Collective structural evolution in neutron-rich Yb, Hf, W, Os and Pt isotopes
K. Nomura, T. Otsuka, R. Rodriguez-Guzman, L. M. Robledo, P. Sarriguren
aa r X i v : . [ nu c l - t h ] N ov Collective structural evolution in neutron-rich Yb, Hf, W, Os and Pt isotopes
K. Nomura, T. Otsuka,
1, 2, 3
R. Rodr´ıguez-Guzm´an, L. M. Robledo, and P. Sarriguren Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Center for Nuclear Study, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824-1321, USA Instituto de Estructura de la Materia, IEM-CSIC, Serrano 123, E-28006 Madrid, Spain Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain (Dated: November 8, 2018)An interacting boson model Hamiltonian determined from Hartree-Fock-Bogoliubov calculationswith the new microscopic Gogny energy density functional D1M, is applied to the spectroscopicanalysis of neutron-rich Yb, Hf, W, Os and Pt isotopes with mass A ∼ − N in a given isotopic chain by calculating excitation energies, B (E2) ratios,and correlation energies in the ground state. It is shown that such transitions tend to occur morerapidly for the isotopes with lower proton number Z , when departing from the proton shell closure Z = 82. The triaxial degrees of freedom turn out to play an important role in describing theconsidered mass region. Predicted low-lying spectra for the neutron-rich exotic Hf and Yb isotopesare presented. The approximations used in the model and the possibilities to refine its predictivepower are addressed. PACS numbers: 21.10.Re,21.60.Ev,21.60.Fw,21.60.Jz
I. INTRODUCTION
The study of the origin of nuclear deformation andits evolution as a function of proton and neutron num-bers has attracted considerable theoretical interest froma large variety of viewpoints [1–13]. Experimentally, low-lying spectroscopy provides a very powerful source of in-formation that allows one to establish signatures correlat-ing the nuclear shape evolution with the energy spectra[14–23].Among many other nuclear structure models, self-consistent mean-field methods, based on microscopic en-ergy density functionals (EDFs), have provided bothaccurate and universal descriptions of different nuclearintrinsic properties including binding energies, ground-state deformations, density distributions, low-lying one-quasiparticle configurations, as well as the way nuclearshapes evolve with the number of nucleons [2, 6, 8–11, 24–28]. Popular EDFs are the non-relativistic Skyrme[6, 29, 30] and Gogny [31, 32] ones, as well as relativisticmean-field Lagrangians [33]. To describe nuclear spec-troscopy one should go beyond the mean-field approxi-mation to take into account the restorations of brokensymmetries and/or the configuration mixing of intrinsicstates in the spirit of the generator coordinate method(GCM) [2, 6, 34–38]. In this kind of studies calculationsmay become computationally much more demanding andtime consuming than the underlying mean field, particu-larly when triaxial degrees of freedom are included in theanalysis.A sound approximation to the full GCM configura-tion mixing and/or the symmetry restoration is the five-dimensional collective Hamiltonian with quadrupole de-grees of freedom where both rotational and vibrational mass parameters are determined from the constrained,self-consistent mean-field calculations with a given EDFand the collective potential is derived by the zero-pointenergy correction to the total mean-field energy (e.g.,[39–41]).Alternatively, nuclear dynamics and spectroscopicquantities can be approximated by introducing appro-priate bosonic degrees of freedom. The interacting bo-son model (IBM) [42] can be regarded as a nice exam-ple for this, and has been exploited in a large numberof phenomenological studies focusing on the low-lyingspectrum of medium-heavy and heavy nuclei [42]. Thesimplest version of the IBM is built on monopole s andquadrupole d bosons, which reflect the collective J π = 0 + and 2 + pairs of valence-shell configurations, respectively[43]. Nevertheless, since the IBM itself should have a cer-tain microscopic foundation, a Hamiltonian of IBM hasbeen derived conventionally from the shell-model config-uration [43], and more recently from EDF-based calcula-tions [44]. These mapping methods have been applied torealistic cases involving a variety of situations coveringfrom nuclei with modest quadrupole deformation includ-ing γ -unstable ones [43–47], to strongly deformed rota-tional nuclei [48–53]. Also quantum-mechanical correla-tion effects in the ground state have been considered [46].Starting from the constrained Hartree-Fock-Bogoliubov(HFB) theory with the D1S [54] parametrization of theGogny functional, the method of [44] was used for thespectroscopic analysis of Pt isotopes [47], and some ofOs and W isotopes [53].In this paper we apply the mapping procedure of [44]to the mass region A ∼ − A ∼ −
200 exhibit a transition between prolate andoblate equilibrium shapes as a function of the nucleonnumber, with the critical point around N ≈
