Phase transitions and critical points in the rare-earth region

Phase transitions and critical points in the rare-earth region J. E. Garcı ´a-Ramos* Departamento de Fı ´sica Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain J. M. Arias Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain J. Barea Departamento de Fı ´sica Ato ´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain and Centro de Ciencias Fı ´sicas, Universidad Nacional Auto ´nomadeMe ´xico, Apartado Postal 139-B, 62251 Cuernavaca, Morelos, Me ´xico A. Frank Instituto de Ciencias Nucleares, Universidad Nacional Auto ´noma de Me ´xico, Apartado Postal 70-543, 04510 Me ´xico, DF, Me ´xico and Centro de Ciencias Fı ´sicas, Universidad Nacional Auto ´nomadeMe ´xico, Apartado Postal 139-B, 62251 Cuernavaca, Morelos, Me ´xico 共Received 3 April 2003; published 15 August 2003兲 A systematic study of isotope chains in the rare-earth region is presented. For chains 60 144⫺154Nd, 62 146⫺160Sm, 64 148⫺162Gd, and 66 150⫺166Dy, energy levels, E2 transition rates, and two-neutron separation energies are described by using the most general 共up to two-body terms兲interacting boson model 共IBM兲Hamiltonian. For each isotope chain a general fit is performed in such a way that all parameters but one are kept fixed, to describe the whole chain. In this region, nuclei evolve from spherical to deformed shapes and a method based on catastrophe theory, in combination with a coherent-state analysis to generate the IBM energy surfaces, is used to identify critical phase transition points. DOI: 10.1103/PhysRevC.68.024307 PACS number共s兲: 21.60.Fw, 27.70.⫹q I. INTRODUCTION Recently, a renewed interest in the study of quantum phase transitions in atomic nuclei has emerged 关1–4兴. A new class of symmetries, which applies to systems localized at the critical points has been proposed. In particular, the ‘‘critical symmetry’’ E(5) 关5兴has been suggested to describe critical points in the phase transition from spherical to ␥ -unstable shapes while X(5) 关6兴is designed to describe systems lying at the critical point in the transition from spherical to axially deformed systems. These are based originally on particular solutions of the Bohr-Mottelson differential equations but are usually applied in the context of the interacting boson model 共IBM兲关7兴since the latter provides a simple but detailed framework in which first and second order phase transitions can be studied. In the IBM language, symmetry E(5) corresponds to the critical point between the U(5) and O(6) symmetry limits while the X(5) symmetry should describe the phase transition region between the U(5) and the SU(3) dynamical symmetries, although the connection is not a rigorous one. Very recently, the O(6) limit itself has also been proposed to correspond to a critical point 关8兴. Usually, the IBM analyses of phase transitions have been carried out using schematic Hamiltonians in which the transition from one phase to the other is governed by a single parameter. It is thus necessary to see how much these predictions vary when a more general Hamiltonian is used. The global approach was first used by Castan ˜ os et al. for the study of series of isotopes 关9–12兴. An alternative procedure is provided by the use of the consistent Q formalism 共CQF兲 关13兴. In this case, although the Hamiltonian is simpler than the general one, the main ingredients are included. Within this scheme, a whole isotope chain is described in terms of a few parameters that change smoothly from one isotope to the next. Because of the possible nonuniqueness of such nucleus by nucleus fits and the restricted parameter space, it is important to study under what circumstances the prediction of the location of critical points in a phase transition is robust. In this paper, we follow Refs. 