Shape phase transition in odd nuclei in a multi- j model: The UB(6) ⊗ UF(12) case

PHYSICAL REVIEW C 75, 064316 (2007) Shape phase transition in odd nuclei in a multijmodel: The UB(6) ⊗UF(12) case C. E. Alonso,1J. M. Arias,1and A. Vitturi1,2 1Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de F´ ısica, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain 2Dipartimento di Fisica Galileo Galilei and INFN, Via Marzolo 8, I-35131 Padova, Italy (Received 2 April 2007; published 22 June 2007) The phase transition in odd nuclei when the underlying even-even core nuclei experience a transition from spherical to deformed γ-unstable shapes is investigated. The odd particle is assumed to be moving in the three single-particle orbitals j=1/2,3/2, and 5/2 . At the critical point in the phase transition, an analytic solution to the corresponding Bohr Hamiltonian, called E(5/12), is worked out. Energy spectra and electromagnetic transitions and moments are presented. The same problem is also attacked in the framework of the interacting boson-fermion model (IBFM). Two different Hamiltonians are used. The first one is constructed ad hoc so as to mimic the situation in the E(5/12) model. The second one leads to the occurrence of the OB(6) ⊗UF(12) symmetry when the boson part approaches the O(6) condition. The entire transition line is studied with this Hamiltonian and, in particular, the critical point. Both IBFM calculations at the critical point are consistent with the E(5/12) results. DOI: 10.1103/PhysRevC.75.064316 PACS number(s): 21.60.Fw, 21.60.Ev I. INTRODUCTION Atomic nuclei have been classified for years in terms of collective deformations that are described by introducing two collective variables [1]. The deformation parameter β measures the axial deviation from sphericity, while the angle variable γcontrols the departure from axial symmetry. Three analytical solutions for the collective Bohr Hamiltonian have been known: the vibrational, rotational, and deformed γunstable limits [1–3]. These situations correspond, respectively, to the spherical, axially deformed, and deformed γindependent ground state shapes. Of course, few nuclei present properties that fit exactly one of these limiting situations, and most of them display a transitional behavior. The occurrence of shape transitional behavior characterizes both even-even and odd nuclei. As the nucleon number changes, the structure of the system changes from one character to another, allowing one to talk about a ground state phase transition (nowadays known as a quantum phase transition). The properties of the nucleus at the vicinity of the critical point vary rapidly, and, consequently, the description of critical nuclei has been thought to be very difficult. Recently, Iachello proposed in a series of papers a new class of symmetries that are formulated in terms of the Bohr Hamiltonian and that can be applied to critical point situations [4–6]. In particular, at the critical point from spherical to γ-unstable shapes, called E(5) [4], at the critical point from spherical to axially deformed shapes, called X(5) [5], and at the critical point from axially deformed shapes to triaxial shapes, called Y(5) [6]. Since the introduction of these limits, many theoretical [7–12] and experimental [13–22] studies have been presented in order to look for nuclei that exhibit the properties of criticality and to classify the corresponding phase transitions. Many studies have extended these original models to more complex situations [23–27]. Because of the further complexity embodied in the description of the coupling of the odd particle to the even core, the study of phase transitions in odd nuclei has only recently been attacked. The first case considered by Iachello [7,28]was the case of a j=3/2 fermion coupled to a boson core that undergoes a transition from spherical to γ-unstable situation. At the critical point, an elegant analytic solution, called E(5/4), was obtained starting from the Bohr Hamiltonian. The possible occurrence of this symmetry in 135Ba has been recently suggested [29]. However, the restriction of the fermion space to a single-jorbit makes difficult the identification of a particular nucleus as critical. A recent work [30] considered the richer case of a collective core coupled to a particle moving in the j=1/2,3/2, and 5/2 single-particle orbits, with the boson part undergoing a transition from sphericity to