116 havinga pronounced γ softness [58–61]. These facts make theregion a potential testing ground to understand the de-formation properties of atomic nuclei. The evolution ofthe nuclear ground states in this mass region has beeninvestigated recently with the constrained self-consistentmean-field method with microscopic EDFs [9, 56, 62].Both the (constrained) Hartree-Fock+BCS (HF+BCS)and the HFB approximations have been used to com-pute energy surfaces with quadrupole degrees of freedomin order to give a microscopic insight into shape transi-tions [9, 55, 56]. It was shown in these studies that thetriaxiality is an important ingredient to describe the evo-lution from prolate to oblate shapes, irrespective of thetypes of the EDFs used.It should be kept in mind that Pt, Hg, and Pb iso-topes are well known [63] for the spectacular coexistenceof different low-lying configurations based on different in-trinsic deformations as observed in their low lying spec-trum. There are a number of works aimed at the un-derstanding of the shape coexistence phenomenon in thisregion in terms of both EDF-based microscopic calcula-tions [36, 64, 65] and phenomenological models [66–68].The paper is organized as follows: In Sec. II, a shortoutline of the theoretical framework is given. Section IIIpresents the energy surfaces, ground-state correlation en-ergies, moments of inertia for the rotational bands, low-lying spectra, and the B (E2) systematics for the consid-ered isotopes chains. Section IV is devoted to the con-cluding remarks and work perspectives. II. THEORETICAL PROCEDURE
The analysis starts with a constrained HFB calculationusing the Gogny-D1M EDF. The constraints in this caserefer to the mass quadrupole moments which are associ-ated with the quadrupole deformation parameters β and γ in the geometrical model [1]. The set of constrainedHFB calculations, for each collective coordinate ( β , γ ),provides the total HFB energy (denoted by E HFB ( β, γ )).For calculation details the reader is referred to [47, 56].In other studies solving the five-dimensional collective Hamiltonian [39–41], the collective potential energy sur-face is obtained by subtracting the zero-point energiesfor both rotational and vibrational motions from the con-strained HFB energy surface. This corrected energy sur-face should be viewed as a collective potential energy sur-face. In the present work, the constrained HFB energysurface and the corresponding boson energy surface arecompared, and they will be referred to simply as energysurface . Note that, as the total energy is considered, allingredients including those relevant to kinetic terms aresupposed to be taken into account to a good extent.Each point of the Gogny-HFB energy surface E HFB ( β, γ ) is mapped onto the corresponding point onthe bosonic energy surface, denoted by E IBM ( β B , γ B )with β B and γ B being the deformation parameters forthe boson system, in such a way that the bosonic energysurface fits the fermionic one [44]. In this paper we con-sider the proton-neutron interacting boson model (IBM-2) [43] because it reflects better the microscopic picturethan the original version of the IBM without distinctionof the proton and the neutron degrees of freedom (of-ten called IBM-1). In what follows we denote the IBM-2simply as the IBM, unless otherwise specified. The IBMenergy surface is obtained as the expectation value ofa given boson Hamiltonian [69] in terms of the coher-ent state | Φ( β B , γ B ) i . The coherent state represents theintrinsic wave function of the boson system, and is char-acterized by the deformation variables β B and γ B . Inprinciple, proton and neutron bosons might have differ-ent values of the deformation parameters, but since pro-ton and neutron systems are supposed to attract eachother strongly in medium-heavy and heavy deformed nu-clei, the deformations of proton and neutron systems canbe taken the same to a good approximation.If the separability of the mapping along the β and the γ directions is assumed, one can consider the relation be-tween the IBM and the geometrical deformation variables[44, 46]. It was shown [69] that, in general terms, thebosonic and the geometrical β s are proportional to eachother and that the proportionality coefficient coincideswith the ratio of the total nucleon number to the valencenucleon number counted from the nearest closed shells.We exploit this relation and assume that β B = C β β , with C β being a numerical coefficient [44]. The typical rangeof the C β value turns out to be approximately 5 ∼ γ , the identi-fication γ B = γ seems valid as indeed both geometricaland IBM γ ’s have the same meaning, ranging from 0 to60 degrees.We adopt the IBM Hamiltonian of the following form:ˆ H IBM = ǫ ˆ n d + κ ˆ Q π · ˆ Q ν + α ˆ L · ˆ L, (1)where the first term ˆ n d = ˆ n dπ +ˆ n dν with ˆ n dρ = d † ρ · ˜ d ρ ( ρ = π or ν ) is identified as the d -boson number operator. Thesecond term on the right-hand side of Eq. (1) stands forthe quadrupole-quadrupole interaction between protonand neutron systems, with ˆ Q ρ = s † ρ ˜ d ρ + d † ρ ˜ s ρ + χ ρ [ d † ρ ˜ d ρ ] (2) being the quadrupole operator for proton or neutron sys-tems. The third term (denoted by LL term, hereafter) isrelevant to the moment of inertia of the rotational band.ˆ L = ˆ L π + ˆ L ν is the angular momentum operator for theboson system with ˆ L ρ = √ d † ρ ˜ d ρ ] (1) .The form of the Hamiltonian ˆ H IBM in Eq. (1) is not themost general, but embodies all essential features of thelow-lying quadrupole collective states. A more generalIBM Hamiltonian with up to two-body interactions con-tains many more terms than those considered here. How-ever, these additional terms are supposed to be of littleimportance, and their implementation would increase thenumber of parameters, which makes the problem quitecomplicated.