关10–12,14,15兴and use a more general oneand two-body IBM Hamiltonian to obtain the model parameters from a fit to energy levels of chains of isotopes. In this way, a set of fixed parameters, with the exception of one that varies from isotope to isotope, is obtained for each isotope chain and the transition phase can be studied in the general model space. The fit to a large data set in many nuclei diminishes the uncertainties in the parameter determination. A possible problem arising from working with such a general Hamiltonian, however, is the difficulty in determining the position of the critical points. Fortunately, the methods of catastrophe theory 关16兴allow the definition of the essential parameters needed to classify the shape and stability of the energy surface 关14,15兴. In this paper, we analyze diverse spectroscopic properties of several isotope chains in the rare-earth region, in which shape transition from spherical to deformed shapes is observed. We combine this study with a coherent-state analysis and with catastrophe theory, in order to localize the critical *Email address: [email protected] PHYSICAL REVIEW C 68, 024307 共2003兲 0556-2813/2003/68共2兲/024307共11兲/$20.00 ©2003 The American Physical Society68 024307-1 points and test the X(5) predictions. Since the introduction of the E(5) and X(5) symmetries, only a small number of candidates 关17–24兴have been proposed as possible realizations of such critical point symmetries. In this paper, we show that the critical points can be clearly identified by means of a general theoretical approach 关14,15兴. The paper is structured as follows. In Sec. II, we present the IBM Hamiltonian used. In Sec. III, the results of the fits made for the different isotope chains are presented. Comparisons of the theoretical results with the experimental data for excitation energies, E2 transition rates and two-neutron separation energies are shown. In Sec. IV, the intrinsic state formalism is used to generate the energy surfaces produced by the parameters obtained in the preceding section. In addition, the location of the critical point in the shape transition for each isotope chain is identified by using catastrophe theory. Also, in this section the alternative description provided by the CQF for the rare-earth region is briefly discussed. Finally, Sec. V is devoted to summarize and to present our conclusions. II. IBM DESCRIPTION In this work we use the IBM to study in a systematic way the properties of the low-lying nuclear collective states in several even-even isotope chains in the rare-earth region. The building blocks of the model are bosons with angular momentum L⫽0(sbosons兲and L⫽2(dbosons兲. The dynamical algebra of the model is U(6). Therefore, every dynamical operator, such as the Hamiltonian or the transition operators, can be written in terms of the generators of the latter algebra. Usually, some restrictions are imposed on these operators, e.g., the Hamiltonian should be number conserving and rotational invariant, and in most cases it only includes up to two-body terms. The most general 共including up to two-body terms兲IBM Hamiltonian, using the multipolar form, can be written as H ˆ⫽A ˜ N ˆ⫹B ˜ N ˆ共N ˆ⫺1兲 2⫹␧dn ˆd⫹ ␬ 0P ˆ†P ˆ⫹ ␬ 1L ˆ•L ˆ⫹ ␬ 2Q ˆ•Q ˆ ⫹ ␬ 3T ˆ3•T ˆ3⫹ ␬ 4T ˆ4•T ˆ4,共1兲 where N ˆand n ˆdare the total boson number operator and the dboson number operator, respectively, and P ˆ†⫽1 2共d†•d†⫺s†•s†兲,共2兲 L ˆ⫽ 冑 10共d†⫻d ˜ 兲(1),共3兲 Q ˆ⫽共s†⫻d ˜ ⫹d†⫻s ˜ 兲(2)⫺ 冑 7 2共d†⫻d ˜ 兲(2),共4兲 T ˆ3⫽共d†⫻d ˜ 兲(3),共5兲 T ˆ4⫽共d†⫻d ˜ 兲(4).