Relationship between X(5) models and the interacting boson model

PHYSICAL REVIEW C 82, 024316 (2010) Relationship between X(5) models and the interacting boson model Jos´ e Barea,1,2,*Jos´ eM.Arias, 2,†and Jos´ e Enrique Garc´ ıa-Ramos3,‡ 1Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain 2Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de F´ ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 3Departamento de F´ ısica Aplicada, Universidad de Huelva, E-21071 Huelva, Spain (Received 19 April 2010; revised manuscript received 8 July 2010; published 19 August 2010) The connections between the X(5) models [the original X(5) using an infinite square well, X(5)-β8, X(5)-β6, X(5)-β4, and X(5)-β2], based on particular solutions of the geometrical Bohr Hamiltonian with harmonic potential in the γdegree of freedom, and the interacting boson model (IBM) are explored. This work is the natural extension of the work presented in Garc´ ıa-Ramos and Arias, Phys. Rev. C 77, 054307 (2008) for the E(5) models. For that purpose, a quite general one- and two-body IBM Hamiltonian is used and a numerical fit to the different X(5) model energies is performed; then the obtained wave functions are used to calculate B(E2) transition rates. It is shown that within the IBM one can reproduce well the results for energies and B(E2) transition rates obtained with all these X(5) models, although the agreement is not so impressive as for the E(5) models. From the fitted IBM parameters the corresponding energy surface can be extracted and, surprisingly, only the X(5) case corresponds in the moderately large Nlimit to an energy surface very close to the one expected for a critical point, whereas the rest of models are situated a little further away. DOI: 10.1103/PhysRevC.82.024316 PACS number(s): 21.60.Fw, 21.60.Ev I. INTRODUCTION In recent years the connection between the Bohr-Mottelson (BM) collective model [1–3] and the interacting boson model (IBM) [4–7] has been the subject of many studies [8–20]. The BM collective model is built on the assumption that the nucleus is composed of a set of strongly interacting fermions that can be treated as a quantum liquid. The surface of such a liquid is characterized in terms of the Hill-Wheeler shape variables (β,γ)[21] and the Euler angles. Under this approximation, nuclear excitations are small-amplitude vibrations and rotations, corrected by the coupling between them [22]. The IBM was designed to describe the collective quadrupole degrees of freedom in medium-mass and heavy nuclei. The IBM Hamiltonian was written from the beginning in second quantization form in terms of the generators of the U(6) algebra, subtended by sand dbosons, which carry angular momenta 0 and 2, respectively [7]. Therefore, the connection between the two models is not evident. An approximated connection comes from considering the IBM as the second quantization of the shape variables (β,γ)[23]. During the 1980s many studies on the connection between the two models were done. The intrinsic state formalism [8–13] was used, but also the complete set of eigenstates [24]was analyzed through a Holstein-Primakoff transformation [25,26] or by an isometric transformation [27]. More recently, the problem of mapping between the two models has been addressed by Rowe and collaborators [14]. Both models present three special limits that can be solved easily and for which the connection between the models is known. *[email protected] †[email protected] ‡[email protected] These three cases are (i) the BM anharmonic vibrator and the dynamical symmetry U(5) IBM limit, (ii) the BM γ-unstable deformed rotor and the dynamical O(6) IBM limit, and (iii) the BM axial rotor and the dynamical symmetry O(6) IBM limit including Q·Q·Qinteractions [14,20,28]. Note that, although traditionally accepted, the correspondence of the dynamical symmetry SU(3) IBM limit to a submodel of the BM model has never been explicitly probed [14]. Each of these limits is assigned to a particular shape using the Hill-Wheeler variables (β,γ)[21]: spherical, deformed with γ-instability, and axially deformed, respectively. For transitional situations the correspondence between the two models is difficult and a possible way to establish a mapping between the BM model and the IBM is through numerical studies. Among the transitional Hamiltonians, an especially interesting case occurs when it describes a critical point in the transition from a given shape to another. In general, for such a situation where the structure of the system can change abruptly by applying a small perturbation, both the BM model and the IBM have to be solved