TABLE I: The parameters for the IBM Hamiltonian ˆ H IBM of Eq. (1), as well as the coefficient C β , obtained from themapping of HFB to IBM energy surfaces for the consideredYb, Hf, W, Os and Pt nuclei with N = 110 − ǫ − κ χ π χ ν α C β (keV) (keV) × × (keV) Yb 212 265 337 -991 -9.06 3.60
Yb 169 265 300 -900 -11.4 3.70
Yb 279 271 302 -548 -9.84 3.87
Yb 418 268 147 -106 -9.54 4.90
Yb 528 265 418 43 -4.68 5.13
Yb 769 267 332 573 -0.185 5.50
Yb 806 271 461 862 21.5 7.20
Hf 124 280 489 -913 -5.61 3.93
Hf 128 282 458 -938 -8.01 4.07
Hf 109 275 400 -700 -4.85 4.40
Hf 250 277 282 -208 -7.90 5.30
Hf 442 280 403 -30 -5.99 5.48
Hf 619 273 388 443 2.79 5.94
Hf 716 277 534 805 18.4 8.20
W 50.4 286 409 -859 -0.400 4.09
W 36.8 285 389 -835 -2.30 4.50
W 69.6 289 401 -662 -1.44 4.80
W 71.3 275 572 -419 -2.72 5.60
W 231 270 189 147 -4.15 6.30
W 627 291 392 536 -5.74 6.87
W 686 281 745 822 15.3 8.50
Os 142 310 331 -689 -0.433 4.40
Os 162 318 352 -672 -2.78 4.83
Os 86.7 303 412 -509 -2.61 5.40
Os 91.5 292 502 -488 -3.09 6.15
Os 289 305 401 -77 -6.04 6.74
Os 541 298 336 513 -5.94 7.64
Os 683 304 573 793 8.50 9.66
Pt 187 328 409 -487 8.16 4.81
Pt 215 336 300 -10 5.93 5.56
Pt 311 362 265 44 -0.117 6.44
Pt 312 366 490 -50 0.214 6.85
Pt 435 356 475 311 1.87 7.28
Pt 489 319 611 565 8.80 7.90
Pt 719 308 467 949 -4.69 8.78
The parameters contained in the first two terms of theHamiltonian ˆ H IBM in Eq. (1), ǫ , κ , χ π and χ ν , as wellas the coefficient C β , are fixed using the fitting methodof Ref. [46]. The LL term contributes to the energy sur-face in the same way as the d -boson number operator,but with a different coefficient, 6 α . Hence, the α coeffi-cient cannot be fixed only by the mapping of the energysurface. A further step is required, in order to incorpo-rate specific non-zero angular frequency features of therotational cranking. The α value is determined by theprocedure of Ref. [52], where the cranking moment of in-ertia was compared between fermion and boson systems.We then calculate the moment of inertia for the 2 +1 excited state by the Thouless and Valatin (TV) formula[70], J TV = 3 /E γ . (2)Here, E γ stands for the 2 +1 excitation energy obtainedfrom the self-consistent cranking method with the con-straint h ˆ J x i = p L ( L + 1), where ˆ J x represents the x -component of the (fermion) angular momentum operator[56]. In [52], the Inglis-Belyaev formula [71, 72] turnedout to be valid for the rotational regime, but the presentTV moment of inertia appears to be more general.For the boson system, we calculate the moment of in-ertia of the intrinsic (coherent) state, denoted by J IBM ,using the cranking formula of Ref. [73] J IBM ( β B , γ B ) = lim ω → ω h Φ( β B , γ B ) | ˆ L x | Φ( β B , γ B ) ih Φ( β B , γ B ) | Φ( β B , γ B ) i , (3)where ω and ˆ L x stand for the cranking frequency and the x -component of the boson angular momentum operator,respectively.While J IBM has six parameters ǫ , κ , χ π , χ ν , C β and α ,all of them but α are already fixed by the energy-surfaceanalysis. The α value for each nucleus is obtained sothat the J IBM value at the equilibrium point, where theboson energy surface E IBM ( β B , γ B ) is minimal, becomesidentical to the J TV value at its corresponding energyminimum.The values of all derived IBM parameters are summa-rized in Table I. When diagonalizing the Hamiltonian inEq. (1), the ǫ parameter is shifted by ∆ ǫ = 6 α . The ǫ value listed in Table I is the one with this shift.The diagonalization of the IBM Hamiltonian, which isparametrized by the set of interaction strengths summa-rized in Table I, generates the energies and the wavefunctions of the excited states. Diagonalization is per-formed in the boson M -scheme basis, where M denotesthe z -component of the boson angular momentum oper-ator. With the eigenvectors of the Hamiltonian ˆ H IBM ,the B (E2) value is calculated: B (E2; L → L ′ ) = 12 L + 1 |h L ′ || ˆ T (E2) || L i| , (4)where L and L ′ are the angular momenta for the initialand the final states, respectively. In the present workthe E2 operator is given as ˆ T (E2) = e π ˆ Q π + e ν ˆ Q ν , whereˆ Q ρ coincides with the quadrupole operator in Eq. (1),and thus the same values of the χ π and χ ν parame-ters as those listed in Table I are used in calculatingthe B (E2) values (so-called consistent-Q formalism (cf.[42])). The boson effective charges for protons and neu-trons are taken the same, namely e π = e ν . III. RESULTS AND DISCUSSIONA. Energy Surfaces
