The Giant Dipole Resonance in Ar-Isotopes in the Relativistic RP A

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1 917 Progress of Theoretical Physics, Vol. 98, No.4, October 1997 The Giant Dipole Resonance in Ar-Isotopes in the Relativistic RP A Zhongyu MA,*'**'*** Hiroshi TOKI,* Baoqiu CHEN** and Nguyen Van GIAI**** * Research Center for Nuclear Physics, Osaka University, Ibaraki 567 ** China Institute of Atomic Energy, P. O. Box, 275/18, Beijing *** Institute of Theoretical Physics, Academia Sinica, Beijing **** Division de Physique Theorique, Institut de Physique Nucleaire F Orsay Cedex (Received June 6, 1997) The isovector giant dipole resonance in Ar-isotopes is studied in the framework of the relativistic random phase approximation starting from effective Lagrangians originally proposed for describing nuclear ground states. The calculations are done with the parameter set TMI which satisfactorily describes the ground states of stable and unstable nuclei up to the drip lines. It is found that the isovector dipole strength in nuclei near drip lines is split into two components: one peaks at the normal giant dipole energy, the other is located at low energy. The lower dipole state is due to the weakly bound neutron (or proton) excitation, which may cause large cross-sections in collisions with heavy nuclei by Coulomb excitation. 1. Introduction The recent development of radioactive nuclear beams has opened a new era for studying unstable nuclei and for exploring new physical phenomena. One of the most interesting discoveries with the radioactive beams is the existence of a neutron halo in nuclei containing neutrons with a very low separation energy and having a far extending neutron distribution. The experiments by Tanihata et al. l ) first showed quite large reaction cross-sections in collisions of 11 Li on high-z targets, an indication that the matter radius of 11 Li is much larger than what is predicted by the usual roa-lj3 law. New types of excitation modes are expected due to the excess of neutrons at the surface. A schematic picture of the new modes has been proposed by various authors. 2 ),3) The distribution of weakly-bound nucleons extends beyond the usual nuclear radius and decouples from the nuclear core. The collective motion is separated into two parts, one being produced by the vibration of the core and the other one originating from the motion of the weakly-bound nucleons against the core. The second type of excitation is predicted to be at much lower energy than the core mode. Recently, radioactive beam facilities have become available in various laboratories, and they have allowed the possibility to experimentally measure such new excitation modes. It is therefore desirable to investigate the new modes theoretically. The relativistic approach based on effective Lagrangians with nucleon-nucleon interactions mediated by the exchange of mesons has achieved great success in re-

2 918 Z. M a, H. Toki, B. Chen and N. V. Giai cent years. The relativistic mean field (RMF) theory with non-linear meson selfinteractions can describe ground state properties of nuclei, not only spherical but also deformed nuclei and nuclei far from the (3 stability line. 4 )-7) The relativistic random phase approximation (RRPA) is the natural extension of the RMF approach for studying nuclear excited states and giant resonances. This has been done first in the framework of Walecka's linear a - w modei 8 )-1l) and more recently with effective Lagrangians having non-linear self-interaction terms. 12 ),13) These investigations show that the collective excitations and giant resonances in closed shell nuclei can be described well by RRPA with non-linear Lagrangians. The purpose of this work is to apply the RRPA aproach to the study of collective excitations in unstable nuclei, and more specifically in the Ar-isotope chain from the proton drip line to the neutron drip line. We limit ourselves to a spherical approximation. In collisions with heavy targets the isovector dipole mode (L1L = 1, L1T = 1) is most strongly excited by Coulomb excitation, and therefore we concentrate on the giant dipole resonance (GDR) in the Ar isotopes. The recently proposed parameter set TM1 6 ) is adopted in our calculations. This Lagrangian has a non-linear w self-energy term in addition to the non-linear a self-energy term. The non-linear w self-coupling is essential for accurately modeling the nucleon self-energy of the relativistic Brueckner-Hartree-Fock theoryl4) and nuclear matter properties in the mean field approximation. We found essentially the same results I2 ),13) with other non-linear parameter sets, such as NL-SH,15) and show only the results with TM1 in this paper. We briefly introduce the RRPA method in 2. The ground state properties of the Ar-isotopes calculated in RMF are summarized in 3. The isovector GDR resonances of the Ar-isotopes are then discussed in 4. Conclusions are drawn in the final section. 2. The relativistic random phase approximation The response function of a quantum system to an external field Q is given by the imaginary part of the polarization operator: 1 R(Q,Qjk,E) = -ImII(Q,Qjk,k,E). 7r The response function describes the distribution of transition strength of the operator Q. Since the unperturbed ground state is treated in the Hartree approximation, i.e., exchange interactions are omitted, the RRPA polarization operator can be obtained by summing the infinite series of ring diagrams. Therefore, the RRPA polarization operator II satisfies the usual RPA integral equation, 16) II(P, Qj k, k', E) = IIo(P, Qj k, k', E) - L g; J (1) d3kld3k2iio(p, Ti; k, kl' E)Di(kl - k2' E)II(Ti' Q; k2' k', E), (2) where IIo is the unperturbed (Hartree) polarization operator. The explicit definitions of II and IIo as well as the expressions of IIo in terms of the Dirac-Hartree single-