共6兲 Symbol •stands for the scalar product, defined as T ˆL•T ˆL ⫽兺M(⫺1)MT ˆLMT ˆL⫺Mwhere T ˆLM corresponds to the M component of operator T ˆL. Operator ␥ ˜ ᐉm⫽(⫺1)m ␥ ᐉ⫺m 共where ␥ refers to sand dbosons兲is introduced to ensure the correct tensorial character under spatial rotations. The first two terms in the Hamiltonian do not affect the spectra but only the binding energy. Therefore, they can be removed from the Hamiltonian if only the excitation spectrum of the system is of interest. However, a complete description of both excitation and binding energies requires the use of the full Hamiltonian 共1兲. The electromagnetic transitions can also be analyzed in the framework of the IBM. In particular, in this work we will focus on E2 transitions. The most general E2 transition operator, including up to one body terms, can be written as T ˆM E2⫽eeff关共s†⫻d ˜ ⫹d†⫻s ˜ 兲M (2)⫹ ␹ 共d†⫻d ˜ 兲M (2)兴,共7兲 where eeff is the boson effective charge and ␹ is a structure parameter. Two-neutron separation energies (S2n) are also studied in the present work. This observable is defined as the difference in binding energy between an even-even isotope and the preceding even-even one: S2n⫽BE共N兲⫺BE共N⫺1兲,共8兲 where Ncorresponds to the total number of valence bosons. Note that if only the first two terms in Eq. 共1兲are considered and A ˜ and B ˜ are assumed to be constant along the isotope chain, S2nwould be given by S2n⫽⫺ 冉 A ˜ ⫺1 2B ˜ 冊 ⫺B ˜ N⫽A⫹BN.共9兲 For a detailed study of this property, we refer to Ref. 关25兴. III. FITS In this section we analyze several isotope chains belonging to the rare-earth region using the most general IBM Hamiltonian Eq. 共1兲and E2 transition operator Eq. 共7兲.As an ansatz for each chain of isotopes, we will assume a single Hamiltonian and a single E2 transition operator. All parameters in these operators are kept fixed for a given isotope chain, except for the single particle energy which is allowed to vary slightly from isotope to isotope. The way of fixing the best set of parameters in the Hamiltonian is to carry out a least-square fit procedure of the excitation energies of selected states (21 ⫹,4 1 ⫹,6 1 ⫹,8 1 ⫹,0 2 ⫹,2 3 ⫹,4 3 ⫹,2 2 ⫹,3 1 ⫹, and J. E. GARCI ´A-RAMOS, J. M. ARIAS, J. BAREA, AND A. FRANK PHYSICAL REVIEW C 68, 024307 共2003兲 024307-2 42 ⫹) and the two-neutron separation energies of all isotopes in each isotopic chain. Once the parameters in the Hamiltonian are obtained, the B(E2) transition probabilities 21 ⫹ →01 ⫹,4 1 ⫹→21 ⫹,2 2 ⫹→01 ⫹,2 3 ⫹→01 ⫹,0 2 ⫹→21 ⫹, and 03 ⫹ →21 ⫹of the set of isotopes are used to fix eeff and ␹ by carrying out a least-square fit. The experimental data for excitation and binding energies and B(E2)’s have been taken from Refs. 关26–38兴. Finally, it is worth noting that in Ref. 关25兴the Hamiltonian parameters were fixed just using the data for excitation energies and then Aand Bwere adjusted to reproduce the experimental values of S2n. In this paper, since we are particularly interested in accurately describing the spectroscopic data associated with shape transitions, both excitation and binding energies are treated on an equal footing, describing the shape transition to determine the set of Hamiltonian parameters in Eq. 共1兲. Tables I and II summarize the parameters obtained for the Hamiltonian and E2 transition operator for each isotope chain. In Figs. 1–4 the systematics of experimental and calculated energies for the states included in the least-square procedure are presented in order to show the goodness of the fitting procedure. In Figs. 5 –8 the systematics of the experimental and calculated B(E2) values are compared. Finally, in Fig. 9 the experimental and calculated S2nvalues are shown. This is a fundamental magnitude for identifying a phase transition since it is directly related to the derivative of the energy surface. First order phase transitions are related with the appearance of a kink in the S2nvalues. As shown in Fig. 9, the calculation matches the experimentally observed behavior. The analysis of the preceding figures for different observables and for several isotope chains shows that the present procedure is appropriate for systematic studies and confirms that it provides a simple framework to describe long chains of isotopes and detect possible phase transitions. An alternative approach to describe long chains of rareearth nuclei is to use the CQF. The CQF Hamiltonian is H ˆ⫽ ⑀ n ˆd⫹ ␬ Q ˆ⬘ •Q ˆ⬘,共10兲 with Q ˆ⬘⫽共s†⫻d ˜ ⫹d†⫻s ˜ 兲(2)⫹ ␹ 共d†⫻d ˜ 兲(2).