations, i.e., lower excitation energies. The number of2qp configurations which include weakly bound protonorbitals increases towards the drip line, resulting in anenhancement of the low-lying BE1 strength.In conclusion, we have employed the self-consistentRHB model and the RQRPA in the analysis of the low-energy dipole response in proton-rich nuclei close to theproton drip line. For nuclei with more protons than neu-trons, in the chain of N  20 isotones the RHB RQRPAcalculation predicts the occurrence of an exotic mode inthe low-energy region below 10 MeV: the proton electricpygmy dipole resonance. The analysis of the correspond-ing proton and neutron transition densities shows thatproton PDR states correspond to the oscillation of theproton excess against an approximately isospin-saturatedcore. The RQRPA amplitudes of the pygmy states display asuperposition of many proton 2qp configurations, whereasthe contribution of neutron excitations is at the level of 1%.The dominant configurations include weakly bound protonorbitals and also the states which are bound only becauseof the presence of the Coulomb barrier. In contrast to therather fragmented GDR, the PDR strength is concentratedin a narrow region of excitation energies well below theregion of giant resonances, and the energy weighted sum ofthe PDR strength corresponds to a few percent of the TRKsum rule. With the decrease of the number of neutrons, forthe chain of proton-rich Ar isotopes the proton PDR islowered in energy, and the integrated BE1 strength in thelow-energy region increases accordingly. For heavier nu-clei the proton drip line is located in the region of neutron-deficient rather than proton-rich nuclei, and therefore nolow-lying dipole strength is calculated in medium-heavyand heavy nuclei close to the proton drip line.This work has been supported in part by the Bundes-ministerium fu¨r Bildung und Forschung Project No. 06 MT193, by the Deutsche Forschungsgemeinschaft (DFG)18250under Contract No. SFB 634, by the Croatian Ministry ofScience Project No. 0119250, and by the Gesellschaft fu¨rSchwerionenforschung (GSI) Darmstadt.1-4[1] A. Leistenschneider et al., Phys. Rev. Lett. 86, 5442(2001).[2] N. Ryezayeva et al., Phys. Rev. Lett. 89, 272502 (2002).[3] J. Enders et al., Nucl. Phys. A724, 243 (2003).[4] T. Hartmann et al., Phys. Rev. Lett. 93, 192501 (2004).[5] G. Colo´ and P. F. Bortignon, Nucl. Phys. A696, 427(2001).[6] D. Sarchi, P. F. Bortignon, and G. Colo´, Phys. Lett. B 601,27 (2004).[7] M. Tohyama and A. S. Umar, Phys. Lett. B 516, 415(2001).[8] M. Matsuo, Nucl. Phys. A696, 371 (2001).[9] N. Tsoneva, H. Lenske, and Ch. Stoyanov, Nucl. Phys.A731, 273 (2004).[10] N. Tsoneva, H. Lenske, and Ch. Stoyanov, Phys. Lett. B586, 213 (2004).[11] D. Vretenar, N. Paar, P. Ring, and G. A. Lalazissis, Phys.Rev. C 63, 047301 (2001).[12] D. Vretenar, N. Paar, P. Ring, and G. A. Lalazissis, Nucl.Phys. A692, 496 (2001).[13] N. Paar, P. Ring, T. Niksˇic´, and D. Vretenar, Phys. Rev. C67, 034312 (2003).[14] N. Paar, T. Niksˇic´, D. Vretenar, and P. Ring, Phys. Lett. B606, 288 (2005).[15] S. Goriely and E. Khan, Nucl. Phys. A706, 217 (2002).[16] S. Goriely, E. Khan, and M. Samyn, Nucl. Phys. A739,331 (2004).[17] T. Niksˇic´, D. Vretenar, P. Finelli, and P. Ring, Phys. Rev. C66, 024306 (2002).[18] J. F. Berger, M. Girod, and D. Gogny, Comput. Phys.Commun. 63, 365 (1991).[19] M. Thoennessen, Rep. Prog. Phys. 67, 1187 (2004).[20] J. Giovinazzo et al., Phys. Rev. Lett. 89, 102501 (2002).[21] M. Pfutzner et al., Eur. Phys. J. A 14, 279 (2002).

The Proton Electric Pygmy Dipole Resonance

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