numerically. However, a few years ago Iachello proposed schematic Bohr Hamiltonians with the intent of describing different critical points and these can be solved exactly in terms of the zeros of Bessel functions. The first of these models is known as E(5) [29]. E(5) is designed to describe the critical point at the transition from spherical to deformed γ-unstable shapes. The potential to be used in the differential Bohr equation is assumed to be γindependent and, for the βdegree of freedom, an infinite square well is taken. Similar models were proposed later by Iachello, called X(5) and Y(5) [30,31], to describe the critical points between spherical and axially deformed shapes and between axial and triaxial deformed shapes, respectively. All these models give rise to spectra and electromagnetic transition rates that depend on a couple of free parameters (including a scale). In 0556-2813/2010/82(2)/024316(12) 024316-1 ©2010 The American Physical Society BAREA, ARIAS, AND GARC´ IA-RAMOS PHYSICAL REVIEW C 82, 024316 (2010) spite of their simplicity, some experimental examples were found [32,33], just after the appearance of these models. In this work, we concentrate on X(5) and related models. (The connection between E(5) and related models and the IBM was already deeply studied in [34].) The formulation of X(5) attracted immediate attention both experimentally and theoretically. Soon after the introduction of the X(5) model, the nucleus 152Sm was proposed by Casten and Zamfir [33] as a realization of it. Other experimental examples proposed are 150Nd, 152–154Gd, 130Ce, 162Yb, 166Hf, 178Os, 226Ra, and 226Th [35,36], although the last two candidates could be better described by combining quadrupole and octupole degrees of freedom [37–39]. Concerning theoretical extensions of X(5), Bonatsos and collaborators studied a sequence of potentials of the type β2n, which allows one to go from the vibrational limit, n=1, to X(5), n→∞[40]. In particular, in Ref. [40] spectra and transition rates for the potentials of the type β2n, with n⩾1, β2,β4,β6, and β8are given explicitly and compared with the original X(5) (infinite square well potential) case. Another extension of X(5) is X(3), which is a rigid version of X(5) [41]. In Ref. [42] the connection between X(5) and a two-parameter free IBM calculation was studied. In Ref. [43] the authors compared X(5)-β2, X(5)-β4, and X(3) also with a restricted two-parameter IBM calculation with a number of bosons N=10. In Refs. [44,45] the authors treated exactly separable versions of the Davidson potential and of the X(5) potential, respectively. Finally, in Ref. [46] the author studied the effect that the β-γcoupling has in solving the BM equation for the X(5) potential. As already mentioned, all these models are produced in the BM scheme and a natural question is to ask for the correspondence of them with the IBM. Is the IBM able to produce the same spectra and transition rates? If yes, does the obtained IBM Hamiltonian correspond to a critical point? This work is intended to answer these questions for the X(5) and related models (−β8,−β6,−β4, and −β2potentials) and to analyze the convergence as a function of the boson number, N. For that purpose, a large set of X(5) and related models results for excitation energies and transition rates are taken as reference for numerical fits of the general IBM Hamiltonian. This procedure will allow us to establish the IBM Hamiltonians that best fit the different X(5)-β2nmodels and their relation with the critical points. The paper is organized as follows: In Sec. II the fitting procedure is described and the obtained results are commented upon. Section III is devoted to study of the energy surfaces of the fitted IBM Hamiltonians and to analyzing them in relation to the critical point. Finally, in Sec. IV a summary and conclusions of this work are presented. II. THE IBM FIT TO X(5) MODELS A. The model The most general, including up to two-body terms, IBM Hamiltonian can be written in multipolar form as ˆ H=εdˆ nd+κ0ˆ P†ˆ P+κ1ˆ L·ˆ L+κ2ˆ Q·ˆ Q +κ3ˆ T3·ˆ T3+κ4ˆ T4·ˆ T4,(1) where ˆ ndis the dboson number operator, and ˆ P†=1 2(d†·d†−s†·s†),(2) ˆ L=√10(d†×˜ d)(1),(3) ˆ Q=(s†×˜ d+d†×˜ s)(2) −√7 2(d†×˜ d)(2),(4) ˆ T3=(d†×˜ d)(3),(5) ˆ T4=(d†×˜ d)(4).