deformed γ-instability. For the critical point in this situation, a new analytic solution, called E(5/12), has been presented. Since this situation is likely to occur in several mass regions, E(5/12) provides a better chance of finding an example of criticality in odd-even nuclei. In this paper we extend the previous study on E(5/12) so as to provide the grounds for experimental studies looking for critical behavior in odd-even nuclei. Spectrum, electric quadrupole, and magnetic dipole transitions along with quadrupole and magnetic moments are worked out in the model. It is worth mentioning that all these studies have been performed in the framework of the collective model. However, a parallel treatment can be done within the interacting boson model (IBM) [31]. Although this model is formulated from the beginning in second quantized form, it is possible to obtain its geometric picture by using coherent states and shape variables [32–34]. Three dynamical IBM symmetries are known: U(5), SU(3), and O(6) corresponding to geometrical spherical, axially deformed, and γ-unstable deformed shapes, respectively [35]. Transitional classes among these limits have been thoroughly studied since the introduction of the model. Even ground state phase transitions were studied soon after [33,36–38], but these studies have been revitalized recently with the above discussed works on quantum phase transitions 0556-2813/2007/75(6)/064316(15) 064316-1 ©2007 The American Physical Society C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) within the Bohr Hamiltonian framework. Numerical studies for even-even nuclei on the critical point from U(5) to O(6) and from U(5) to SU(3) have been presented, and the analysis of the phase transition characteristics has been the object of many papers [39–61]. A recent review on the topic can be found in Ref. [62]. For the case of odd-even nuclei, the simplest IBM extension is called the interacting boson-fermion model (IBFM) [63]. The IBFM has been used to study the phase transition along the U(5)-O(6) line for the case of a fermion in a single j=3/2 orbital [64,65], which is the situation studied in the E(5/4) model. The choice of the j=3/2 orbital allows one to recover at the γ-soft extreme the OB(6) ⊗SUF(4) boson-fermion symmetry. This is because in this case the full boson-fermion quadrupolequadrupole term in the Hamiltonian can be expressed in terms of the Casimir operators of the general UBF(5), OBF(5), SpinBF(6), and SpinBF(5) algebras. The case of the j=3/2 is not, however, unique. It is known in the literature [63,66] that also in the multi-jcase with j=1/2,3/2,5/2, with a proper choice of the boson-fermion Hamiltonian, one can obtain a transitional Hamiltonian expressed in terms of Casimir operators associated with the OBF(6) algebra. This has allowed one to study within the IBFM a similar situation to that studied in the E(5/12) model. The IBFM study has been shown to be useful to check the E(5/4) [64] and E(5/12) [30] cases. In this paper, we present for j=1/2,3/2,5/2the equivalent E(5/12) results for selected IBFM Hamiltonians. The confrontation of E(5/12) and IBFM at the corresponding critical point will provide useful guidelines for experimental studies. The structure of the paper is as follows. In Sec. II,the E(5/12) model is presented in detail and the wave functions are used to calculate different spectroscopic properties of experimental interest. In Secs. III and IV, two IBFM model Hamiltonians produced to study the critical point within the IBM framework are presented. We will therefore consider within the IBFM the case of the coupling of an odd particle moving in the j=1/2,3/2, and 5/2 orbitals to a boson core undergoing a transition from U(5) to O(6) situations. We follow along the transition the evolution of the spectrum and study in detail energies and transitions in correspondence to the critical point. We will show that the results of IBFM and E(5/12) models give comparable behaviors producing robust results indicating criticality in odd-even nuclei. II. THE BOHR HAMILTONIAN AT THE CRITICAL POINT AND THE E(5/12) MODEL In this section, we study the coupling of an even core to a particle moving in the 1/2, 3/2, and 5/2 orbitals. We show that it can also display analytic solutions within the Bohr collective model, in analogy to the case of the j=3/2 particle, described by Iachello within the E(5/4) critical point symmetry. We