3 The Giant Dipole Resonance in Ar-Isotopes in the Relativistic RPA 919 particle states can be found in Ref. 9). In the relativistic approach, the residual particle-hole interactions are just given by the meson propagators Di and coupling constants gi' In Eq. (2) the index i runs over it, wand p mesons, and the values of the coupling constants gi are specified by the effective Lagrangian. The operators r i are T'j = 1, r w =,,/1-' and r p =,,/1-'7 for the three mesons of the TMI Lagrangian. We start from an effective Lagrangian of the form I:- = ifi(hl-'{}i-' - MN - grrit - gw"/i-'wl-' - gpta,,/l-'p:)1f/ + ~{}l-'it{}l-'it - Urr(iT) - ~ WI-'VWI-'V + Uw(w) al-' a 1 Ral-'vRa ;r; I-'A 1 (1 ),Tr 1 Fl-'vF 2,mpp PI-'-4 I-'l'-' e,,/ 1-'2, -T3' -4 I-'l" (3) where ( ) WI-'V == {}I-'W l' - {}V wi-', Ral-'l' == {}I-' pal' _ {}V pal-' + gpfabc pbl-' pcv, FI-'l' == {}I-' Al' _ {}l' AI-', _~ 2 2 ~ 3 ~ 4 )_~ 2 I-' ~ I-' 2 Urr it - 2mrriT + 3g2iT + 4g3iT, Uw(W - 2mww WI-' + 4C3(w WI-'). (5) The nucleon field If/ interacts with it, W and P~ meson fields and the photon field AJl" The coupling constants grr, gw and gp, the parameters g2, g3 and C3 appearing in the self-interaction terms U rr and U w, and the it meson mass are adjusted to reproduce the ground state properties of finite nuclei as well as bulk properties of nuclear matter. The parameter set TM1 can be found in the corresponding reference. 6 ) In field theory, the equations of motion for fermion and boson fields are obtained by variations of the action of the system with respect to the corresponding fields. The first order variation of the action with respect to a given meson field cp gives the field equation (Klein-Gordon equation) satisfied by this field. Self-consistently solving the set of Dirac and Klein-Gordon equations determines the classical value cpo of the field cp which makes the action stationary. Then, the second order variation of the action around cpo will lead to the equation satisfied by the meson propagator. 17) For example, for the scalar meson propagator we have (4) (6) It is clear that the meson propagators with the non-linear self-energies in the effective Lagrangian (3) are no longer simple Yukawa functions. Taking the Fourier transform of Eq. (6) we obtain the expression of the propagator in momentum space, (E2 - k 2 )Drr(k - k', E) - (2:)3 J 8 rr (k - kddrr(k1 - k', E)d 3 k1 = (27r)38(k - k'), (7)