Figure 1 shows the mapped IBM energy surfaces forYb, Hf, W, Os and Pt isotopes with 112 N β (= β B /C β )and γ (= γ B ) up to 2 MeV from its absolute minimum,since most of the quadrupole collective states are withinthis range. Note that the IBM energy surfaces for N =110 and 122 are not drawn as they are similar to thosefor N = 112 and 120 nuclei, respectively. The Gogny-D1M energy surfaces are not shown as they do not differsubstantially from the ones depicted in [56] with Gogny-D1S.For all the isotopes but the Pt ones, the energy min-imum shifts from the prolate ( γ = 0 ◦ ) to the oblate( γ = 60 ◦ ) sides as the number of neutrons increases, pass-ing through the most notable γ -soft nuclei with N ≈ χ π and χ ν values for many N = 116 isotonesthen satisfy χ π + χ ν ≈
0, as summarized in Table I. Thischoice of the χ parameters is at the origin of the almosttotally flat topology of the energy surface in the IBM-2, as seen for example in Os nucleus in Fig. 1. Thechange in the topology of the energy surface is an evi-dence of prolate-to-oblate shape/phase transition, whichbecomes sharper for smaller Z . The Gogny-D1S energysurfaces reported in [47, 56] were somewhat steeper inboth β and γ directions than the present Gogny-D1Mones.A difference is apparent between the energy surfacesof the Pt isotopes and those of the others. For the Ptisotopes, the variation of the energy surface takes placemuch moderately. Such slow structural transition in Ptisotopes was also observed in the case of the D1S func-tional [47, 55]. While a certain quantitative difference isobserved between the two Gogny functional results, theconclusion does not change.It should be noted that the Gogny-HFB calculationsuggested shallow triaxial wells for the transitional, N =116 Os and W nuclei [56]. In contrast, the mapped IBMenergy surfaces in Fig. 1 are flat in the γ direction, asthe only γ -dependent term of the bosonic energy surfaceis proportional to cos 3 γ . This is the case as long as theboson Hamiltonian contains up to two-body interactions.Only when a three-body (so-called cubic) term is consid-ered, a stable minimum at a γ value different from γ = 0and 60 degrees is obtained [74, 75]. B. Correlation energies
We next discuss a signature for a shape transition froma simple perspective. To do this we consider the followingquantity that will be called correlation energy hereafter,which was already introduced in Ref. [46]: E Corr = E IBM (0 +1 ) − h ˆ H IBM i min , (5)where the first term E IBM (0 +1 ) is the eigenenergy of theIBM Hamiltonian, Eq. (1), for the L π = 0 + ground state,and the second term h ˆ H IBM i min denotes the minimumvalue of the IBM energy surface, that is obtained by thevariation with respect to β and γ .In the self-consistent mean-field calculation with agiven EDF (e.g., Ref. [34]), the quantum-mechanical ef-fect can be extracted by comparing the minimum valueof the total energy surface of the mean field with the L π = 0 + eigenenergy resulting from the restoration ofthe broken symmetries and the configuration mixing.For calculations of correlation energies by mapping theEDF theory into shell model like interactions, includ-ing quadrupole and pairing correlations, the reader isreferred to [76].In the present study, all correlation effects can be in-cluded by the diagonalization of the boson Hamiltonian,and the energies and the wave functions of the stateswith good angular momentum and particle number canbe generated. Thus, the quantity defined in Eq. ( 5) con-tains correlation energies coming from symmetry restora-tion and configuration mixing and is similar to the equiv-alent quantity discussed in GCM studies.The behavior of E Corr with neutron number correlateswell with the underlying shape transition. Figure 2 showsthat for each considered isotopic chain the correlationenergy is maximal in magnitude at the neutron number N ∼ N = 126 is approached. This isconsistent with the overall systematic trend of the under-lying energy surface in Fig. 1. These features have beenrecognized in the GCM studies (e.g., in [38]) also. Forthe Pt isotopes, the magnitude of E Corr decreases with N , indicating that a clear transition is not expected forthese nuclei.Compared with the analysis by the GCM configurationmixing using e.g., a Skyrme functional [34] for the samemass region as considered here, the magnitude of thepresent correlation energy E Corr is rather small, whereasthe qualitative features mentioned above do not contra-dict the GCM results.In comparison to some rare-earth nuclei such as Nd-Sm-Gd isotopes, where a distinct first-order shape tran-sition is observed [13], the shape transition occurs rathermoderately in the considered mass region. Thus, con-trary to E Corr in Fig. 2, any drastic change with nucleonnumber is not expected in some other quantities in theground state, like two-nucleon separation energies.