共11兲 For each nucleus, parameters ⑀ , ␬ , and ␹ are determined in order to fit the excitation energies and B(E2)’s. In particular, in Ref. 关39兴the parameters of the Hamiltonian are calculated within the CQF framework with the ansatz that the strength of the quadrupole term of the Hamiltonian remains constant along a wide region of the mass table. As in the present paper, they compare experimental data and theoretical values for excitation energies and B(E2) transition rates. Both methods provide a consistent description of the rare-earth region with a similar number of parameters, as can be observed in Fig. 10 and in Table III where the case of 152Sm is analyzed. Note that in the present work the results come from a global analysis, therefore the B(E2) transition rates are not normalized to transition B(E2:21 ⫹→01 ⫹) in a particular isotope. If in Table III the results are normalized so as to reproduce the observed value for B(E2:21 ⫹→01 ⫹)in152Sm, the results of this work and CQF are basically the same. IV. ENERGY SURFACES AND PHASE TRANSITIONS The study of phase transitions in the IBM requires the use of the so called intrinsic-state formalism 关40–42兴although other approaches can be used 关3,43兴. This formalism is very useful to discuss phase transitions in finite systems because it provides a description of the behavior of a macroscopic system up to 1/Neffects. To define the intrinsic or coherent state, it is assumed that the dynamical behavior of the system can be described in terms of independent bosons 共‘‘dressed bosons’’兲moving in an average field 关44兴. The ground state of the system is a condensate 兩 c 典 of bosons occupying the lowest-energy phonon state ⌫c †: TABLE I. Values of ␧din the Hamiltonian 共in keV兲for each isotopic chain as a function of the neutron number. Neutron number Element 84 86 88 90 92 94 96 98 100 60Nd 1686.3 1606.7 1645.4 1602.9 1536.1 1595.9 62Sm 1427.3 1393.5 1289.3 1210.8 1158.6 1192.5 1312.2 1452.0 64Gd 1479.3 1508.7 1409.0 1300.4 1221.5 1174.4 1162.0 1176.5 66Dy 1558.8 1607.6 1562.4 1503.9 1461.0 1427.7 1413.4 1409.2 1443.1 TABLE II. Rest of the parameters in the Hamiltonian and in the E2 transition operator. Isotopes A ˜ 共MeV兲B ˜ 共MeV兲 ␬ 0共keV兲 ␬ 1共keV兲 ␬ 2共keV兲 ␬ 3共keV兲 ␬ 4共keV兲eeff (e•b) ␹ 60 144⫺154Nd 16.75 ⫺0.51 83.753 ⫺13.928 ⫺17.151 ⫺101.27 ⫺187.57 0.119 ⫺1.43 62 146⫺160Sm 18.05 ⫺0.46 53.209 ⫺11.267 ⫺14.674 ⫺31.769 ⫺131.24 0.119 ⫺1.69 64 148⫺162Gd 22.55 ⫺0.76 45.207 ⫺7.932 ⫺13.129 ⫺35.224 ⫺156.24 0.110 ⫺1.77 66 150⫺166Dy 25.06 ⫺0.80 38.651 ⫺6.416 ⫺13.638 ⫺59.165 ⫺163.05 0.103 ⫺1.60 PHASE TRANSITIONS AND CRITICAL POINTS IN . . . PHYSICAL REVIEW C 68, 024307 共2003兲 024307-3 兩 c 典 ⫽1 冑 N!共⌫c †兲N 兩 0 典 ,共12兲 where ⌫c †⫽1 冑 1⫹ ␤ 2 冉 s†⫹ ␤ cos ␥ d0 †⫹1 冑 2 ␤ sin ␥ 共d2 †⫹d⫺2 †兲 冊 共13兲 and ␤ and ␥ are variational parameters related with the shape variables in the geometrical collective model. The expectation value of the Hamiltonian in intrinsic state 共12兲provides the energy surface of the system, E(N, ␤ , ␥ )⫽ 具 c 兩 H ˆ 兩 c 典 . The energy surface in terms of the parameters of Hamiltonian 共1兲 and the shape variables can be readily obtained 关45兴: 具 c 兩 H ˆ 兩 c 典 ⫽N ␤ 2 共1⫹ ␤ 2兲 冉 ␧d⫹6 ␬ 1⫺9 4 ␬ 2⫹7 5 ␬ 3⫹9 5 ␬ 4 冊 ⫹N共N⫺1兲 共1⫹ ␤ 2兲2 冋 ␬ 0 4⫹ ␤ 2 冉 ⫺ ␬ 0 2⫹4 ␬ 2 冊 ⫹2 冑 2 ␤ 3 ␬ 2cos共3 ␥ 兲 ⫹ ␤ 4 冉 ␬ 0 4⫹ ␬ 2 2⫹18 35 ␬ 4 冊 ,共14兲 where the terms which do not depend on ␤ and/or ␥ 关corresponding to A ˜ and B ˜ in Eq. 