4 920 Z. Ma, H. Toki, B. Chen and N. V. Giai For the effective La where S~(k - k') is the Fourier transform of a2u~(cy)/acy2. grangian (3) the functions S~ and Sw are S~(k - k') = J e-i(k-k').r(m~ + 292cy(r) + 393cy 2 (r))d 3 r, Sw(k - k') = J e-i(k-k').r(m: + 2c3w~(r))d3r, (8) cy(r) and wo(r) being the classical values of the cy and w fields at point r. In the limit of the linear model, we recover the usual meson propagator, which is a local function in momentum space. 3. Ground state properties of Ar isotopes In this section we briefly present the main results of ground state calculations performed in the RMF approximation. The ground state properties are not the main focus of this work, but some aspects are nevertheless interesting since they can shed light on the behaviour of the dipole response functions which will be presented in the next section. In any case, it is necessary to determine the self-consistent mean field if one wishes to build the unperturbed and RRPA polarization operators IIo and II. We have studied the chain of Ar isotopes (Z = 18) from 30 Ar to 52 Ar. Spherical symmetry is assumed for single-particle orbitals. Since most of these nuclei do not correspond to closed subshells, the usual filling approximation has been employed. Each orbital i = (nlj) is occupied by ni nucleons with 1 ::::; ni ::::; 2j + 1, and the nuclear densities are expressed as weighted sums over occupied orbitals. For instance, the baryonic density is (9) f'!.. ii d.:l r (1m) Fig. 1. Proton (dashed) and neutron (solid) density distributions in 30 Ar.

5 The Giant Dipole Resonance in Ar-Isotopes in the Relativistic RPA 921 where ~ Vi = V 2] + 1 ' and cpi(r) is the 4-component radial wave function of the state (i). The isotope 30 Ar has 12 neutrons, where the 1d 5 / 2 neutron orbit is partially occupied. The last partially occupied Id 3 / 2 proton state has a very small binding energy (c = MeV). The rms radius of the neutron distribution (2.921 fm) is much smaller than that of the proton distribution (3.353 fm), and this nucleus clearly has a proton skin and a far extending proton tail. This is shown in Fig. 1. At the other side of the chain, the isotope 52 Ar has 34 neutrons, where the 2P3/2 and 2Pl/2 neutron states are fully occupied. These states with relatively low binding energy and small angular momentum produce a thick neutron skin with a long tail. The density distributions of 52 Ar are shown in Fig. 2. When we move through the isotope chain from the proton rich nuclei to the neutron rich nuclei, the neutron skin gradually builds up as a result of filling more f' 10-3!..,. i r (1m) Fig. 2. Proton (dashed) and neutron (solid) density distributions in 52 Ar. (10) Ar-isotopes! t e A Fig. 3. Root-mean-square radii for Ar-isotopes, protons (dashed curve) and neutrons (solid curve).

