The extraction of the nuclear incompressibility from the isoscalar giant
monopole resonance (GMR) measurements is analysed. Both pairing and mutually
enhanced magicity (MEM) effects play a role in the shift of the GMR energy
between the doubly closed shell $^{208}$Pb nucleus and other Pb isotopes.
Pairing effects are microscopically predicted whereas the MEM effect is
phenomenologically evaluated. Accurate measurements of the GMR in open-shell Pb
isotopes are called for.Comment: 4 page

E. Khan

M. Harakeh

U. Garg

English

arXiv

4 pagesInternational audienceThe extraction of the nuclear incompressibility from the isoscalar giant monopole resonance (GMR) measurements is analysed. Both pairing and mutually enhanced magicity (MEM) effects play a role in the shift of the GMR energy between the doubly closed shell $^{208}$Pb nucleus and other Pb isotopes. Pairing effects are microscopically predicted whereas the MEM effect is phenomenologically evaluated. Accurate measurements of the GMR in open-shell Pb isotopes are called for

Khan, E.

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Giant monopole resonance in Pb isotopesElias KhanTo cite this version:Elias Khan. Giant monopole resonance in Pb isotopes. Physical Review C, American PhysicalSociety, 2009, 80 (5), pp.057302. <10.1103/PhysRevC.80.057302>. <in2p3-00405679>HAL Id: in2p3-00405679http://hal.in2p3.fr/in2p3-00405679Submitted on 20 Jul 2009HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s..The Giant Monopole Resonance in Pb isotopesE. Khan11 Institut de Physique Nucle´aire, Universite´ Paris-Sud, IN2P3-CNRS, F-91406 Orsay Cedex, FranceThe extraction of the nuclear incompressibility from the isoscalar giant monopole resonance(GMR) measurements is analysed. Both pairing and mutually enhanced magicity (MEM) effectsplay a role in the shift of the GMR energy between the doubly closed shell 208Pb nucleus and otherPb isotopes. Pairing effects are microscopically predicted whereas the MEM effect is phenomeno-logically evaluated. Accurate measurements of the GMR in open-shell Pb isotopes are called for.PACS numbers: 21.10.Re,24.30.Cz,21.60.JzI. INTRODUCTIONIt has been recently shown, using the Tin isotopicchain, that pairing has an effect on the nuclear incom-pressibility [1, 2]. In Ref. [1], including pairing effectsin the description of the isoscalar giant monopole reso-nance (GMR) allows to explain part of the so-called Snsoftness [3, 4]: pairing decreases the predicted centroidof the GMR, located at ∼ 16 MeV, by few hundreds ofkeV. In Ref. [2] an explicit decreasing correlation is foundbetween the nuclei incompressibility, obtained from theenergy of the GMR, and the magnitude of the pairinggap.With the recent advent of accurate microscopic modelsin the pairing channel, such as fully self-consistent Quasi-particle Random Phase Approximation (QRPA) [1, 5], itis now possible to predict the GMR position with accu-racy and to study small but non-negligible effects suchas pairing. Using a microscopic approach is crucial: theGMR is mainly built from particle-hole configurations lo-cated far from the Fermi level, where pairing do not playa major role. However giant resonances are known to bevery collective [6] and pairing can still have a sizable ef-fect on the GMR properties: around 10 % on the centroid[2], which is the level of accuracy of present analysis onthe extraction of K∞ [7, 8].Experimentally, the measurement of the GMR on anisotopic chain facilitates the study of superfluidity on theGMR properties [3], and the possibility to measure theGMR in unstable nuclei emphasizes this feature [9]. Itis therefore necessary to go towards the measurement ofthe GMR on several nuclei, such as an isotopic chain.The overused method of precise GMR measurements ina single nucleus, such as 208Pb, may not be the relevantapproach. Other nuclei have been used such as 90Zr and144Sm. Indeed when considering the available GMR datafrom which the K∞ value has been extracted,208Pb isstiffer than the Sn, Zr and Sm nuclei: K∞ is about 20MeV larger, both in non-relativistic and in relativisticapproaches [7, 8]. The question may not be ”why Tin areso soft ?”