FIG. 1: (Color online) The IBM energy surfaces for the considered Yb, Hf, W, Os and Pt isotopes with N = 112 − β . ◦ γ ◦ up to 2 MeV excitationfrom the minimum. Contour spacing is 100 keV. C. Moments of inertia
Based on the analysis in Sec. III B, we discuss to whatextent the moment of inertia is affected by the configu-ration mixing due to the diagonalization of IBM Hamil-tonian. The effect is most nicely illustrated in the Wisotopes, for which relatively many experimental spec-troscopic data are available.We show in Fig. 3 the moments of inertia of W iso-topes, calculated by the cranking formula for the coher-ent state J IBM in Eq. (3) and those taken from the 2 +1 eigenenergies of the IBM and the experimental 2 +1 ex-citation energies [77] using the rotor formula L ( L + 1).Note that the cranking moment of inertia of the IBM is,due to the correction by the LL term, set identical to theTV moment of inertia. Thus the TV moment of inertiais not depicted in Fig. 3.The experimental moment of inertia decreases with N and the slope of this decrease appears to change at N = 116. This change suggests a gradual shape transi-tion. The moment of inertia of the IBM intrinsic state,in contrast, decreases smoothly with the exception of the kink at N = 114. Perhaps such a kink reflects a detailedshell structure irrelevant to the present work. However,the kink is eliminated in the moment of inertia after di-agonalization, which falls on the same systematics as theexperimental data.It appears that, from Fig. 3, the cranking moment ofinertia still works for the nuclei N =110 and 112, forwhich one cannot see any difference from the moment ofinertia taken from the IBM eigenenergies. In the tran-sitional region of 114 N D. Excited states
We now discuss in Figs. 4 and 5 the low-lying statesfor the considered isotopic chains.Experimentally [77–79], the excitation energies of the
110 114 118 1220.511.52 Neutron Number − E C o rr ( M e V ) OsPtWHfYb
Correlation energy
FIG. 2: (Color online) Correlation energy E Corr defined inEq. (5) for Yb, Hf, W, Os and Pt isotopes.
110 114 118 1220102030 Neutron Number M o m en t o f i ne r t i a ( M e V − ) W IBM (from E(2 +1 ))IBM (coherent state)Expt. FIG. 3: (Color online) Moments of inertia of W isotopes,computed by the cranking formula for the coherent state, bythe the rotor formula L ( L + 1) using the 2 +1 eigenenergies ofthe IBM and of the experimental 2 +1 excitation energies [77] ground-state band shown in Fig. 4, namely the 2 +1 , 4 +1 ,6 +1 and 8 +1 yrast states, increase as the neutron shell clo-sure N = 126 is approached. The increase of these yrastlevels with neutron number N becomes more rapid withsmaller Z , when departing from the proton shell closure Z = 82. The present results follow the overall experi-mental isotopic trend for those nuclei. For Pt, Os and Wisotopes, the same systematics have been observed withthe Gogny-D1S functional [47, 53].The LL term has a remarkable influence on the ground-state band at the quantitative level. Without this term, the experimental yrast spectra would not be reproducedwith that precision. This is particularly the case withlighter W (Hf) isotopes with N = 110 and 112, whichfollow the rotor formula L ( L + 1) with their respectiveexperimental ratios being E +1 /E +1 =3.27 (3.29) and3.23 (3.26) [77]. For these nuclei, the results shown inFigs. 4(c) and 4(d) compare rather well with the exper-iments.We now turn to the description of the side-band ener-gies in Fig. 5. To begin with, we discuss the excited 0 + ( 0 +2 ) state. It is well known that the intruder configura-tions may play a role for mid-shell Pt isotopes, where theoblate-prolate shape coexistence is observed [10, 63]. Thephenomenological IBM study (see Ref. [66], for instance)considers particle-hole excitations across the Z = 82 pro-ton shell. In this kind of work one needs to extend theboson model space as to take into account the intruderconfiguration with additional proton bosons, arising from(mainly) the 2 p -2 h excitation. The normal and the in-truder configurations are mixed, and the model Hamilto-nian should be then diagonalized in such enlarged config-uration space. The validity of this mixing calculation hasbeen discussed extensively [67, 68], and is thus of greatinterest.The mixing in general becomes more significant whenapproaching the middle of the major shell. In Fig. 5(a),the calculated 0 +2 excitation energies for N
116 Ptisotopes, as well as those with Gogny-D1S [47], seem tocompare reasonably well with the data, even without tak-ing into account the mixing between normal and intruderstates. Furthermore, the original HFB energy surfaces forPt isotopes do not exhibit clear coexisting minima. Dueto this, the present framework cannot fix the parametersfor both the normal and the intruder configurations aswell as those for the operators mixing the two configu-rations. Although such a mixing calculation is a rathersubtle problem, it is very interesting to study the extentto which the intruder configuration plays a role when in-troduced in the present mapping method.It was shown experimentally [16–19] that, in the non-yrast states of lighter W, Os and Pt nuclei, the bandmixing could arise more or less from the coexistence ofthe different intrinsic states mentioned above, and