(6) The symbol ·stands for the scalar product, defined as ˆ TL·ˆ TL= M(−1)Mˆ TLM ˆ TL−M, where ˆ TLM is the Mcomponent of the operator ˆ TL. The operator ˜γm =(−1)mγ−m(where γrefers to sand dbosons) is introduced to ensure the correct tensorial character under spatial rotations. The electromagnetic transitions can also be analyzed in the framework of the IBM. In particular, in this work we will focus on the E2 transitions. The most general E2 transition operator including up to one-body terms is written as ˆ TE2 M=eeff(s†×˜ d+d†×˜ s)(2) M+χ(d†×˜ d)(2) M,(7) where eeff is the boson effective charge and χis a structure parameter. In this work χwill be kept fixed to the SU(3) value of −√7/2. Although one could use the most general IBM Hamiltonian (1) to describe the X(5) models a natural question is whether it is possible or not to reduce the number of free parameters. Apriori, it is not obvious which terms of the Hamiltonian can be taken out. To answer this question one can rewrite the Hamiltonian (1) in terms of Casimir operators (with the definitions for the Casimir operators having been taken from [47]): ˆ H=κ0 4N(N+4) +εd+18 35κ4 ˆ C1[U(5)] +18 35κ4ˆ C2[U(5)] +κ1−3 8κ2−κ3 10 −κ4 14ˆ C2[O(3)] +κ2 2 ˆ C2[SU(3)] +κ3 2−3 14κ4ˆ C2[O(5)] −κ0 4 ˆ C2[O(6)].(8) It is expected that for the description of the X(5) models one will need the contribution of the three IBM dynamical symmetry chains. In the Hamiltonian (8) two contributions come from the U(5) algebra, the linear and the quadratic Casimir operators. Therefore, it becomes a reasonable “antsatz” to remove the U(5) quadratic Casimir operator. That implies fixing κ4=0. Note that its contribution to the rest of the Casimir operators is absorbed by the rest of the parameters. B. The fitting procedure In this section we describe the procedure for getting the IBM Hamiltonian parameters that best fit the different X(5) models. 024316-2 RELATIONSHIP BETWEEN X(5) MODELS AND THE ... PHYSICAL REVIEW C 82, 024316 (2010) The χ2test is used to perform the fitting. The χ2function is defined in the standard way, χ2=1 Ndata −Npar Ndata  i=1 [Xi(data) −Xi(IBM)]2 σ2 i ,(9) where Ndata is the number of data, from a specific X(5) model, to be fitted, Npar is the number of parameters used in the IBM fit, Xi(data) is an energy level [or a B(E2) value] taken from a particular X(5) model, Xi(IBM) is the corresponding calculated IBM value, and σiis an arbitrary error assigned to each Xi(data). At this point it is necessary to explain how we treat the γ bandhead in the X(5) model. The position of this band is not determined by the model; therefore an extra parameter should be introduced for determining the γbandhead, specifically the location of the bandhead 2+ γstate. Consequently, besides the IBM Hamiltonian parameters, there is this extra free parameter, γ, in the fit. This parameter gives the optimum position of the γband in a given X(5) model for which the best IBM fit is obtained. To perform the fit, we minimize the χ2function for the energies, using εd,κ0,κ1,κ2, and κ3as free parameters of the IBM Hamiltonian and γas a free parameter for the position of the γbandhead. Note that besides the qualitative argument presented in Sec. II A to justify the election of κ4=0, we have extensively explored other possibilities, as for example κ4= 0[48]. For this election a little improvement in χ2is obtained and for some particular cases it even has a negative impact. In addition, in this case the χ2function is very flat and the correlation that exists between the free parameters generates nonphysical oscillations in the value of the fitted parameters. We have also explored the case κ3=0 and κ4=0, which produces a important increase in the χ2value. The κ4=0 selection produces a smooth and consistent behavior of the fitted parameters, as can be observed in Figs. 2–6. For doing the fit and the minimization of the χ2function the MINUIT [49] code has been used. It allows us to minimize any multivariable function. The labels for the energy levels follow the usual notation introduced for the X(5) model [29]: senumerates the zeros of the βpart of the wave function, and nγenumerates the number of γphonons. The set of levels included in the fit for the different X(5) models are as follows: (i) For the ground-state band, s=1,n γ=0, the set includes all the states with angular momentum smaller than 10. An arbitrary σ=0.001 is used for these states except for the 2+ 1state for which σ=0.0001 is used. This latter value allows us to normalize all the IBM energies to E(2+ 1)=1. Note that the energy of the state 2+ 1is fixed arbitrarily to 1 (where we recall that the spectrum is calculated up to a global scale factor). (ii) For the βband, s=2,n γ=0, the set includes all the states with angular momentum smaller than 10. An arbitrary σ=0.01 is used for these states. (iii) For the s=3,n γ=0 band, which can be identified with the ββ band, the set includes all the states with angular momentum smaller than 10. An arbitrary σ= 0.01 is used for these