will name this case as E(5/12). But first, we would like to present briefly the E(5) case for even-even nuclei, since it will be an important reference for our study. The E(5) critical point situation corresponds to the solution for the Bohr Hamiltonian H=− ¯h2 2B1 β4 ∂ ∂β β4∂ ∂β +1 β2sin 3γ ∂ ∂γ sin 3γ∂ ∂γ −1 4β4 κ Q2 κ sin2γ−2 3πκ+U(β,γ),(2.1) in which the potential energy Uis γindependent and modeled as an infinite square well in β U(β)=0,β<β w, U(β)=∞,β⩾βw. (2.2) This situation is supposed to mimic the critical point from spherical (minimum of the energy surface at β=0) to deformed γunstable (minimum of the energy surface at β= 0 and γindependent). In this situation, analytical solutions are found. Energy levels are directly related to zeros of appropriate Bessel functions [4,7]. In Fig. 1, the spectrum and a few selected E2 transition probabilities are displayed. For notation and more details, we refer to Refs. [4,7]. We must mention that in this situation, because of the symmetry of the problem, states are classified by the totally symmetric irreducible representations of the O(5) and O(3) groups, (τ,0) and L, respectively. In addition, an extra label ξis introduced to mark different solutions for the radial (β) equation. Thus, states are labeled by |ξ,τ,L. The solutions of Eq. (2.1)are ψ(β,γ,θi)=fξ,τ(β)τµLM(γ,θi),(2.3) where the functions were determined in Refs. [3,67]. These functions are eigenstates of any quadrupole Hamiltonian which displays O(5) ⊃O(3) symmetry, i.e., for U(β,γ)=U(β). The f(β) functions depend on the selection of the βpotential. For the infinite square well proposed, the wave functions in β take the simple form fξ,τ(β)=Cξ,τβ−3/2Jτ+3/2xξ,τ βω β,(2.4) 0+ 0+ 2+ 6+4+3+0+ 2+ 4+ 4+ 2+ 2+ 0 1 2.20 3.59 3.03 4.80 6.78 0 1 2 3 0 1 2 τ E τ E 100 167 217 114 75 124 167 103 155 217 87 124 62 E(5) ξ=1 ξ=2 FIG. 1. Energy levels (normalized to the energy of the first excited state) and B(E2) transition probabilities for E(5). 064316-2 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) where τ=0,1,2,... labels the SO(5) group, ξis an index that enumerates the successive zeros of the Bessel functions, xξ,τ is the ξth zero for a given τ,Cξτ is a normalization, and βωis the range of the square potential. The full states in Eq. (2.3) can then be denoted by |ξτµLM,(2.5) where µis a label that distinguishes between repeated L for a given value of τ. Once the wave functions are known, the transition probabilities are related to the reduced matrix element of the relevant electromagnetic operator, B(E/M,λ)=1 2Li+1ξfτfµfLf||T(λ) C||ξiτiµiLi2,(2.6) its calculation includes an integral of products of f(β) functions on the βvariable, plus integrals of products of τµLM(γ,θi) functions in the γvariable and the Euler angles. Details of these integrations can be found in Ref. [7]. Next we go to our case of interest, in which a single particle that can occupy the j=1/2,3/2, and 5/2 orbitals is coupled to an E(5) even-even core. In this situation, one can use the separation of the single-particle angular momentum into a pseudospin and pseudo-orbital angular momentum (0 and 2) and assume no coupling due to the pseudospin. In this situation, an E(5) core is coupled to a two-level system representing the odd particle. The lower level with LF=0(j=1/2) has zero energy, and the upper one with LF=2(j=3/2,5/2 degenerate) has some κenergy. The pseudo-orbital part transforms as the representations [τF,0] of OF(5), with τFbeing either 0 or 1. One can use in this case a model Hamiltonian of the form H=− ¯h2 2B1 β4 ∂ ∂β β4∂ ∂β +1 β2sin 3γ ∂ ∂γ sin 3γ∂ ∂γ −1 4β4 κ Q2 κ sin2γ−2 3πκ+u(β)+kg(β) ×[2 ˆ LB◦ˆ LF]+kg(β)ˆ L2 F, (2.7) with u(β)=0,β<β w, u(β)=∞,β⩾βw, g(β)=¯h2 2Bβ2. (2.8) Note that ˆ LBand ˆ LFare the five-dimensional boson and fermion angular momenta, and that the dot ◦indicates the five-dimensional scalar product. Here and in the following we will use the calligraphic letter ˆ Lto denote the five-dimensional O(5) angular momenta and the italic letter Lto denote the usual three-dimensional O(3) angular momenta. It is worth noting that the Hamiltonian (2.7) is a generalization of the one proposed for the E(5/4) case. Since in E(5/12) there are two OF(5) representations involved, in Eq. (2.7)thelast term has been introduced to include the energy difference between j=1/2 and j=3/2,5/2 single-particle orbitals. The coupling scheme is presented in Fig. 2.Inthelowest part of the