共1兲兴 have not been included. The equilibrium values of variational parameters ␤ and ␥ are obtained by minimization of ground state energy 具 c 兩 H ˆ 兩 c 典 . As mentioned above, these parameters are related to the parameters of the geometrical collective model and provide an image of the nuclear shape for a given IBM Hamiltonian. A spherical nucleus has a minimum in the energy surface at ␤ ⫽0, while for a deformed one the energy surface has a minimum at a finite value of ␤ and ␥ ⫽0共prolate nucleus兲or ␥ ⫽ ␲ /3 共oblate nucleus兲. Finally, a ␥ -unstable nucleus corresponds to the case in which the energy surface has a minimum at a particular value of ␤ and is independent of the value of ␥ . The equilibrium values of ␤ and ␥ are the order parameters to study the phase transition of the system although in the case under consideration 共IBM1兲, only ␤ has to be taken into account since the minima in ␥ are well defined. 0 1 2 3 60Nd 21 exp 21 theo 41 exp 41 theo 61 exp 61 theo 81 exp 81 theo 0 1 2 E (MeV) 02 exp 02 theo 22 exp 22 theo 42 exp 42 theo 144 146 148 150 152 154 A 0 1 223 exp 23 theo 31 exp 31 theo FIG. 1. Excitation energies of Nd isotopes. 0 1 2 321 exp 21 theo 41 exp 41 theo 61 exp 61 theo 81 exp 81 theo 0 1 2 3 E (MeV) 62Sm 02 exp 02 theo 22 exp 22 theo 42 exp 42 theo 146 148 150 152 154 156 158 160 A 0 1 2 23 exp 23 theo 31 exp 31 theo FIG. 2. Excitation energies of Sm isotopes. J. E. GARCI ´A-RAMOS, J. M. ARIAS, J. BAREA, AND A. FRANK PHYSICAL REVIEW C 68, 024307 共2003兲 024307-4 In Fig. 11 the energy surfaces for the isotopes of the different isotope chains studied in this paper are plotted as a function of ␤ . The figure on the right is a zoom of the region close to ␤ ⫽0. The classification of phase transitions that we follow in this paper and that is followed traditionally in the IBM is the Ehrenfest classification 关46兴. In this context, the origin of a phase transition resides in the way the energy surface 共their minima positions兲is changing as a function of the control parameter that, in this work, is a combination of parameters of the Hamiltonian 关see Eq. 共21兲兴. 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 order parameter ␤ . 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 ␤ ⫽0 and the surface satisfies condition (d2E/d ␤ 2) ␤ ⫽0⫽0. With the introduction of the E(5) and X(5) symmetries to describe phase transitional behavior, diverse attempts to identify nuclei that could be located at the critical points have been made. The theoretical approaches have been mainly performed with restricted IBM Hamiltonians. In particular, within the CQF, or other restricted Hamiltonians, the location of the critical point is obtained by imposing d2E/d ␤ 2⫽0at ␤ ⫽0, where Eis the energy surface 关2兴. This condition leads to a flat surface in a region of small values of ␤ , with a single minimum in limit ␹ ⫽0 and two almost degenerate minima 共one of them in ␤ ⫽0) in the other cases. In the CQF approximation it can be said that (d2E/d ␤ 2) ␤ ⫽0⫽0 corresponds approximately to a ‘‘very flat energy surface,’’ as happens for the E(5) and X(5) critical point models. Following this approach, both 150Nd and 152Sm have been found to be close to critical. However, when studying a transitional region in which the lighter nuclei are spherical and the heavier are well deformed, the a priori restriction of the parameter space could play a crucial role in the identification of a particular isotope as critical. It is thus important to perform a general analysis in order to check whether the predictions obtained within the CQF for those nuclei close to a critical point are robust. We present below such an analysis in the region of the rare earths. We follow closely the approach introduced in Refs. 关14,15兴using catastrophe theory. In the following section the main ingredients of the theory are summarized and the relevant equa0 1 2 364Gd 21 exp 21 theo 41 exp 41 theo 61 exp 61 theo 81 exp 81 theo 0 1 2 E (MeV) 02 exp 02 theo 22 exp 22 theo 148 150 152 154 156 158 160 162 A 0 1 2 23 exp 23 theo 31 exp 31 theo 42 exp 42 theo FIG. 3. Excitation energies of Gd isotopes. 