states. TABLE I. States included in the energy fit. In the states labeled with two subindexes, the first one corresponds to X(5), whereas the second corresponds to the rest of models. Band Error States s=1,n γ=0σ=0.0001 2+ 1 σ=0.001 0+ 1,4+ 1,6+ 1,8+ 1 s=2,n γ=0σ=0.01 0+ 2,2+ 2,4+ 2,6+ 2,8+ 2 s=3,n γ=0σ=0.01 0+ 3,2+ 3,4,4+ 4,6+ 4,8+ 4 s=1,n γ=1σ=0.01 2+ 4,3,3+ 1,4+ 3,5+ 1,6+ 3,7+ 1,8+ 3 (iv) For the γband, s=1,n γ=1, the set includes all the states with angular momentum smaller than 10. An arbitrary σ=0.01 is used for these states. With this selection, the number of energy levels included into the fit, Ndata, is equal to 21. Note that the state 0+ 1is not an actual data point to be reproduced because we are interested just in excitation energies and therefore the ground state is naturally fixed to zero in both the X(5) models and the IBM. In Table Ithe states included in the fit are explicitly given. The ordering index for the even Lstates in the γ,β, and ββ bands is unknown apriori, because of the undetermined position of the X(5) γbands. However, our best fit always provides a γband above the βband, but below the ββ band, that generates the same ordering independently of the particular X(5) model and number of bosons. The only exception happens for X(5), where the 2+of the γband is above the 2+stateintheββ band (i.e., they correspond to 2+ 4and 2+ 3, respectively). Once the IBM Hamiltonian is fixed for each X(5) model by fitting the energy levels, the χ2function for the B(E2) values is calculated without any additional fitting. The two parameters in the E2 operator (7)areeeff , which is fixed to give B(E2; 2+ 1→0+ 1)=100, and χ, which is fixed to the SU(3) value of −√7/2. The computed transitions are listed in Table II. Note that only transitions between states with nγ=0 states are considered. C. The results We have performed fits of the IBM Hamiltonian (1) parameters plus γfor different values of the number of TABLE II. B(E2) transitions to be calculated. nγ=0 in all the cases. sisfsisf B(E2:2 + 1→0+ 1)1 1B(E2:2 + 2→0+ 1)2 1 B(E2:4 + 1→2+ 1)1 1B(E2:2 + 2→0+ 2)2 2 B(E2:6 + 1→4+ 1)1 1B(E2:4 + 2→2+ 1)2 1 B(E2:8 + 1→6+ 1)1 1B(E2:4 + 2→4+ 1)2 1 B(E2:0 + 2→2+ 1)2 1B(E2:4 + 2→2+ 2)2 2 B(E2:2 + 2→2+ 1)2 1 024316-3 BAREA, ARIAS, AND GARC´ IA-RAMOS PHYSICAL REVIEW C 82, 024316 (2010) 10 20 30 40 50 60 70 N 4000 5000 6000 7000 8000 9000 10000 11000 12000 χ2 X(5) X(5)-β8 X(5)-β6 X(5)-β4 X(5)-β2 FIG. 1. (Color online) χ2for the IBM fit to the energy levels of the different X(5) models, as a function of N. bosons, N, so as to reproduce as well as possible the energies of the states given in Table I. These states are generated by the different X(5) models: X(5), X(5)-β8, X(5)-β6, X(5)-β4, and X(5)-β2. As mentioned before, εd,κ0,κ1,κ2,κ3, and γare free parameters in a χ2fit to the energy levels produced by the different X(5) models (where κ4was fixed to zero as discussed in the preceding section). In Fig. 1the value of the χ2for the best fit to the different X(5) models as a function of Nis shown. The different lines in Fig. 1correspond to the fit to different X(5) models as stated in the legend. It is clearly observed that for any Nthe best agreement is obtained for the X(5)-β6and X(5)-β8cases, which present a χ2function almost identical for any value of N.Theχ2function increases for X(5)-β4and X(5)-β2, up to reach X(5), which has the higher χ2value and therefore the worst degree of agreement. Anyway, the differences among the different X(5) models are smaller than a factor 2inχ2. It is worth noting that these results change slowly with the boson number and in all cases the χ2function saturates to a given value in the large-Nlimit. The situation presented here is somehow different from the analysis of the E(5) models [34] where there is a monotonic behavior in the χ2values in passing from E(5)-β4, which presents the lowest value, to E(5)-β6, E(5)-β8, and E(5), where the maximum appears. In Figs. 2–6the Hamiltonian parameters and the excitation energies for the ground, βand γbands are plotted for the cases of X(5), X(5)-β8, X(5)-β6, X(5)-β4, and X(5)-β2, respectively. The Hamiltonian parameters present clear analogies in their behavior in all the X(5) models. εdincreases continuously on the whole range of Nfor all the X(5) models. κ1, in contrast, exhibits a relatively small and modest value. κ0and κ2show a very smooth variation until reaching a value of saturation, with a tendency to increase in all cases, except in the case of κ0for X(5)-β2, which tends to decrease. κ3in almost all the cases tends to increase all the way. In the X(5) case (Fig. 2), κ3exhibits a decreasing behavior for small