figure the even-even degrees of freedom, given by the E(5) model, have to be coupled to the odd-particle degrees of freedom. These last ones are separated into a pseudo-orbital part (two-level system) plus a pseudospin part. First, the core E(5), whose states are classified by (ξ,τB,L B) as explained above, is coupled to the pseudo-orbital part of the odd particle, labeled (τF,L F). This gives rise to a BFS (0) 0 (τΒ) LB(τF) LF ⊗⊗ s (1) 2 (2) 4,2 (0) 0 (1) 2 1/2 τΒ,τF(τ1,τ2) LBF 0,0(0,0) 0 1,0(1,0) 2 2,0(2,0) 4,2 5/2,3/2 5/2,3/2 0,1(1,0) 2 1,1(2,0) 4,2 1,1(1,1) 3,1 1,1(0,0) 0 9/2,7/2,5/2,3/2 1/2 9/2,7/2,5/2,3/2 7/2,5/2,3/2,1/2 1/2 full BF pseudo-orbital BF JFIG. 2. Coupling scheme for E(5/12) model. Lower part: an even-even E(5) core (represented by the lowest three states within the ground state band, ξ=1) is coupled first to a two-level system that represents the pseudo-orbital angular momentum of the odd particle. The resulting total orbital angular momentum for the odd-even system is finally coupled to the spin part of the odd particle. 064316-3 C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) boson-fermion pseudo-orbital coupling presented in the upper left panel. The states are labeled τBcharacterizing the OB(5) group and τFcharacterizing the OF(5) group. These two representations are coupled to a common OBF(5) group whose irreducible representations are labeled (τ1,τ 2). If the coupling is with τF=0,L F=0, all the obtained states are as in the E(5) even-even case. When the coupling is with τF=1, the allowed coupled representations are (τ1,τ 2)=(τB+1,0),(τB,1), and (τB−1,0). Once the OBF(5) representations are known, the usual reduction to angular momentum is performed. In the upper left panel of Fig. 2, such a pseudo-orbital coupling is represented for the ξ=1 states in the E(5) part, the left band comes from the coupling of the E(5) states to the (τF=0,L F=0) state and then, as mentioned above, are as in the E(5) model. The other three bands come from the coupling of the E(5) part to the (τF=1,L F=2) state. This coupling for the τB=0 produces just one state (τ1=1,τ 2=0) at the same energy as the excitation energy of the τF=1 with respect to the τF=0 state for the odd particle. This state is labeled τB,τ F(τ1,τ 2)LBF =0,1(1,0)2 (LBF is the boson-fermion pseudo-orbital angular momentum). The coupling of the boson τB=1 states to the single-particle pseudo-orbital τF=1 state give three possibilities (τ1,τ 2)=(2,0),(1,1),and (0,0); the corresponding reduction rules give LBF =4,2,L BF =3,1, and LBF =0, respectively. This is represented schematically in Fig. 2(upper left panel). Finally, these states are coupled to the pseudospin part and give the spectrum for the full boson-fermion system, presented in the upper right part of Fig. 2. Each line represents degenerate states with total angular momentum given by J, produced by the coupling of the boson-fermion pseudo-orbital angular momentum LBF to the pseudospin s. With this choice of the coupling, the total wave function can be factorized in the form =F(β)[(γ,θi,η orb)⊗χ(ηspin)]JBF ,(2.9) where ηorb and ηspin represent the particle coordinates associated with the pseudo-orbital and pseudospin spaces. The wave function is characterized by good quantum numbers (τB,τ F,τ 1,τ 2,L BF,M L) and is given by τB,τF,(τ1,τ2),LBF ,ML(γ,θi,η orb) = LB,MLB,LF,mLF(τB,0) (τF,0) |(τ1,τ 2) LBLF|LBF  ×LBLF|LBF MLBmLF|MLφτB,LB,MLB(γ,θi) ×XτF,LF,mLF(ηorb),(2.10) where the symbols after the summation are isoscalar factors for the O(5)-O(3) and the O(3)-O(2) reductions. The function X(ηorb) describes the five-dimensional pseudo-orbital part, while χ(ηspin) is the pseudospin wave function, which does not contribute to the energy. The equation for F(β) then acquires the familiar form −¯h2 2B 1 β4 ∂ ∂β β4∂ ∂β +¯h2 2B  β2+u(β)F(β)=EF(β), (2.11) with =τB(τB+3) +k[τ1(τ1+3) +τ2(τ2+1) −τB(τB+3) −τF(τF+3)] +kτF(τF+3).(2.12) The solutions of this equation, as in the case of the E(5/4) model, are given in terms of Bessel functions of order ν= √+9/4, in the form Fξ,{τB,τF,(τ1,τ2),LBF }(β) =cξ,{τB,τF,(τ1,τ2),LBF }β−3/2Jν(xξ,{τB,τF,(τ1,τ2),LBF }β/βw), (2.13) where cξ,{τB,τF,(τ1,τ2),LBF }is a normalization constant, and xξ,{τB,τF,(τ1,τ2),LBF }the ξth zero of Jν(z). The corresponding eigenvalues are given as Eξ,{τB,τF,(τ1,τ2),LBF }=¯h2 2Bxξ,{τB,τF,(τ1,τ2),LBF } βw2 .