makesit rather difficult to identify the clear band structure bya model prediction. The band-mixing feature should beoutside of the model space of bosons with low-spin onwhich the IBM is built, and may be somewhat difficultto be reproduced. It is yet not clear whether the similarcomplicated band mixing will be observed in the exoticYb and Hf isotopes.The 2 +2 level, which is normally the band-head of the K π = 2 + (so-called quasi- γ ) band, is a good test for theevolving triaxiality in a given isotopic chain. Figure 5shows that the calculated 2 +2 level of the N = 116 nucleiis lowest among each of Yb, Hf, W and Os isotopes. Ex-perimental excitation energies keep steady (decrease) inPt (Os, W) isotopes as N increases from 110 to 116.In our calculations, the decrease of the energies of the
110 114 118 12201234 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (a) Pt +1 +1 +1 +1
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (b) Os +1 +1 +1 +1
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (c) W +1 +1 +1 +1 +1 +1 +1 +1
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (d) Hf +1 +1 +1 +1
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (e) Yb +1 +1 +1 +1 FIG. 4: (Color online) Theoretical (curves) and experimental [77–79] (symbols) low-lying spectra of Yb, Hf, W, Os, and Ptisotopes with 110 N
122 for the 2 +1 , 4 +1 , 6 +1 and 8 +1 states. Symbols for the experimental levels are defined in the panel(c). +2 , 3 +1 , 4 +2 and 5 +1 states occurs more rapidly for lower Z isotopes, which have a larger number of active bosons.Around N = 116 a change in this tendency occurs andthese excitation energies increase. This is in agreementwith the only experimental measurement available in Osisotopes.A remarkable difference between the theoretical andthe experimental quasi- γ -band structure observed in Ptand Os isotopes is that the calculated 3 +1 and the 4 +2 states, and the 5 +1 and the 6 +2 states as well, form dou-blets, which are absent in the data. Since all the statesin Figs. 5(a) and 5(b) except the 0 +2 ones, are supposedto be the quasi- γ band states, the appearance of thesedoublets points to the emergence of the γ -unstable [80]or O(6) dynamical symmetry [42], in which the spectrabelonging to the same family of the quantum number τ are nearly degenerated. Since the rigid triaxial rotormodel with γ = 30 ◦ [81] predicts the doublets (2 + , 3 + ),(4 + , 5 + ), etc, in the γ band, the experimental data inFig. 5 for (a) Pt, (b) Os, and (c) W isotopes suggest asituation rather in between the γ -unstable rotor and the rigid-triaxial rotor pictures. The discrepancy of the γ -band energies occurs probably because the IBM energysurface does not show the triaxial minimum which is,however, seen in the original HFB energy surface.There are several possible effects which may eliminatethis staggering in the γ -band spectra and improve theagreement with the experiments at the quantitative level.In the present paper, however, we do not look into thedetails of this issue due to the large number of additionalparameters to be introduced and the lack of experimentaldata for the Yb and Hf nuclei. First, a three-body (cubic)term, which partially breaks O(6) symmetry, may correctthe deviation. This has been done mainly in the IBM-1[74, 75]. For the present case some type of cubic termappears to be necessary mainly for W, Os and Pt nuclei,where the Gogny HFB energy surface exhibits a shallow,but stable triaxial minimum [56]. While the calculatedexcitation energies of the quasi- γ band for Yb and Hf inFig. 5(d,e) look like that of pure O(6) limit as well, thevalidity of this term seems to be marginal in these cases.Indeed for the Yb and Hf isotopes the original Gogny-
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (a) Pt +1 +2 +2 +1 +2
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (b) Os +2 +2 +1 +2 +1
110 114 118 1220123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (c) W +2 +2 +1 +2 +1 +2 +2 +2 +1 +1
110 114 118 122123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (d) Hf +2 +2 +1 +2 +1
110 114 118 122123 Neutron Number E xc i t a t i on E ne r g y ( M e V ) (e) Yb +2 +2 +1 +2 +1 FIG. 5: (Color online) Same as Fig. 4, but for the 0 +2 , 2 +2 , 3 +1 , 4 +2 and 5 +1 states. D1M energy surface indicates the discrete change of theminimum point from the oblate ( γ = 60 ◦ ) to the prolate( γ = 0 ◦ ) sides, similarly to the Gogny-D1S energy surface[56].The second possibility would be to relax the con-straint on the deformation parameters γ π and γ ν so thatthey could take different values. As the IBM-2 can beviewed as a two-fluid system consisting of proton andneutron bosons, the phase-structure analysis would beexploited in the context of the coherent-state formalism[82], whereas it is not obvious to define a consistent map-ping procedure for realistic cases.The third would be the inclusion of higher-spin bosons,like the g -boson. It is not independent of the first pos-sibility involving the cubic term, since the cubic termcan be derived effectively from the renormalization ofthe g boson into the sd -boson sector [74]. This would, ofcourse, make the problem more complicated.We now address the problem of why the side-bandspectra, particularly for Pt in Fig. 5(a) and Os inFig. 