values of Nbut, as already mentioned, exhibits an increasing behavior for larger Nvalues. A similar behavior is also observed for γ.Forthe 8000 12000 16000 0 2 4 6 8 -800 -600 -400 -200 0 parameter (arb. units) εd κ0 κ1 κ2 κ3 0 5 10 15 20 Ener gy ( arb. units ) 10 20 30 40 50 60 70 N 17 17.2 17.4 17.6 17.8 ∆γ 10 20 30 40 50 60 70 N 20 25 01 + 21 + 41 + 61 + 81 + 0β + 2β + 4β + 6β + 8β + 2γ + 3γ + 4γ + 5γ + 6γ + 7γ + 8γ + (a) (b) (c) (d) (e) (f) FIG. 2. (Color online) Parameters [(a), (b), and (c)] and excitation energies of the ground (d), β(e), and γ(f) bands for the X(5) case, as a function of N. In the right panels, the continuous red lines are the X(5) results, and the dashed black lines are the fitted results. 024316-4 RELATIONSHIP BETWEEN X(5) MODELS AND THE ... PHYSICAL REVIEW C 82, 024316 (2010) 6000 8000 10000 12000 14000 0 2 4 6 8 -600 -400 -200 0 parameter (arb. units) ε κ0 κ1 κ2 κ3 0 5 10 15 20 Energy (arb. units) 10 20 30 40 50 60 70 N 10 10.2 10.4 10.6 10.8 Gap 10 20 30 40 50 60 70 N 10 15 01 + 21 + 41 + 61 + 81 + 0β + 2β + 4β + 6β + 8β + 2γ + 3γ + 4γ + 5γ + 6γ + 7γ + 8γ + (a) (b) (c) (d) (e) (f) FIG. 3. (Color online) Same as Fig. 2but for the X(5)-β8case. energy levels, it is remarkable that the plotted energies have almost constant values regardless the value of N, except for the γband in the low-Nregion where the energies smoothly move to a saturation value. The agreement for the ground-state band is very good but for the βand γbands the agreement is poorer. In the X(5)-β8case (Fig. 3), κ3also exhibits a decreasing behavior for very low values of N, but from there on it monotonically increases. γexhibits an almost constant value, except for the largest values of N, where a increase (note the small energy scale) is shown. The energies remain almost constant in the full range of N, except for the γband, where the increase of the γvalue generates, for large values on N, a corresponding increase. The agreement of the energies in the ground-state band is perfect and reasonable in the βand γbands. The X(5)-β6case (Fig. 4) is very similar to that of X(5)-β8, although here γexhibits a rather flat behavior with 4000 8000 12000 0 2 4 6 8 -800 -400 0 parameter (arb. units) εd κ0 κ1 κ2 κ3 0 5 10 15 Energy (arb. units) 10 20 30 40 50 60 70 N 8.6 8.8 9 9.2 9.4 9.6 ∆γ 10 20 30 40 50 60 70 N 10 15 01 + 21 + 41 + 61 + 81 + 0β + 2β + 4β + 6β + 8β + 2γ + 3γ + 4γ + 5γ + 6γ + 7γ + 8γ + (a) (b) (c) (d) (e) (f) FIG. 4. (Color online) Same as Fig. 2but for the X(5)-β6case. 024316-5 BAREA, ARIAS, AND GARC´ IA-RAMOS PHYSICAL REVIEW C 82, 024316 (2010) 6000 7000 8000 9000 10000 0 2 4 6 8 -800 -600 -400 -200 0 200 parameter (arb. units) εd κ0 κ1 κ2 κ3 0 5 10 15 Energy (arb. units) 10 20 30 40 50 60 70 N 7 7.2 7.4 7.6 7.8 ∆γ 10 20 30 40 50 60 70 N 10 15 01 + 21 + 41 + 61 + 81 + 0β + 2β + 4β + 6β + 8β + 2γ + 3γ + 4γ + 5γ + 6γ + 7γ + 8γ + (a) (b) (c) (d) (e) (f) FIG. 5. (Color online) Same as Fig. 2but for the X(5)-β4case. a minimum around N=40. Once more, the agreement of the energies in the ground-state band is perfect and reasonable in the βand γbands. The X(5)-β4case shown in Fig. 5 shows a monotonic increase of κ3and a behavior for γ almost identical to that for X(5)-β6. Here also the agreement of the energies in the ground-state band is good whereas the description of the βand γbands is poorer. Finally, X(5)-β2 (Fig. 6) also exhibits a monotonic increase of κ3whereas κ0 has a smooth decreasing behavior till a saturation value. In γ the behavior of X(5) is recovered, with a smooth decrease up to 5000 6000 7000 8000 0 2 4 6 8 -600 -400 -200 0 200 parameter (arb. units) εd κ0 κ1 κ2 κ3 0 5 10 Ener gy ( arb. units ) 10 20 30 40 50 60 70 N 5.2 5.3 5.4 5.5 ∆γ 10 20 30 40 50 60 70 N 5 10 01 + 21 + 41 + 61 + 81 + 0β + 2β + 4β + 6β + 8β + 2γ + 3γ + 4γ + 5γ + 6γ + 7γ + 8γ + (a) (b) c) ) (d) (e) (f) FIG. 6. (Color online) Same as Fig. 2but for the X(5)-β2case. 024316-6 RELATIONSHIP BETWEEN X(5) MODELS AND THE ... PHYSICAL REVIEW C 82, 024316 (2010) TABLE III. Parameters (in arbitrary units) of the IBM Hamiltonian plus the excitation energy for the γbandheads, γ,thatbestfit the different X(5) models for N=50. εdκ0κ1κ2κ3γ X(5) 15420 −31.77.5−77.1−454.817.5 X(5)-β811426 53.617.8−43.3−369.210.4 X(5)-β610661 58.519.9−38.5−372.79.2 X(5)-β48942 72.919.6−27.8−301.47.4 X(5)-β26595 65.918.2−17.6−216.65.4 N=15 and a smooth increase from there on. The agreement of the energies in the ground-state band is very good and reasonable for the βand γbands. In panels (e) of Figs. 2–6one can see that the moment of inertia of the βband increases from X(5), where it reaches the smallest value, till X(5)-β2, where it