0 1 2 3 66 Dy 2 1 exp 2 1 theo 4 1 exp 4 1 theo 6 1 exp 6 1 theo 8 1 exp 8 1 theo 0 1 2 E (MeV) 0 2 exp 0 2 theo 2 2 exp 150 152 154 156 158 160 162 164 166 A 0 1 2 2 3 exp 2 3 theo 0 1 2 2 2 theo 4 2 exp 4 2 theo 150 152 154 156 158 160 162 164 166 0 1 2 3 1 exp 3 1 theo FIG. 4. Excitation energies of Dy isotopes. PHASE TRANSITIONS AND CRITICAL POINTS IN . . . PHYSICAL REVIEW C 68, 024307 共2003兲 024307-5 tions are particularized for the IBM Hamiltonian written in multipolar form, Eq. 共1兲. A. The separatrix plane For the study of phase transitions in the IBM within the framework of catastrophe theory, we already have the basic ingredients: the Hamiltonian of the system, Eq. 共1兲, and the intrinsic state, Eq. 共12兲. With them, we have generated the corresponding energy surface Eq. 共14兲in terms of the Hamiltonian parameters and the shape variables. It is our purpose to find the values of the parameters of the Hamiltonian that correspond to critical points. In principle, this analysis involves the six parameters of the Hamiltonian but a first simplification occurs since the energy surface only depends on five parameters: 具 c 兩 H ˆ 兩 c 典 ⫽N␧ ˜ ␤ 2 共1⫹ ␤ 2兲⫹N共N⫺1兲 共1⫹ ␤ 2兲2 ⫻ 冉 a1 ␤ 4⫹a2 ␤ 3cos共3 ␥ 兲⫹a3 ␤ 2⫹u0 2 冊 , 共15兲 where ␧ ˜ ⫽␧d⫹6 ␬ 1⫺9 4 ␬ 2⫹7 5 ␬ 3⫹9 5 ␬ 4 a1⫽1 4 ␬ 0⫹1 2 ␬ 2⫹18 35 ␬ 4 a2⫽2 冑 2 ␬ 2 a3⫽⫺ 1 2 ␬ 0⫹4 ␬ 2 u0⫽ ␬ 0 2.共16兲 Fortunately, it is possible to reduce the number of relevant 共or essential兲parameters to just two and study all phase transitions by using catastrophe theory 关16兴. We refer the reader to Refs. 关14,15兴for details of the application of this theory to the IBM case. The idea is to analyze the energy surface and obtain all equilibrium configurations, i.e., to find all the critical points of Eq. 共15兲. First, the critical point of maximum degeneracy has to be identified. In our case, it corresponds to ␤ ⫽0. Next, the bifurcation and Maxwell sets are constructed 关14,16兴. Finally, the separatrix of the IBM is ob0.0 1.0 60Nd 21→01 exp 21→01 theo 41→21 exp 41→21 theo 0.00 0.01 0.02 BE2 (e2b2) 22→01 exp 22→01 theo 23→01 exp 23→01 theo 144 146 148 150 152 154 A 0 0.2 0.4 02→21 exp 02→21 theo 03→21 theo FIG. 5. B(E2) transition rates for Nd isotopes. 0.0 0.5 1.0 1.5 2.0 21→01 exp 21→01 theo 41→21 exp 41→21 theo 0.00 0.01 0.02 0.03 0.04 BE2(e2b2) 22→01 exp 22→01 theo 23→01 theo 146 148 150 152 154 156 158 160 A 0 0.2 0.4 0.6 62Sm 02→21 exp 02→21 theo 03→21 theo FIG. 6. B(E2) transition rates for Sm isotopes. J. E. GARCI ´A-RAMOS, J. M. ARIAS, J. BAREA, AND A. FRANK PHYSICAL REVIEW C 68, 024307 共2003兲 024307-6 tained by the union of Maxwell and bifurcation sets. In general, a bifurcation set, corresponding to minima, limits an area where two minima in the energy surface coexist. A second order phase transition develops when these minima become the same. The crossing of a Maxwell set corresponding to minima leads to a first order phase transition. In order to follow this scheme, one has to identify the catastrophe germ of the IBM, which is the first term in the expansion of the energy surface around the critical point of maximum degeneracy that cannot be canceled by an arbitrary selection of parameters. In our case, one finds that the first derivative in ␤ ⫽0 is always 0 because of the critical character of the point for any value of the parameters. The second and third derivatives can also be canceled with an appropriate selection of parameters. However, if one imposes the cancellation of the fourth derivative, the energy becomes a constant for any value of ␤ . This means that the catastrophe germ is ␤ 4and the number of essential parameters is equal to two, which can be defined, following reference 关14,15兴,as r1⫽a3⫺u0⫹␧ ˜ /共N⫺1兲 2a1⫹␧ ˜ /共N⫺1兲⫺a3 ,共17兲 r2⫽⫺ 2a2 2a1⫹␧ ˜ /共N⫺1兲⫺a3 ,共18兲 where ␧ ˜ ,a1,a2, and a3are defined in Eq. 共16兲. The denominator in both expressions fixes the energy scale, which means that when it becomes negative, the energy surfaces are inverted. The essential parameters r1and r2can also be written in terms of the parameters appearing in Eq. 