(2.14) The corresponding spectrum is given in Fig. 3for the choice of the parameters k=−1/4 and k=5/2. The states are labeled according to the quantum numbers ξ,τB,τ F,(τ1,τ 2),L BF. One should remember that each state actually represents a multiplet of degenerate levels, whose angular momenta can be extracted from Table I. The parameters kand kcontrol the order of the bands. In Fig. 4we present the evolution of the bandheads as a function of the kand kparameters. Once the wave functions are known, one can calculate any other spectroscopic observable of interest. In general, the electromagnetic operator for the odd system will have both contributions from the collective core and from the single fermion. We will have to calculate reduced matrix elements f||T(λ||i,(2.15) where are given by Eq. (2.9). Since in this formulation the total angular momentum for the fermion is not a good quantum number, we prefer to perform a simple change of coupling transformation given by |(LB,L F),L BF,s;J = j (−1)LB+LF+1/2+J(2LBF +1)(2j+1) ×   LBLFLBF 1/2Jj   |LB,(LF,s),j;J.(2.16) With this it is easy to use standard angular momentum coupling to calculate both collective, T(λ) C, and single particle, T(λ) F, parts of any electromagnetic operator: 064316-4 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) TABLE I. Reduction (branching rules) from the OBF(5) irreducible representations (τ1,τ 2) (first column) to the OBF(3) irreducible representations LBF (second column). They can be found, for instance, in Ref. [63] (p. 73, Eqs. 3.85 and 3.86). In the last column, the angular momentum content JBF in each OBF(5) representation is presented. It comes from the coupling of LBF to the fermion pseudospin s=1/2. The superscript on a given LBF (or J) means the number of times that LBF (or J) is obtained. OBF(5) irrep OBF(3) irrep Total ang. mom. (τ1,τ 2)LBF J (0,0) 0 1/2 (1,0) 2 5/2,3/2 (2,0) 4,2 9/2,7/2,5/2,3/2 (3,0) 6,4,3,0 13/2,11/2,9/2,(7/2)2,5/2,1/2 (4,0) 8,6,5,4,2 17/2,15/2,13/2,(11/2)2,(9/2)2,7/2,5/2,3/2 (1,1) 3,1 7/2,5/2,3/2,1/2 (2,1) 5,4,3,2,1 11/2,(9/2)2,(7/2)2,(5/2)2,(3/2)2,1/2 (3,1) 7,6,52,4,32,2,1 15/2,(13/2)2,(11/2)3,(9/2)3,(7/2)3,(5/2)3,(3/2)2,1/2 ξ,τB,τ F,(τ1,τ 2),L BF ,1/2; J||T(λ) C||ξ,τ B,τ F,(τ 1,τ 2),L  BF ,1/2; J = LB,LF,L B,L F(τB,0) (τF,0) |(τ1,τ 2) LBLF|LBF (τ B,0) (τ F,0) |(τ 1,τ 2) L BL F|L BF  × j,j (−1)L B+J+j+λ(2LBF +1)(2L BF +1)(2J+1)(2J+1)(2j+1)(2j+1) ×LBLFLBF 1/2Jj L BL FL BF 1/2JjLBJj JL Bλξ,τB,L B||T(λ) C||ξ,τ B,L  BδτF,τ FδLF,L Fδj,j,(2.17) where ξ,τB,L B||T(λ) C||ξ,τ B,L  Bis the collective part already calculated for the even-even system. For the fermion part ξ,τB,τ F,(τ1,τ 2),L BF ,1/2; J||T(λ) F||ξ,τ B,τ F,(τ 1,τ 2),L  BF ,1/2; J = LB,LF,L B,L F(τB,0) (τF,0) |(τ1,τ 2) LBLF|LBF (τ B,0) (τ F,0) |(τ 1,τ 2) L BL F|L BF  × j,j (−1)L B+J+j+λ(2LBF +1)(2L BF +1)(2J+1)(2J+1)(2j+1)(2j+1) ×LBLFLBF 1/2Jj L BL FL BF 1/2JjjJL B Jjλ ×j||T(λ) F||jδξ,ξδτB,τ BδLB,L B,(2.18) where j||T(λ) F||jis a single-particle matrix element that is obtained from j||(a† j1טaj2)(λ)||j=− √2λ+1δj2,jδj1,j .(2.19) With these general expressions, any multipole transition and moment can be evaluated. In particular, we are studying E2 and M1 transitions and moments below. 064316-5 C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) IBFM E(5/12) ξ=1 ξ=1 ξ=2 ξ=2 (τ1 B+1,0) (τ1 B,1) (τ1 B-1,0) 0,0(0,0) 0,0(0,0) 1,0(1,0) 2,0(2,0) 1,0(1,0) 2,0(2,0) 3,0(3,0) 0,1(1,0) 0,1(1,0) 1,1(2,0) 2,1(3,0) 3,1(3,1) 2,1(2,1) 1,1(1,1) 2,1(1,0) 1,1(0,0) κ=−1/4 κ′=5/2 0,0(0,0) 1,0(1,0) 2,0(2,0) 3,0(3,0) 0,0(0,0) 1,0(1,0) 2,0(2,0) 0,1(1,0) 1,1(2,0) 2,1(3,0) 1,1(1,1) 2,1(2,1) 3,1(3,1) 1,1(0,0) 2,1(1,0) 0,1(1,0) 167 100 145 145 63 88 143 161 90 60 81 108 100 100 217 75 124 52 31 24 0.5 0.7 118 179 145 211 111 74 45 25 110 9 11 10 41 12 16 30 8 35 18 29 17 12 0.07 0.2 2 2 FIG. 3. Upper frame: Energy levels in the odd system at the E(5/12) critical point (normalized to the energy of the first excited state). Each (degenerate) state is characterized by the ξ,τB,τ F,(τ1,τ 2) quantum numbers. The values k=−1/4andk= 5/2 were used in the Hamiltonian (2.7). The values of the angular momenta for each degenerate state can be extracted from Table I. Some selected values of B(E2), corresponding to the transitions between the highest spins of initial and final multiplets, are given in the figure [the B(E2) associated with the lowest 5/2-1/2 transition is normalized to 100]. Lower frame: Energy levels in the odd system within the IBFM Hamiltonian (3.6) at the critical point. The number of bosons N=7 has been assumed. The values of kand kare the same as in the E(5/12) Hamiltonian. For B(E2), see the upper frame. A. E2 transition and moments Let us first consider the electric quadrupole operator. This can be written as the sum of the collective contribution plus -2 -1 0 1 2 κ -2 -1 0 1 2 3 4 5 Energy (arbitrary units) -1 0 1 2 3 4 κ ′ κ ′=5/2 κ =−1/4 0,1 (1,0) 0,0 (0,0) 0,0 (0,0) 0,0 (0,0) 0,0 (0,0) 1,1 (1,1) 1,1 (0,0) 1,0 (1,0) 1,0 (1,0) 0,1 (1,0) 1,1 (0,0) 1,1 (1,1) FIG. 4. Behavior of the few lowest energy E(5/12) bandheads as a function of parameters kand kin the Hamiltonian. the odd-particle one: T(E2) =T(E2) C+T(E2) F.