6 922 Z. Ma, H. Toki, B. Chen and N. V. Giai and more neutron subshells. In Fig. 3 we can see that the rms radii of neutron distributions increase regularly up to 46 Ar, and then the increase becomes faster because of the outer 2p3/2 and 2pl/2 orbitals. At the same time the proton rms radii do not vary much, with a slight decrease from A = 30 to A = 36 and a small positive slope from A = 36 to A = 52. This is a result of two opposite effects, the neutron-proton symmetry energy which makes the proton orbitals more bound when the neutron excess becomes larger, and the fact that the outer neutrons tend to pull out the protons. 4. Giant dipole resonances in Ar-isotopes We have calculated the response of Ar nuclei to the isovector dipole operator Q = l' ryioto by solving the RRPA equation (2). The particle-hole residual interactions for isovector modes are due mainly to the p meson exchange in the effective lagrangian. The isoscalar cr and w mesons play little role. The details of the procedure of solving the RRPA equation (2) can be found in Refs. 9) and 10). Here we only indicate the specific approximations used for the open shell nuclei and the space truncation. Since the Fermi level for some isotopes may be partially occupied, the excitation of a particle from a deeper state to this partially occupied state is allowed. In order to take account of such excitations, each wave function of a hole state is multiplied by the factor Vi of Eq. (10) whereas the wave function of a particle state is multiplied by a factor Ui: 2j ni 2j + 1 This prescription for building the particle-hole space insures the correct sum rules. To represent the bound unoccupied states and the single-particle continuum, we diagonalize the Dirac-Hartree equation in a basis of spherical harmonic oscillator functions with the major shell quantum number up to N = 12 and oscillator length b = 1.86 fm for all isotopes. The single-particle energy cutoff Emax = M MeV is adopted in all cases, where M is the nucleon mass. In the calculation of response functions, an averaging parameter.1 = 2 Me V is used to smooth out the strength distributions. This is done by replacing the excitation energy E by E + i.1/2 in all expressions, The GDR response functions for Ar-isotopes from A = 30 to A = 52 are shown in Figs. 4(a) and (b). In order to show the collectivity of the GDR, the unperturbed and RPA strengths for the nuclei 30 Ar at the proton drip line and 52 Ar at the neutron drip line are plotted in Fig. 5 in comparison with the stable nuclei 38,40 Ar. The strengths are pushed up slightly to high energy due to the collectivity, and a pronounced peak is located around MeV. The strengths of the isovector GDR in Figs. 4(a) and (b) show broad structures and a well marked giant resonance peak. This is consistent with the experimental finding for light nuclei. An interesting feature of Figs. 4 and 5 is that the strengths of the isovector GDR modes are split into two peaks for the nuclei near drip lines, such as Ar (11)

7 The Giant Dipole Resonance in Ar-Isotopes in the Relativistic RPA 923 TMl GDR (a) o'"""-~'-'-'-...w'-'-'--==<_-...j o E (liev) (b) ~ I }! E (liev) Fig. 4. GDR response function in Ar-isotopes. (a) for 30 Ar to 40 Ar; (b) for 42 Ar to 52 Ar. near the neutron drip line and Ar near the proton drip line. One peak is at the normal GDR position, while the other is located at lower energy, around 6 MeV. The dipole strength at low energy becomes stronger when approaching the drip line. The strengths at low energy observed here are due to the excitation of the weakly bound states. They are mainly due to (p s h - and (p dh - excitation of nucleons at the Fermi surface. By comparing with the unperturbed strength one can see that the lower peaks show less collectivity and the RPA strength nearly keeps the shape of the unperturbed strength (see Fig. 5). Thus, the general picture is that in nuclei near the two ends of the isotopic chain there are particle-hole excitations with relatively small excitation energies (the hole is not very bound and the particle is low in the continuum) and exhausting an appreciable fraction of the dipole strength. The isovector GDR in Ar-isotopes calculated in the present approach have a well marked peak, and the peak energy moves to slightly lower energies as the neutron number increases. The peak energies of the GDR as a function of the atomic number are plotted in Fig. 6 with diamond signs. The peak energies of the Lorentzian GDR