共1兲as 0.0 0.5 1.0 1.5 21→01 exp 21→01 theo 41→21 exp 41→21 theo 0.00 0.03 0.05 0.08 BE2(e2b2) 22→01 exp 22→01 theo 23→01 exp 23→01 theo 148 150 152 154 156 158 160 162 0 0.5 1 64Gd 02→21 exp 02→21 theo 03→21 theo FIG. 7. B(E2) transition rates for Gd isotopes. 0 1 2 21→01 exp 21→01 theo 41→21 exp 41→21 theo 0.000 0.100 BE2(b2e2) 22→01 exp 22→01 theo 23→01 exp 23→01 theo 150 152 154 156 158 160 162 164 166 A 0.0 0.5 66Dy 02→21 theo 03→21 theo FIG. 8. B(E2) transition rates for Dy isotopes. 146 148 150 152 154 A 12 13 14 15 16 17 18 S2n (MeV) 148 150 152 154 156 158 160 A 150 152 154 156 158 160 162 A 152 154 156 158 160 162 164 166 A exp theo 60Nd 62Sm 64Gd 66Dy FIG. 9. S2nvalues for Nd, Sm, Gd, and Dy isotopes. PHASE TRANSITIONS AND CRITICAL POINTS IN . . . PHYSICAL REVIEW C 68, 024307 共2003兲 024307-7 r1⫽␧ ˜ ⫺共N⫺1兲共 ␬ 0⫺4 ␬ 2兲 ␧ ˜ ⫹共N⫺1兲 冉 ␬ 0⫺3 ␬ 2⫹36 35 ␬ 4 冊 ,共19兲 r2⫽⫺ 4 冑 2 ␬ 2共N⫺1兲 ␧ ˜ ⫹共N⫺1兲 冉 ␬ 0⫺3 ␬ 2⫹36 35 ␬ 4 冊 .共20兲 A property of the parametrization used in this work is that the different chains of isotopes are located on a straight line that crosses the point corresponding to the U(5) limit. The equation of this line is given by r1⫽ 2 ␬ 0⫺7 ␬ 2⫹36 35 ␬ 4 4 冑 2 ␬ 2 r2⫹1. 共21兲 It should be remarked that the derivation of the essential parameters has nothing to do with catastrophe theory. The application of this theory begins once those parameters are obtained. The basic point is to translate every set of Hamiltonian parameters to the plane formed by the essential parameters r1and r2. This plane is divided into several sectors by the bifurcation set that form the geometrical place in the parameter space where d2E/d ␤ 2⫽0 for a critical value of ␤ , and the Maxwell sets, the geometrical place in the space of parameters where two or more critical points are degenerate 关16兴. Both sets form the separatrix of the system; in this case, of the IBM. In Refs. 关14,15兴the IBM bifurcation (r2axis, r2⫽0 and r1⬍0 semi-axis, r11 , and r12) and Maxwell 共negative r1semi-axis, r13 ⫹, and r13 ⫺) sets were obtained. They are all indicated in Fig. 12. In this representation, it is required that the denominator in Eqs. 共17兲and 共18兲be positive. The separatrix for r1⬎0 is associated with minima while for r1⬍0 it is associated with maxima 共except the negative r1semi-axis兲. In order to clarify the figure on the separatrix, the energy surfaces corresponding to each set are plotted as insets. The half plane with r2⬎0 corresponds to prolate nuclei while the one with r2⬍0 corresponds to oblate nuclei. Note that expressions 共19兲and 共20兲are only valid for prolate nuclei but can be readily obtained for the oblate case. TABLE III. Relevant transition rates for 152Sm 共in W.u.兲. Expt. X(5) This work CQFa B(E2:21 ⫹→01 ⫹)144 144 128 144 B(E2:41 ⫹→21 ⫹)209 228 193 216 B(E2:61 ⫹→41 ⫹)245 285 215 242 B(E2:81 ⫹→61 ⫹)285 327 218 248 B(E2:101 ⫹→81 ⫹)320 376 210 242 B(E2:02 ⫹→21 ⫹)33 91 53 57 B(E2:22 ⫹→41 ⫹)19 52 14 20 B(E2:22 ⫹→21 ⫹)613 5 11 B(E2:22 ⫹→01 ⫹)1 3 0 0.1 B(E2:42 ⫹→61 ⫹)440 7 14 B(E2:42 ⫹→41 ⫹)59 2 8 B(E2:42 ⫹→21 ⫹)1 13 0 0.1 aFollowing Ref. 关2兴. 0 1 2 E (MeV) 01 + 21 + 41 + 61 + 81 + 101 + 02 + 22 + 42 + 01 + 21 + 41 + 61 + 81 + 101 + 42 + 22 + 02 + 01 + 21 + 41 + 61 + 81 + 101 + 42 + 22 + 02 + 101 + 81 + 61 + 41 + 21 + 01 + 02 + 22 + 42 + (a) (b) (c) (d) FIG. 10. Spectrum of 152Sm: 共a兲experimental, 共b兲X共5兲symmetry, 共c兲this work, and 共d兲using CQF 关2兴. -2 0 2 4 -4 -2 0 2 4 β -2 -1 0 1 2 β Nd Sm Gd Dy Nd Sm Gd Dy E(β,0) (MeV) E(β,0) (MeV) N=6 78 9 10 11 N=7 9 8 10 11 12 13 14 -2 -5 -5 0 0 0 2 14 13 12 N=8 910 15 11 15 13 16 17 N=9 10 11 12 14 5 5 6 0 0 0 0 1 -1 1 1 1 -1 -1 -1 78 9 10 11 789 10 11 12 1314 8910 11 12 13 14 15 910 11 12 13 14 15 16 17 FIG. 11. Energy surfaces for the different chain of isotopes. -4 -2 0 2 4 r 2 -4 -3 -2 -1 0 1 2 r 1 U(5) SU(3) O(6) SU(3) r 12 r +13 r -13 r 11 FIG. 12. Separatrix plane with a positive energy scale. J. E. GARCI ´A-RAMOS, J. M. ARIAS, J. BAREA, AND A. FRANK PHYSICAL REVIEW C 68, 024307 共2003兲 024307-8 In this figure, the symmetry limits and the correspondence with Casten’s triangle 关7兴are also represented. For completeness, one should consider the case where the denominator of Eqs. 