(2.20) For the collective part, usually a quadrupole operator depending linearly on βis used in the collective models T(E2) Cµ=t(2) Cα2,µ =t(2) CβD(2) µ0(θi) cos γ +1 √2D(2) µ2(θi)+D(2) µ−2(θi)sin γ,(2.21) where t(2) Cis a scale factor. However, the importance of the next term quadratic in βhas been shown. Because of that we will include the next order in the quadrupole coordinates, which is given by [67] T(E2) Cµ=t(2) Cα2µ+χ 2√7[α2×α2](2) µ =t(2) C m D(2)∗ µm ¯ Q(2) m,(2.22) where ¯ Q(2) 0=βcos γ−1 14χβ2cos (2γ), ¯ Q(2) 2=Q(2) −2=1 √2βsin γ+1 14χβ2sin (2γ), ¯ Q(2) 1=¯ Q(2) −1=0. (2.23) The 2√7 denominator in Eq. (2.22) is introduced here for convenience to match the expression based on the interacting boson model (IBM) quadrupole operator. The χparameter controls the weight of the quadratic relative to the linear term. The α2µvariable behaves as the O(5) ⊃O(3) tensor (τ,L)= (1,2), while the second-order term [α×α](2) µis associated with the tensor (2,2) [67]. This means that they fulfill strict selection rules in the φτµLM(γ,θi) collective space: τB=±1 for the former and τB=0,±2 for the latter, with the usual restrictions in angular momentum coupling [31,67]. Thus the first term leads to “allowed” transitions in first order, and the second to nonzero quadrupole moments and to small values for “forbidden” ones, in the E(5) scheme of Ref. [4]. For the single-particle part, in general, T(E2) F= j,j t(2) j,j(a† jטaj)(2),(2.24) where t(2) j,jare parameters. To reduce the number of parameters, the following form dictated by the U(6/12) symmetry has been used: T(E2) F=t(2) F−4 5(a† 1/2טa3/2+h.c.)(2) −6 5(a† 1/2טa5/2+h.c.)(2).(2.25) 064316-6 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) 1 1 1 1 5 5 5 106 5 3 33 3 53 3 7 9 977 167 17 151 134 33 50/117 100 100 118 31 15 24 52/35 0.50.5 3412 95 34/6/13 35/83 25/99 75 75 B(E2) 74/50 86/37 5 49/74 111/12 ∆τ = ±1 B E(5/12) FIG. 5. B(E2) strengths induced by the linear term of the collective E2 operator for the E(5/12) model. Spins given are twice the actual value. Transitions obey the selection rule τB=±1. The transition B(E2; 5/21→1/21) is normalized to 100. To get absolute values, all the strengths have to be multiplied by (0.07453)(t(2) C)2β2 w. For the odd system, contributions from both collective and single-particle operators are to be calculated. In the upper panel of Fig. 3,someB(E2) values are presented for the collective contribution and transitions between the largest spin in each degenerate multiplet. In Figs. 5and 6,B(E2) values for τB=±1 and τB=0,±2 are presented. In the first case, only the linear term is effective; while in the second case, only the quadratic contribution plays a role. 1 1 1 1 5 5 55 5 3 33 3 53 3 7 9 977 56 12 14 122 110 100 61/15 23/52 73 E(5/12) B(E2) ∆τB = 0, ±2 0 23 56 50 118 45 26 3 116 90 254/23/177/20 21/257/39/158 355/236 0/160/45 60/91 449/300 310 310 FIG. 6. B(E2) strengths induced by the quadratic term of the collective E2 operator for the E(5/12) model. Spins given are twice the actual value. Transitions obey the selection rule τB=0,±2. The transition B(E2; 5/21→3/21) is normalized to 100. To get absolute values, all the strengths have to be multiplied by (0.0007855)χ2(t(2) C)2β4 w. 1 1 1 1 1 5 55 5 0.7 33 3 5 3 7 9 977 53 1.73 1.58 -0.37 -0.26 0.86 0.60 00 1.11 1.01 -0.24 -0.17 0.29 0.25 -0.41 E(5/12) collective Q 3 FIG. 7. Quadrupole moments induced by the collective quadratic term in T(E2) for the E(5/12) model. Spins given are twice the actual value. The quadrupole moment of the state 5/21is normalized to 1. To get absolute values, all the quadrupole moments have to be multiplied by (0.12229)χt(2) Cβ2 w. Collective quadrupole moments induced by the quadratic term in the E2 operator can be calculated as eQ(ξ,τB,τ F,(τ1,τ 2),L BF,1/2; J) =16π 5J(2J−1) (J+1)(2J+1)(2J+3) ×ξ,τB,τ F,(τ1,τ 2),L BF,1/2; J||T(E2) C+T(E2) F|| ×ξ,τB,τ F,(τ1,τ 2),L BF,1/2; J.