is maximum. In panels (d) and (f) of these figures one can also observe how the moment of inertia of ground and γbands is roughly stable for all the models. This tendency is correctly reproduced by the IBM fits. It is worth remarking that the agreement of the energies in the βand the γbands does not clearly deteriorate when ascending up in the band. In particular, in the γband the best agreement is obtained for angular momenta values 4, 5, and 6. Similar conclusions are obtained for the ββ band. This fact has a important consequence for the value of χ2: Contrary to what might be expected, the higher states included in the fitting procedure for the β,γ, and ββ bands produce smaller contributions to χ2than the low-lying states of these bands. To have a clearer idea of the values of the Hamiltonian parameters obtained in the fits we present in Table III the parameters of the IBM Hamiltonians that best fit the different X(5) models for N=50. Two facts are apparent: 1. The parameters for X(5)-β8and X(5)-β6are amazingly similar and 2. The parameters change almost monotonically when going from X(5) to X(5)-β2. As a test for the produced wave functions with the fitted IBM Hamiltonian, they are used for calculating E2 transition probabilities, B(E2). The effective charge in the E2 operator (7)isfixedsoastogiveB(E2; 2+ 1→0+ 1)=100, 10 20 30 40 50 60 70 N 0 10 20 30 40 50 60 70 80 90 χ2 X(5) X(5)-β8 X(5)-β6 X(5)-β4 X(5)-β2 FIG. 7. (Color online) χ2values for the E2 transition rates for the different X(5) models, as a function of N, calculated using the IBM electromagnetic transition operator ˆ TE2 M=eeff[(s†˜ d+d†˜ s)− √7/2(d†×˜ d)(2) M]. and thus no free parameters are left in this calculation. For the B(E2) values calculated (not a fit) a χ2value has been obtained for each X(5) model with an arbitrary σ=10. In Fig. 7the corresponding χ2value is plotted as a function of N for all the X(5) models considered. Figure 7shows a smooth dependence of χ2on N.Theχ2value decreases monotonically as Nincreases for all the X(5) models. The best agreement is obtained for X(5) while the worst is for X(5)-β2. For a quantitative comparison, the B(E2) values for selected transitions with N=50 are shown in Table IV.Inthis table, the remarkable agreement between the IBM calculations TABLE IV. B(E2) values (in arbitrary units) obtained, for N=50, for the fitted IBM Hamiltonians (see text) compared with those provided by the different X(5) models. X(5) IBM X(5)-β8IBM X(5)-β6IBM X(5)-β4IBM X(5)-β2IBM B(E2:2 + 1→0+ 1) 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 B(E2:4 + 1→2+ 1) 159.9 155.4 163.4 159.1 165.3 145.7 169.0 164.9 177.9 174.6 B(E2:6 + 1→4+ 1) 198.2 181.6 208.8 189.2 214.6 160.8 226.2 199.9 255.2 217.7 B(E2:8 + 1→6+ 1) 227.6 199.1 247.3 210.3 258.1 166.2 279.9 224.9 337.1 249.2 B(E2:0 + 2→2+ 1)62.443.474.755.781.011.293.274.6 121.9 106.7 B(E2:2 + 2→2+ 1)8.24.49.77.310.30.911.315.413.433.4 B(E2:2 + 2→0+ 1)2.10.22.20.02.23.52.00.61.64.4 B(E2:2 + 2→0+ 2)79.556.197.267.5 106.068.3 122.069.3 155.749.2 B(E2:4 + 2→2+ 1)0.90.20.80.00.75.20.50.70.12.1 B(E2:4 + 2→4+ 1)6.13.47.76.88.40.79.616.612.427.7 B(E2:4 + 2→2+ 2) 120.0 111.9 149.1 121.7 162.9 108.1 187.7 110.9 240.399.3 024316-7 BAREA, ARIAS, AND GARC´ IA-RAMOS PHYSICAL REVIEW C 82, 024316 (2010) and those from the X(5) model become clear. However, as soon as we move to the rest of models the agreement starts getting worse. Some comments on these results are in order. First, one observes that the B(E2) values within the ground-state band are calculated qualitatively correct in all models. This agreement is quantitatively good for the X(5) case and gets worse for the other models. The X(5)-β2 case is the worst, showing differences of a factor of 2 for some transitions. The IBM seems to saturate to too small a value as the angular momentum increases; therefore the deviation increases with the angular momentum. The intraband transitions in the βband exhibit the same kind of agreement, where the larger discrepancy is for B(E2:4 + 2→2+ 2). The interband transitions once more agree well qualitatively. In summary, one can say that the structure of the wave functions is correctly captured by the IBM fit, especially for X(5), and the calculations are able to reproduce the sequence of large and small values (with few exceptions), confirming the appropriate structure of the wave functions. III. THE CRITICAL HAMILTONIAN One of the most attractive features of the X(5) models treated in this work is that they are supposed to describe, at different approximation levels, the critical point in the transition from spherical to rigid axially deformed shapes. Since