5(b) isotopes, are overestimated in the present cal-culation when approaching the N = 126 shell closure.The direct reason would be that the microscopic Gogny energy-surface calculation predicts mostly oblate defor-mations with small quadrupole moment but with ratherlarge amount of deformation energy characterized by thedepth of the potential minimum [56]. Such a topology ofthe HFB energy surface is not well described by the IBMHamiltonian close to the end of the major shell Z = 82.Nearby the closed shell one has a relatively small num-ber of bosons. The deviation of the spectra seems to bedue to this limited degrees of freedom. The problem onthe description of the side-band energies was observed inother cases of shape transitions in different mass regions[44, 46], and is still an open problem. According to theabove argument it may be expected that the predictedlevels for exotic Yb and Hf isotopes in the vicinity of theshell closure N = 126 might be overestimated.To further examine the problem, it is interesting toconsider the relevant energy ratios, as they nicely tracethe underlying shape transition. Figure 6 depicts theenergy ratios (a) R / ≡ E (4 +1 ) /E (2 +1 ) and (b) R γ ≡ E (4 +1 ) /E (2 +2 ) as functions of N . The ratio R / is prob-ably the simplest and best-studied measure for the evo-lution of collectivity. The ratio R γ presents the locationof the band-head of the quasi- γ band 2 + γ (2 +2 ) relative to
110 114 118 1222.42.83.2 Neutron Number E ( + ) / E ( + ) (a)SU(3)O(6) 110 114 118 12200.40.81.2 Neutron Number E ( + ) / E ( + ) (b) PtOsWHfYb FIG. 6: (Color online) Theoretical (curves) and experimental (symbols) [77] energy ratios (a) R / = E (4 +1 ) /E (2 +1 ) and (b) R γ = E (4 +1 ) /E (2 +2 ) as functions of N . Definitions of the theoretical curves and the symbols for the experimental data appearin panel (b). the 4 +1 excitation energy. Since in many γ -soft nuclei the2 +2 level lies quite close to the 4 +1 level, the overall trendof ratio R γ can help to measure the γ softness.In Fig. 6(a), the experimental R / ratios for Os andW isotopes exhibit a gradual decrease as a function of N from the rotor limit of R / = 3 . N = 110 toward the O(6) limit of R / = 2 .
5. This re-flects the transition from the axially deformed rotor tothe γ -unstable shape. Also of particular interest is thedifference of the R / ratio between Pt isotopes and theother isotopes. The experimental R / ratio for all thePt isotopes studied remains practically constant all theway, being close to the O(6) limit of 2.5. The presentcalculation follows the decrease of the experimental R / value from N = 110 to 116 in Os and W isotopes, whilean increase is suggested for not only Os and W but Hfand Yb isotopes for N > R / occurs quite rapidly for Hf and Yb isotopes incomparison to W and Os isotopes. The discrepancy ofthe tendency for N >
118 for Os nuclei could be the con-sequence of the unexpectedly large χ π and χ ν values withpositive sign, as seen in Table I, since the correspondingIBM energy surfaces exhibit notable oblate deformation.The same would hold for explaining the overall devia-tion in Pt isotopes. In this context, to describe all theobserved data including those for N >
118 regime, thetriaxial dynamics needs to be correctly incorporated inthe present model.The energy ratio R γ is depicted in Fig. 6(b). Theexperiment shows that in the lighter Pt, Os and W iso-topes with N = 110, 112 and 114, the ratio is belowunity. While for Pt isotopes the experimental ratio R γ remains all the way with values close to unity, for Os and W isotopes the γ softness gradually develops with N asthe ratio R γ increases for 110 N
116 and over-passes R γ = 1 at N = 116. The overall trend of thisexperimental ratio for W and Os isotopes is reproducedin the present calculation, and the same systematic trendis predicted for Yb and Hf isotopes. For Os, the experi-mental R γ ratio decreases from N = 116 to 118, whichis reproduced by the calculation. In the heavier isotopeswith N >
118 there is a new tendency that the calculatedratio shows overall decrease, being much below the unity,whereas the experimental ratio for Os isotopes keeps in-creasing, being larger than unity. The results presentedhere do not differ much from the case of D1S functionalalready studied in [47, 53]. E. B (E2) systematics Lastly, we examine the B (E2) systematics for a few es-sential cases corresponding to the shape transition. The B (E2) ratios relevant to the band-head of quasi- γ band,2 +2 state, can be the stringent tests.We show in Fig. 7 the ratio (a) B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) and the branching ratio (b) B (E2; 2 +2 → +1 ) /B (E2; 2 +2 → +1 ) for the considered iso-topes in comparison with the data [83, 84].The 2 +2 → +1 E2 transition rate shows a certain sensi-tivity to the neutron number N and thus it is useful as asignature of the structural evolution involving the γ soft-ness. The B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) ratios forPt isotopes differ notably from those of other isotopes.For Yb, Hf, W, and Os isotopes, the calculated ratio ispeaked at N = 116. This confirms that in each of theseisotopic chains the N = 116 nucleus is softest in γ direc-0
110 114 118 122012 B ( E ) r a t i o s (a) B(E2;2 −>2 )/ B(E2;2 −>0 ) O(6)
Neutron Number
SU(3)