共17兲and 共18兲is negative. It implies that the energy scale becomes negative and the energy surface should be inverted. The separatrix for this case is plotted in Fig. 13 and corresponds to the inversion of Fig. 12. Again, the schematic energy surfaces corresponding to each branch of the separatrix are shown as insets. Note that in this case the symmetry limits do not appear in the figure because they correspond to positive denominators for r1and r2. In our analysis only prolate nuclei are considered because of which a new figure, Fig. 14, is included. In this figure, the right panel corresponds to positive denominators for r1and r2while the left panel shows the case of negative denominator for r1and r2. In the following, we will follow the convention presented in this figure. A set of parameters in the Hamiltonian corresponds to a point in the separatrix plane. The location of the point in that plane provides the required information on its transitional phase character. As mentioned above, it follows that points located on a separatrix line correspond to critical points. Note that the dynamical behavior of the system is controlled by the lowest minimum in the energy surface. In this sense, we are adopting the Maxwell convention in the catastrophe theory language 关16兴and the only relevant branches of the separatrix are r13 ⫹and r2⫽0 with r1⭐0. All these branches correspond to first order phase transitions except for the single point (r1⫽0,r2⫽0) that corresponds to a second order phase transition. The rest of Maxwell lines do not correspond to a phase transition because they are related to maxima. The interest of the bifurcation set, corresponding to minima, arises from the fact that it defines regions where two minima exits. In the following section the transitional isotope chains studied in this paper are analyzed in the separatrix plane. B. Rare-earth region on the separatrix plane The fits presented in Sec. III provide the parameter sets given in Tables I and II for the four isotope chains studied in this paper. In this section, we plot the corresponding sequences of points representing the isotopes in each chain on the separatrix plane. As can be observed in the previous tables, all the parameters for each chain are fixed except the value of ␧dthat changes along the chain. In Fig. 15 the positions of the different isotopes in the chains studied are plotted in the separatrix plane. The interpretation of these lines is given in Fig. 14. As mentioned above, all isotopes in a chain lie on a straight line. The lighter ones are close to the U(5) point 共spherical shapes兲 while as the number of neutrons is increased the corresponding points get increasingly away. For the heavier isotopes of Gd and Dy, the denominator of r1and r2becomes negative, which means that the left panel in Fig. 14 has to be used. The main feature we find is that some nuclei are close to Maxwell set r13 ⫹: the closest are 148Nd 共boson number N -4 -2 0 2 4 r 2 -2 -1 0 1 2 3 4 r 1 r 12 r +13 r -13 r 11 FIG. 13. Separatrix plane with a negative energy scale. -4 -2 0 -4 -3 -2 -1 0 1 2 02 4 r 12 r +13 r -13 r 11 r1 r2 FIG. 14. Separatrix plane for prolate nuclei 共 ␹ ⬍0兲. -10 -5 0 5 10 r2 -4 -2 0 2 4 r1 9 6 Nd Sm Gd Dy 12 14 8 11 13 7 12 11 14 17 13 15 FIG. 15. Representation of isotopes in the separatrix plane 共with ␹ ⬍0兲. The numbers on the isotopes correspond to the number of bosons. PHASE TRANSITIONS AND CRITICAL POINTS IN . . . PHYSICAL REVIEW C 68, 024307 共2003兲 024307-9