they are connected to a given IBM Hamiltonian, as shown in the preceding section, this should correspond to the critical point in the transition from spherical to axially deformed shapes, that is, this Hamiltonian should produce an energy surface with degenerated spherical and deformed minima. Is this the case for the fitted IBM Hamiltonians obtained in the preceding section? Before starting with the discussion it is necessary to establish a measure of how close is a given IBM Hamiltonian to the critical point. An energy surface can be associated with a given IBM Hamiltonian by using the intrinsic state formalism [8,9,11], which introduces the shape variables (β,γ)intheIBM.To define the intrinsic state one has to consider that the dynamical behavior of the system can be approximately described in terms of independent bosons moving in an average field [50]. The ground state of the system is written as a condensate, |c, of bosons that occupy the lowest energy phonon state, † c: |c= 1 √N!(† c)N|0,(10) where † c=1 1+β2s†+βcos γd† 0+1 √2βsin γ(d† 2+d† −2). (11) Here βand γare variational parameters related with the shape variables in the geometrical collective model [11]. The expectation value of the Hamiltonian (1) in the intrinsic state (10) provides the energy surface of the system, E(N,β,γ)=c|ˆ H|c. This energy surface in terms of the parameters of the Hamiltonian (1) and the shape variables can be readily obtained [51] (where we keep the κ4variable for completeness) as c|ˆ H|c= Nβ2 (1 +β2)εd+6κ1−9 4κ2+7 5κ3+9 5κ4 +N(N−1) (1 +β2)2κ0 4+β2−κ0 2+4κ2 +2√2β3κ2cos(3γ)+β4κ0 4+κ2 2+18 35κ4. (12) The shape of the nucleus is defined through the equilibrium value of the deformation parameters, βand γ, which are obtained by minimizing the ground-state energy, c|ˆ H|c. A spherical nucleus has a global minimum on the energy surface at β=0, whereas a deformed one has the absolute minimum at a finite value of β. The parameter γrepresents the departure from axial symmetry; that is, γ=0 and γ=π/3 stand for a prolate and oblate, respectively, axially deformed nucleus, while any other value corresponds to a triaxial shape. An additional situation arises when the energy surface is independent of γbut shows a minimum at a finite value of β; in this case the nucleus is γ-unstable. It has to be noted that for a general IBM Hamiltonian including up to two-body terms, such as the one considered in this work, triaxiality is forbidden. With the tools just described one can study phase transitions in the IBM [8]. First, the parameters that define the Hamiltonian are the control parameters and these are usually chosen in such a way that only one of them is a variable, while the rest remain constant. The deformation parameters βand γbecome the order parameters, although in our case the order parameter is just β. Roughly speaking, a phase transition appears when there exists an abrupt change in the shape of the system when changing smoothly the control parameter. The phase transitions can be classified according to the Ehrenfest classification [52]. First-order phase transitions appear when there exists a discontinuity in the first derivative of the energy with respect to the control parameter. This discontinuity appears when two degenerate minima exist in the energy surface for two values of the order parameter β. Second-order phase transitions appear when the second derivative of the energy with respect to the control parameter displays a discontinuity. This happens when the energy surface presents a single minimum for β=0 and the surface satisfies the condition (d2E dβ2)β=0=0. In a modern classification, secondorder phase transitions belongs to the high-order or continuous phase transitions [52]. The X(5) situation was designed to describe first-order phase transitions. To determine whether a given Hamiltonian corresponds to a critical point or not, the flatness or the existence of two degenerate minima in the energy surface should be investigated. For the case of a one-parameter IBM Hamiltonian (e.g., consistent Q (CQF) Hamiltonians [53]), it is simple to find an analytical expression for the critical control parameter in the Hamiltonian. However, for a general IBM Hamiltonian, such as the one used in this work, it is necessary to rewrite the energy surface in a special way, as proposed first by L´ opez-Moreno and Casta˜ nos [15]. There, the authors manage 024316-8 RELATIONSHIP BETWEEN X(5) MODELS AND THE ... PHYSICAL REVIEW C 82, 024316 (2010) to write the energy surface of a general IBM Hamiltonian in terms of two parameters. They made use of some concepts from catastrophe theory [54] to define the two