110 114 118 12200.20.40.6 B ( E ) r a t i o s (b) B(E2;2 −>0 )/B(E2;2 −>2 )Neutron Number U(5),O(6)SU(3)
PtOsWHfYb
FIG. 7: (Color online) The B (E2) ratio (a) B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) and the branching ratio (b) B (E2; 2 +2 → +1 ) /B (E2; 2 +2 → +1 ) for relevant low-lying states of the considered Yb, Hf, W, Os, and Pt isotopes with Gogny-D1M EDF.Experimental data for W, Os, and Pt isotopes are taken from Ref. [84]. Definitions of symbols and theoretical curves appearin panel (b). tion. On the other hand, for Pt isotopes the calculated B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) value keeps increasingtoward N = 110 to approach the O(6) limit, rather thantaking a maximum at N = 116. This tendency appearsto be consistent with that expected from the topologyof the HFB energy surface [55] and from the predictedsystematics of the quasi- γ band-head in Fig. 5(a), whichreflects that the γ softness persists for rather wide regionin the Pt isotopic chain.When compared with the D1S case [53], the presentD1M result suggests that the ratio B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) is rather sensitive to the iso-topic chains. In fact, in Fig. 7(a), the B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) values below and above N =116 appear to have a certain Z dependence when theD1M functional is used. For instance, the B (E2; 2 +2 → +1 ) /B (E2; 2 +1 → +1 ) value for W isotopes is generally farfrom the O(6) limit all the way. It has been noticed inRef. [53], however, that the calculated value of this B (E2)ratio is practically the same for Os and W isotopes whenthe D1S functional is taken. It would be interesting tosee if this Z dependence is observed experimentally.The branching ratio B (E2; 2 +2 → +1 ) /B (E2; 2 +2 → +1 )in Fig. 7(b) also presents a clear signature of the struc-tural evolution involving triaxiality. For Yb, Hf and Wisotopes with 110 N γ -soft nuclei asconfirmed by the experimental data on Os and W iso-topes. At this point, one can observe the increase from N = 116 toward the shell closure N = 126. The increaserepresents the deviation from the γ -soft character, as the corresponding mapped energy surface in Fig. 1 exhibitsnotable oblate deformation. The change in the branchingratio occurs more slowly than the D1S case [53]. This isconsistent with our general finding that the D1M energysurfaces for these nuclei show less pronounced quadrupolecorrelation than the D1S ones. As observed in Fig. 7(b),the branching ratios for Pt isotopes remain always muchcloser to zero, which is compatible with their sustained γ -soft character. IV. SUMMARY
In summary, the method of deriving the Hamiltonian ofthe interacting boson model from the constrained HFBcalculations with the Gogny functional D1M has beenapplied to the spectroscopic analysis of the neutron-richYb, Hf, W, Os, and Pt isotopes. The microscopic energysurface obtained from the constrained HFB calculationturns out to be a good starting point for both reproduc-ing and predicting the ground-state shape of the consid-ered nuclei. Spectroscopic observables that characterizethe underlying shape transitions, such as excitation en-ergies, B (E2) ratios and correlation energies, have beencalculated.It has been shown that the Pt isotopes largely differfrom the other isotopes in the rapidity of the shape tran-sition. For most of the considered Pt nuclei the mappedIBM energy surfaces are γ soft. The transition occursmore rapidly when departing from Z = 76 (Os) through Z = 70 (Yb). The triaxial deformation helps to un-derstand the prolate-to-oblate shape transition that oc-curs in the considered isotopes. The N = 116 nuclei can1be commonly identified as the transition points. This ismost noticeably seen in the overall systematic trend ofthe band-head of the γ band 2 +2 , as well as in energy and B (E2) ratios. Predicted spectra have been presented forthe neutron-rich Yb and Hf isotopes, where a quite rapidstructural evolution is suggested. When compared to theresults from the standard Gogny-D1S parametrization[47, 53], the D1M functional seems to be equally validto describe the physics involved.On the other hand, the present work aims at inves-tigating the possible ways of refining the current modeland clarifying its limitations when applied to the consid-ered mass region. First, as discussed in Sec. III D, thediscrepancy in the level structure of the quasi- γ bandturns out to be a major limitation. It is likely that thisdiscrepancy is mainly due to the use of the IBM Hamil-tonian not reproducing the triaxial energy minimum. Aspecific three-body (cubic) term may improve the agree-ment. Second, the boson effective charges need to bedetermined in a microscopic way and effects beyond themean field, like core polarization, should be taken intoaccount. It would also be meaningful to compare thespectra and the electromagnetic transition rates result- ing from the present method directly with those obtainedfrom full configuration-mixing and symmetry-conservingcalculations including triaxial degrees of freedom. Thiswould help to quantify the predictive power of the em-ployed model when applied to heavy exotic nuclei. Workalong these directions is in progress. Acknowledgments
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