(2.26) In Fig. 7, the collective quadrupole moments induced by the quadratic term in T(E2) Care presented normalized to the quadrupole moment of the 5/21state. The fermionic quadrupole operator (2.25) also contributes to some transitions. However, the structure (2.25) provides a selection rule τF=±1, and consequently there are no contributions of the fermion part to the quadrupole moments. For the same reason, there are no transitions where interference between the collective and the fermionic parts exists (for the collective part, τF=0). In addition, the fermionic E2 operator provides a selection rule τB=0. In Fig. 8,afew fermionic E2 transitions induced by the operator (2.25)are presented. B. M1 transition and moments The magnetic dipole operator can be written as the sum of the collective contribution plus the odd-particle one: T(M1) =T(M1) C+T(M1) F.(2.27) The collective contribution is T(M1) C=t(1) C1ˆ LB √10 +t(1) C2α2׈ LB √10(1) +.... (2.28) 064316-7 C. E. ALONSO, J. M. ARIAS, AND A. VITTURI PHYSICAL REVIEW C 75, 064316 (2007) 1 1 1 1 5 5 5 5 E(5/12) 3 fermionic B(E2) 3 3 0.1/0.9 ∆ τF = ± 1 3 97 7 1 1 0.8/0.2 0.3/0.7 0.2/0.8 0.9/01 0.7/0.3 0.4/0.6 0.6/0.4 FIG. 8. B(E2) strengths induced by the fermionic operator (2.25) in T(E2) for the E(5/12) model. Spins given are twice the actual value. The strength from the state [ξ=1,τ B=0,τ F=1,(τ1=1,τ 2= 0)LBF =2,J =5/2] to the ground state is 1. To get absolute values, all strengths have to be multiplied by (t(2) F)2. The first term is proportional to the collective angular momentum ˆ LBand is diagonal for the even-even core but not for the odd system. The transition strengths and dipole moments arising from this term are presented in Figs. 9and 10. The fermionic term is proportional to the fermion angular momentum j. For the case we are studying, the appropriate operator is T(M1) F=−t(1) F1 2(a† 1/2טa1/2)(1) +√5(a† 3/2טa3/2)(1) +35 2(a† 5/2טa5/2)(1).(2.29) The fermionic magnetic dipole operator (2.29) also contributes to some transitions with the selection rule τB=0. Since 1 1 1 1 5 5 55 5 3 33 3 53 3 7 9 97 7 111 0 100 0 0 100 5 28 100 E(5/12) B(M1) ∆τB = 0, ±1 0 25 27 21 875 321/29/400 1000/500 788/88 313/9/429 53 100* 7* 96*/11* FIG. 9. B(M1) transition strengths induced by the collective M1 operator (2.28) for the E(5/12) model. Spins given are twice the actual value. Both τB=0andτB=±1 transitions are included. The τB=0 transitions are normalized to the B(M1; 5/21→3/21)= 100. In order to get absolute values B(M1) with τB=0haveto be multiplied by (0.04)(t(1) C1)2.TheτB=±1 are marked with an ∗ and are normalized to the B(M1; 5/22g→5/21)=100 (5/22gis the second 5/2 within the ground state band). To get absolute values, B(M1) with τB=±1 have to be multiplied by (0.02445)(t(1) C2)2. this selection rule is identical for this part and the collective part of the dipole operator, there will be interference between both terms. Table II presents combined strengths for selected transitions in which interference between collective and singleparticle terms are important. Figures 3–10 and Table II give detailed spectroscopic information for the E(5/12) model. In the next two sections, calculations at the critical point in the transition from spherical TABLE II. T(M1) reduced matrix elements for B(M1) and magnetic moments involving interference effects between collective and single-particle contributions for some selected E(5/12) states. Initial |iFinal |fReduced matrix element |ξ,τB,τ F(τ1,τ 2)LBF,J|ξ,τB,τ F(τ1,τ 2)LBF,Jf||T(M1)||i 1,1,0 (1,0)2,5/2 1,1,0 (1,0)2,3/2 23 5t(1) F−0.4899t(1) C1 1,1,0 (1,0)2,5/2 1,1,0 (1,0)2,5/2 21 10 t(1) F+1.833t(1) C1 1,1,0 (1,0)2,3/2 1,1,0 (1,0)2,3/2 −3 5t(1) F+1.4697t(1) C1 1,2,0 (2,0)4,9/2 1,2,0 (2,0)4,7/2 2√10 3t(1) F−0.6667t(1) C1 1,2,0 (2,0)2,5/2 1,2,0 (2,0)2,3/2 23 5t(1) F−0.4899t(1) C1 1,2,0 (2,0)4,9/2 1,2,0 (2,0)4,9/2 55 18 t(1) F+4.4222t(1) C1 1,2,0 (2,0)4,7/2 1,2,0 (2,0)4,7/2 −√14 3t(1) F+3.9441t(1) C1 1,2,0 (2,0)2,5/2 1,2,0 (2,0)2,5/2 21 10 t(1) F+1.8330t(1) C1 1,2,0 (2,0)2,3/2 1,2,0 (2,0)2,3/2 −3 5t(1) F+1.4697t(1) C1 064316-8 SHAPE PHASE TRANSITION IN ODD NUCLEI IN A . . . PHYSICAL REVIEW C 75, 064316 (2007) 1 1 1 1 1 5 55 5 0.9 33 3 5 3 7 9 977 53 2 1.94 1 0.9 10.9 00 1 0.97 0.5 0.45 0.75 0.71 0.25 E(5/12) collective µ 3 3 0.17 0 FIG. 10. Magnetic moments induced by the leading collective term in