essential parameters, (r1,r 2) of the problem. In terms of these two essential parameters they found expressions for the locus, in the essential parameter space, that gives a critical point at the origin in β, called the bifurcation set, and for the locus that gives rise to two degenerate minima, called the Maxwell set. The essential parameters r1and r2can be written as r1=a3−u0+˜ε/(N−1) 2a1+˜ε/(N−1) −a3 ,(13) r2=− 2a2 2a1+˜ε/(N−1) −a3 ,(14) where ˜ε=εd+6κ1−9 4κ2+7 5κ3+9 5κ4, a1=1 4κ0+1 2κ2+18 35κ4, a2=2√2κ2,(15) a3=−1 2κ0+4κ2, u0=κ0 2. By using the essential parameters, the energy surface can be written as E∗(β,γ)=c|ˆ H|c−N(N−1)u0/2 N(N−1)(2a1+˜ε/(N−1) −a3) =1 1+β2[β4+r1β2(β2+2) −r2β3cos3γ].(16) Note that E∗(β,γ) does not depend on the number of bosons, allowing us to compare fairly energy surfaces corresponding to different boson numbers. In Fig. 8we display the plane of the essential parameters where the fitted Hamiltonians to the different X(5) models are plotted with different symbols. In this plane a critical first-order Hamiltonian corresponds to a point over the dashed line (Maxwell set, i.e., two degenerated minima). The r1=0 line corresponds to (d2E dβ2)β=0=0 (i.e., to the appearance of the spherical minimum called antispinodal point). The curved full line above the Maxwell set corresponds to (d2E dβ2)β0=0, with β0= 0 (i.e., it corresponds to the appearance of a deformed minimum called spinodal point). The area below the curved full line and above r1=0 is the coexistence region. In this region two minima, one spherical and one deformed, coexist. We also represent in the plane the second-order critical point (r1=r2=0). Note that all the points above the dashed line correspond to spherical shapes, whereas those below that line represent deformed shapes. We only plot the r2>0 semiplane because our IBM Hamiltonian always corresponds to prolate shapes and therefore to r2>0. The semiplane r2<0 is identical to the one presented in Fig. 8, but for oblate shapes. The different symbols in the figure correspond to the IBM Hamiltonians fitted to one of the X(5) models (see legend) for Nranging from 10 to 70. The idea is to see whether 0 0.5 1 1.5 r2 -0.1 0 0.1 0.2 r1 X(5) X(5)-β8 X(5)-β6 X(5)-β4 X(5)-β2 2ndP.T. 1stP.T. FIG. 8. (Color online) Position in the plane r1-r2of the energy surfaces extracted from the fitted IBM Hamiltonians to the different X(5) models as a function of N(10, 20, 30, 40, 50, 60, and 70). The largest value of Nis placed on the left-hand side whereas the smallest one is on the right part of the figure. in the large-Nlimit the obtained IBM Hamiltonians go to the Maxwell set (dashed line), which is the line of first-order phase transitions. For all the models, the Nvalue of the different points increases from the right to the left. Several important features can be extracted from this figure. The main one is that the X(5) case, for the whole range of N, is always very close to the Maxwell set (i.e., to the first-order phase transition line), being closer as the value of Nincreases. This is not exactly the case for the rest of fits. X(5)-β8, X(5)-β6, and X(5)-β4are situated on the coexistence region for small or moderate Nvalues, but they go toward the prolate deformation region as Nincreases, although they are always very close to the antispinodal line. Finally, the case X(5)-β2lies in the deformed region but it approaches the coexistence area as Nincreases. In general, one observes that the mapped IBM energy surfaces move further away from the coexistence region as one changes from X(5) through X(5)-β8, X(5)-β6, X(5)-β4, and X(5)-β2. In general, the higher the value of nis [X(5)-βn], the closer to the coexistence region is the IBM energy surface. In all the cases one observes that the different models always move in the direction of the second-order phase transition point as Nincreases. To illustrate graphically the shapes of the energy surfaces obtained with the different IBM Hamiltonians, we plot in Fig. 9 the axial IBM energy surfaces E∗(β,0) [Eq. (16)], along the prolate leg, extracted from the fit to the five analyzed X(5) models as a function of the deformation βfor three values of N: 10, 40, and 70. All the panels show a rather similar aspect: For N=10 there exists a more pronounced minimum, while passing from N=40 to N=70 the energy surface flattens, moving into a shape with two approximately degenerated minima. As was studied in Fig. 8,inFig.9(a), which is for X(5), the three curves present two minima, in Figs. 9(b),9(c), and 9(d), corresponding to X(5)-β8, X(5)-β6, and X(5)-β4, respectively, only the line for N=10 has two minima, whereas 024316-9