[4] but rather ”why 208Pb is so stiff ?”. Alsorecent results in Cadmium isotopes confirm that open-shell nuclei provide a value of K∞ which is lower thanthe one extracted from 208Pb [10].It should be underlined that it is not possible to de-scribe the GMR both in Pb and in other open-shell nucleiwith the same functional [2]. This is valid both for nonrelativistic and relativistic calculations. In Ref. [2], ithas been shown how pairing effects play a role in theSn isotopic chain: the energy of the GMR is increasedfor doubly magic 132Sn, due to the vanishing of pairingin this nuclei. The aim of the present work is to lookfor a possible similar effect in the Pb isotopic chain. In-deed 204,206Pb nuclei are stable, but almost all the ex-perimental efforts in the past decades were devoted tothe measurement of the GMR in 208Pb. It would beinteresting to perform accurate GMR measurement onopen-shell 204,206Pb nuclei by inelastic alpha scatteringin direct kinematics.Even with the inclusion of pairing effects, the SLy4functional [11], which accurately describes the GMR in208Pb, is still overestimating the GMR in Sn isotopes(see Fig. 1 of Ref. [2]). Hence an additional shell ef-fect is at work, to explain the discreapency of the ex-tracted K∞ values between208Pb and Sn isotopes: theSLy4 (K∞=230 MeV) [11] functional allows to describethe 208Pb GMR whereas the SkM* (K∞=215 MeV) [12]functional is in agreement with the Sn data.The puzzle of the stiffness of 208Pb may come from itsdoubly magic behaviour: a possible explanation is thatthe experimental EGMR data is especially increased inthe case of doubly magic nuclei, as observed in 208Pbcompared to the GMR data available in other nuclei(such as in the Tin isotopic chain). This difficulty todescribe with a single functional both doubly magic andother nuclei has already been observed on the masses,namely the so-called ”mutually enhancement magicity”(MEM), described in Ref. [13, 14]: functionals designedto describe masses of open-shell nuclei cannot predict themasses of doubly magic nuclei such as 132Sn and 208Pb,which are systematically more bound that predicted. Inorder to evaluate the MEM effect, it may be necessaryto take into account quadrupole correlation effects dueto the flatness of the mean-field potential for open-shellnuclei [15]. The incompressibility being related to thesecond derivative of the energy with respect to the den-sity, it would be useful to find a way to predict the GMR2beyond QRPA by taking into account quadrupole cor-relations. However such a microscopic approach is farbeyond the scope of the present work.Therefore the MEM effect shall be phenomenologicallyevaluated by using a functional which describes well theGMR in 208Pb (SLy4), and in the case of open-shell Pbisotopes, a functional (SkM*) which describes well theGMR in open shell nuclei, such as Sn isotopes. The aimis to predict a value of the GMR centroid in the Pb iso-topic chain which could be useful to compare with exper-imental data.II. CONSTRAINEDHARTREE-FOCK-BOGOLIUBOVCALCULATIONSIn order to consider pairing effects on the GMR, itis necessary to use a fully microscopic method includingan accurate pairing approach. We use the constrainedHartree-Fock method, extended to the Bogoliubov pair-ing treatment (CHFB) [2, 16]. It should be noted thatthe extension of the CHF method to the CHFB case hasbeen recently demonstrated in Ref. [16]. The CHF(B)method has the advantage to very precisely predict thecentroid of the GMR using the m−1 sumrule [17, 18].The whole residual interaction (including spin-orbit andCoulomb terms) is taken into account and this methodis by construction the best to predict the GMR centroid[18]. Introducing the monopole operator Qˆ as a con-straint :Hˆcons = Hˆ + λQˆ (1)withQˆ =A∑i=1r2i , (2)the m−1 value is obtained from the derivative of themean value of this