8 924 Z. Ma, H. Toki, B. Chen and N. V. Giai , f'5 >,'it ".!l 0 ;:: I>: 10 N E ( MeV) E (MeV) Fig. 5. GDR response function 30 Ar, 38 Ar, 40 Ar and 52 Ar. Unperturbed strengths and RPA strengths are indicated by dashed curves and solid curves, respectively. parametrizations which fit the experimental data 18) are plotted with octagons. The latter points can be reproduced well by the expression, 19) which is indicated by a dashed curve. The predictions for the peak energy of isovector GDR modes in the present approach with the parameter set TM1 are slightly lower than the experimental data. For instance, the calculated peak position for 40Ca was found to be about 1.8 MeV lower than the experimental valuey) We therefore increase the GDR peak energies in Ar-isotopes by 1.8 MeV. The corrected peak energies are plotted with crosses, which are found to be described well by Eq. (12). Thus, the GDR peaks of the Ar-isotopes, even at the nuclear drip lines seem to have,. '= :.0 a "' Ar OEzp () Peak EnerlY ):( Corrected Peak EneraY (12) A Fig. 6. Peak energies of the GDR as functions of the atomic number. The experimental data are indicated by octagons. These data are described well by Eq. (12) with a dashed curve. The solid curve is for 80A- 1 / 3. The theoretical results with TM1 calculated in this work'are indicated by diamonds. Since these values slightly underestimate the experimental peak energies, the energies corrected by 1.8 MeV are depicted with crosses.

9 The Giant Dipole Resonance in Ar-Isotopes in the Relativistic RPA 925 normal behavior. 5. Conclusion We have studied the CDR of the Ar-isotopes in the framework of the relativistic RPA with the nonlinear model TMl. From the static Dirac-Hartree spectrum we can already see that, near the proton and neutron drip lines, there are particlehole configurations at low energy produced by the jump of a loosely bound nucleon into the nearby continuum. These configurations are well separated from the usual 1fiw configurations which carry the major part of dipole strength. The two types of configurations do not couple to each other and should lead to separated dipole peaks. This is confirmed by the RRPA calculations, which show that the strength of the GDR is split into two parts for nuclei near neutron and proton drip lines. One peak is at the normal CDR excitation energy and this is reproduced well by an empirical two-parameter formula. The other peak is located at lower energy and is due to the unperturbed particle-hole excitations involving loosely bound occupied states. The lower dipole strength becomes stronger when the isotopes move to the drip lines. One may expect that a manifestation of this would be an enhancement of the observed cross-sections in Coulomb excitation experiments involving these nuclei. Acknowledgements Z. M. and N. V. G. acknowledge the COE program for enabling them to stay at RCNP-Osaka, where this work has been carried out. DPT of IPN-Orsay is a Unite de Recherche des Universites Paris XI et Paris VI associee au CNRS. This work was also supported in part by the National Natrural Science Foundation of China, under contract No References 1) I. Tanihata et a!., Phys. Rev. Lett. 55 (1985), ) P. G. Hansen and B. Jonson, Europhys. Lett. 4 (1987), ) K. Ikeda, Nucl. Phys. A538 (1992), 355c. 4) Y. K. Gambhir, P. Ring and A. Thimet, Ann. of Phys. 198 (1990), ) D. Hirata, H. Toki, T. Watabe, I. Tanihata and B. V. Carlson, Phys. Rev. C44 (1991), ) Y. Sugahara and H. Toki, Nue!. Phys. A579 (1994), ) Z. Z. Ren, B. Q. Chen, Z. Y. Ma and W. Mittig, J. of Phys. G: Nuc!. Part. Phys. 21 (1995), ) K. Wehrberger and F. Beck, Phys. Rev. C37 (1988), ) M. L'Huillier and ~guyen Van Giai, Phys. Rev. C39 (1989), ) J. F. Dawson and R. J. Furnstahl, Phys. Rev. C42 (1990), ) D. S. Oakley,.J. R. Shepard and N. Auerbach, Phys. Rev. C45 (1992), ) Z. Y. Ma, N. Van Giai, H. Toki and M. L'Huillier, Phys. Rev. C55 (1997), ) Z. Y. Ma, N. Van Giai and H. Toki, to be published in Nue!. Phys. (1997). 14) R. Brockmann and R. Machleidt, Phys. Rev. C42 (1990), ) M. M. Sharma, M. A. Nagarajan and P. Ring, Phys. Lett. B312 (1993), ) A. L. Fetter and.j. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971). 17) B. D. Serot and J. D. Walecka, Adv. Nuc!. Phys. 16 (1986),1.

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