T(M1) for the E(5/12) model. Spins given are twice the actual value. The magnetic moment of the state 5/21is normalized to 1. To get absolute values, all the magnetic moments have to be multiplied by (1.29442)t(1) C1. to deformed γ-unstable shapes within the IBFM framework are presented. Two different IBFM Hamiltonians for describing the critical point are used in order to see whether the signs of criticality produced by E(5/12) are robust. Before ending this section, we would like to note that our E(5/12) model yields an infinite number of levels, it corresponds to solving the Bohr equation for a γ-independent infinite square well in βpotential. This will not be the case for the IBFM calculations, which are associated with a finite number of bosons and consequently characterized by a finite number of levels. In this latter case, in addition, the spectrum itself will change according to the chosen number of bosons, in particular, its higher part. The comparison between the two approaches will therefore necessarily be significant only for the lower part of the spectrum as in the previous E(5/4) model for odd-even nuclei, in the E(5), X(5) models for even-even nuclei concerning critical points, or, in general, when comparing the collective model with the IBM. III. AN INTERACTING BOSON-FERMION MODEL HAMILTONIAN THAT MIMICS E(5/12) AT THE CRITICAL POINT For the odd-even system, the IBFM Hamiltonian is written as H=HB+HF+VBF,(3.1) where HBis the bosonic part, HFthe fermionic one, and the term VBF couples the boson and fermion degrees of freedom. For the bosonic part, we will use a parametrized Hamiltonian that produces a transition between spherical and γ-unstable shapes of the form HB=xˆnd−1−x Nˆ QB·ˆ QB,(3.2) where the operators in the Hamiltonian are given by ˆnd= µ d† µdµ,(3.3) ˆ QB=(s†×˜ d+d†×˜s)(2),(3.4) and Nis the total number of bosons. This Hamiltonian can be recast into the form HB=xC1(UB(5)) −1−x 2NC2(OB(6)) −C2(OB(5)), (3.5) where C1(UB(5)) is the linear Casimir operator of the UB(5) group, while C2(OB(6)) and C2(OB(5)) are the quadratic Casimir operators of the OB(6) and OB(5) groups. We recall that this Hamiltonian produces, when varying the parameter xfrom 1 to 0, a phase transition between the two extreme situations characteristic of U(5) and O(6) symmetries, at xc=4N−8 5N−8. Note that for any value of xthis Hamiltonian maintains the degeneracies typical of the O(5) symmetry. Consistent with this, within the IBM coherent state formalism [32–34], this Hamiltonian always produces an energy surface which is independent of the γdegree of freedom. In the β variable, the energy surfaces display a spherical minimum in β=0forxlarger than the critical value, while having a deformed minimum for values of xsmaller than the critical values [4,41,58]. It is worth mentioning that when considering the combined boson-fermion system, the energy surface has contributions from both bosons and fermions. The contribution from the fermion is of the order 1/N compared with the one from the bosons. Thus, usually it is accepted that the energy surface for the combined system is governed by the bosons with slight contributions from the fermionic degrees of freedom. However, close to the critical point, the boson energy surface is rather flat, and the contribution from the odd fermion and its coupling to the boson core will play a fundamental role in determining the minimum in the potential energy surface. This is an important topic that remains to be investigated in detail. For the purposes of the present paper, i.e., checking the analytical results obtained for the E(5/12) model, we will accept that the critical point of the combined system is just the critical point of the boson part and the fermion is coupled to it without changing the equilibrium configuration. Depending on the selection of HF+VBF in Eq. (3.1), different model Hamiltonians to study the critical point are produced. In this section HF+VBF is selected so as to have a IBFM Hamiltonian as close as possible to the E(5/12) one. Thus, our proposal is HF+VBF =k 2N[C2(OBF(5)) −C2(OB(5)) −C2(OF(5))] +k 2NC2(OF(5)),(3.6) where C2(OF(5)),C 2(OBF(5)) are the quadratic Casimir operators of the OF(5) and OBF(5) algebras, respectively. As mentioned above this Hamiltonian is designed to mimic as much as possible the corresponding Hamiltonian in E(5/12), since the terms 2 ˆ LB◦ˆ LFand ˆ L2 Fin Eq. (2.7) can be written in 064316-9