operator:m−1 = −12[∂∂λ〈Qˆ〉]λ=0(3)The m1 sumrule is extracted from the double commu-tator using the Thouless theorem [19]:m1 =2~2A〈r2〉 (4)Finally the GMR centroid is given byEGMR=√m1/m−1. All details on the CHF(B)method can be found in [7, 16, 17].The present work uses the HFB approach in coordinatespace [20] with Skyrme functionals and a zero-range sur-face pairing interaction:Vpair = V0[1−(ρ(r)ρ0)]δ (r1 − r2) (5)This interaction is known to describe a large variety ofpairing effects in nuclei [21]. The magnitude of the pair-ing interaction V0 is taken as -735 MeV.fm−1 for SLy4and -700 MeV.fm−1 for SkM*: it is adjusted so to de-scribe the trend of the neutron pairing gap Pb isotopes.The single quasiparticle spectrum is considered until 60MeV. Previous CHFB calculations in Sn isotopes are de-scribed in Ref. [2].III. RESULTSFig. 1 displays the GMR energy (times A1/3 to cor-rect for the slow lowering of the GMR with the nu-clear mass [6]) for 204−212Pb nuclei obtained from mi-croscopic CHFB predictions using two functionals: SLy4[11] (K∞=230 MeV, which describes well the208Pb GMRdata), and SkM* [12] (K∞=215 MeV, which describeswell the open-shell GMR data such as Tin isotopes). Asexpected the GMR energy is predicted higher in the SLy4case than in the SkM* case. For both interactions, thestriking feature of Fig. 1 is the increase of the GMR cen-troid located at the doubly magic 208Pb nucleus. Thisindicates that pairing effects should be considered to de-scribe the behaviour of nuclear incompressibility, andthat vanishing of pairing make the nuclei stiffer to com-press, confirming our previous statement on the stiffnessof 132Sn compared to open-shell Sn isotopes [2]. Pairingeffects (CHFB calculations) decrease the centroid of theGMR as observed in open-shell Pb isotopes, compared to208Pb. This confirms again the results of [1, 2] in the Tindata, and show that the effect of pairing on the GMRmay be universal.200 204 208 212 216 220A7778798081828384A1/3 EGMR (MeV)SLy4SkM*FIG. 1: Excitations energies of the GMR (times A1/3) in204−212Pb isotopes calculated with constrained HFB methodand the SLy4 and SkM* interactions3We now study the hypothesis that both pairing andMEM effects are contributing to the GMR position whencomparing open-shell nuclei with doubly magic ones.Pairing effects are treated microscopically, using theCHFB approach, as described above. However there isno present model to take microscopically into accountthe MEM effect on the GMR centroid. It should benoted that the MEM effect is not well understood yetin the case of nuclear masses. Nevertheless its effect onthe GMR can be evaluated phenomenologically by calcu-lating the predicted position of the GMR centroid withSLy4 in the 208Pb case, and with SkM* in the open-shell204,206,210,212Pb nuclei: SkM* allows for a good descrip-tion of the GMR in open-shell nuclei such as the Tin iso-topes, where the MEM effect is at work. As stated above,the GMR position predicted with SLy4 is in agreementwith the measurements on 208Pb, but for open-shell nu-clei, a functional with lower incompressibility should beused, as showed by the previous analysis on Sn [2] andCd isotopes [10]. We therefore use SkM* for open-shellPb isotopes. The aim is to provide values of the GMR inthe Pb isotopes to be compared with measurements.Fig. 2 displays the predicted energy of the GMR for204−212Pb. The solid lines corresponds to CHFB calcu-lations using the SLy4 functional: it displays the pairingeffect on the GMR. The dashed line corresponds to theCHFB calculations using the SLy4 functional for 208Pband the SkM* functional for open-shell Pb isotopes: ittakes into account both pairing and MEM effects. Oneobserves that the main contribution of the increase ofthe GMR centroid for 208Pb comes from the MEM ef-fect. However pairing effects still induce a decrease ofthe GMR centroid for open-shell Pb isotopes. We expectmeasurements on 204Pb and 206Pb to be compared withthe values displayed on Fig. 2 in order to test both thepairing and MEM effects on the GMR centroid.200 204 208 212 216 220A13.213.413.613.81414.2E GMR (MeV)PairingPairing+MEMExp.FIG. 2: Excitations energies of the GMR in 204−212Pb iso-topes calculated with constrained HFB method, taking intoaccount the MEM effect (see text). The experimental data istaken from Ref. [22]It should be noted that the GMR has been measuredin 206Pb several decades ago [23], providing a centroidvalue of 14.0 ± 0.3 MeV. Therefore no deviation is foundwith respect to 208Pb. However this measurement hasbeen performed above 12 deg.: at such large angle, theGMR cross section is very weak, compared to the giantquadrupole resonance (GQR) cross section and it is del-icate to extract a value of the GMR centroid since boththe GQR and the high energy background are important.Hence the measurement was not optimal, especially whenlooking for typical 500 keV effects. Therefore it would beof particular interest to measure the GMR in 206Pb at 0degree, allowing for a larger GMR cross section. Further-more, there is no GMR data for 204 Pb. The accuracyof future GMR measurement in 204,206Pb will be crucial.Moreover, it should be mentioned that a somewhat lowervalue for the GMR centroid of 208Pb has been found inthe RCNP experiment (13.5 ± 0.2 [24]), compared to theTexas A&M one [22]: there is a current debate about thereason such variations. An accurate experiment on Pbisotopes, including 208Pb should also help to solve thisissue.Another concern is related to the MEM effect. If thiseffect is due to quadrupole correlations coming from theflatness of the potential, as stated in Ref. [15], it is ex-pected smaller in the Pb case than in the Tin one: 204,206Pb are in the vicinity of a doubly magic nuclei and theirmean-field potentials are still sharp, allowing for a smallMEM effect. In the case of 112−124Sn nuclei, the poten-tial is much flatter [25]: the 132Sn doubly magic nucleusis several neutrons away on the nuclear chart. Thereforethe MEM effect is expected larger in the Sn case than inthe Pb one and it may be possible that only pairing ef-fects play a role for the Pb isotopes, which are about 100keV to 200 keV (solid line in Fig. 2): it is crucial thatfuture measurement are performed within this accuracy.In the case of the interpretation of the MEM effect fromRef. [15] is correct, taking it into account microscopicallywould necessitate to predict the GMR with quadrupolecorrelations due to the flatness of the mean-field potentialin the case of open-shell nuclei. We expect it to lowerthe predicted GMR value, as stated above. Therefore,the value K∞=230 MeV extracted from the analysis ofthe 208Pb data should be correct, whereas the apparentsoftness of open-shell nuclei like Tin, may be artificial(K∞ typically 20 MeV lower), because the MEM effecthas not been included in all previous analyses.IV. CONCLUSIONThe GMR energy of open-shell Pb nuclei is predictedsignificantly lower than in the case of 208Pb. This is re-lated to 2 effects: pairing and mutually enhanced magic-ity. These two effects may also be the explanation ofthe apparent softness of Sn isotopes, compared to 208Pb.The pairing effect is evaluated microscopically whereasthe MEM effect is evaluated phenomenologically, sinceit is still not well characterised. A measurement of the4GMR in 204Pb and 206Pb compatible with the presentpredictions would solve this softness problem. It wouldalso mean that K∞ should not be determined by mea-suring the GMR in a single nuclei such as 208Pb, but thewhole isotopic chain should be measured in order to pro-vide a general view on the various effects on the GMR.Additional theoretical investigations are called for inorder to predict the GMR including the mutually en-hancement magicity effect in a microscopic way. Thiswould necessitate an ambitious microscopic approach,trying to link QRPA and GCM approaches. 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The Giant Monopole Resonance in Pb isotopes

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