Nuclear Shape Transition Between Spherical U ( 5 ) and γ-Unstable O ( 6 ) Limits of the Interacting Boson Model

The interacting boson model (IBM) with intrinsic coherent state (characterized by and ) is used to describe the nuclear second order shape phase transition (denoted E(5)) between the spherical oscillator U(5) and the -soft rotor O(6) structural limits. The potential energy surfaces (PES's) have been derived and the critical points of the phase transition have been determined . The model is examined for the spectra of even-even neutron rich xenon isotopic chain. The best adopted parameters in the IBM Hamiltonian for each nucleus have been adjusted to reproduce as closely as possible the experimental selected numbers of excitation energies of the yrast band,  by using computer simulated search program.Using the best fitted parameters , the  energy ratios for the  levels are calculated and compared to those of the O(6) and U(5) dynamical symmetry limits. 122 Xe and 132 Xe are considered as examples for the two O(6) and U(5) dynamical symmetry limits


1.Introduction
The interacting boson model (IBM) [1] was widely used for describing the quadrupole collective states of the medium and heavy nuclei .In the original version of sd-IBM-1,the model includes s-and d-bosons and no distinction is made between proton and neutron bosons.The algebraic structure of this model is based upon U(6), and three dynamical symmetries arise, involving the subalgebras U(5), SU (3) and O(6) corresponding to spherical oscillator, axially symmetric rotor and gamma soft rotor.These three symmetry limits form a Casten triangle [2], that represents the nuclear phase diagram [3].The intermediate between these three limits are of great interest.It was shown by Iachello [4][5][6], that new dynamical symmetries, called E(5),X (5) and Y (5) hold , respectively at the critical point between spherical shape and -unstable deformation, between spherical and axially symmetric shape and between deformed axial and triaxial shape.In all these cases, critical points are defined in the context of the collective geometric BohrHamiltonian [7].Soon, thereafter the introduction of this concept of critical point symmetries different nuclear shapes and phase transitions between them are studied [8][9][10][11][12][13][14][15] .
Even mass Xenon nuclei54Xe were received much attentions.The Xe nuclei with the mass number A ~ 120 -130 was studied experimentally [23,24] and interpreted theoretically by the general Bohr Hamiltonian(GBH) [25] and by acquiring special solutions in the E(5) and X(5) critical limits by using Davison potential Bohr Hamiltonian [26,27].Also Montica et al [28] carried out the IBM2 calculations for the Xe isotopic chain.
The purpose of this paper is to analyze the potential energy surfaces (PES's) to investigate the evolution of nuclear shape transition in Xenon nulei from deformed -soft O(6) to spherical vibrator U(5) in framework of the sd-IBM-1 with intrinsic coherent state formalism [29,30].
The outline of the paper is as follows: In section 2 wedescribe the formalism of the IBM 1 Hamiltonian under study, its intrinsic coherent state and the PES's.Comment on equilibrium deformation and critical points are considered in section 3. The U(5)-O(6) shape transition with more than Hamiltonian form is produced in section 4.Section 5 presents numerical calculations and discussion for Xe isotopic chain.Finally a conclusion and some remarks on our study are given in section 6.
In terms of the parameters  and , the expectation value of the Hamiltonian H is easily obtained from the evolution of the expectation values of each single term given by : with these values the expectation value of the Hamiltonian for  = 0 can be written in the form: where with

3.Equilibrium Deformation and Critical Points
Minimization of the energy with respect to  for given values of the parameters gives the equilibrium value  0 defining the phase of the system, 0 = 0corresponds to the symmetric phase, and  0 ≠ 0 to the broken symmetry phase.To determine the critical values of the order parameters of any nucleus, one needs to determine the locus of points for which conditions 2 = 0aresatisfied.The minima of E as a function of  can be estimated by equating the first order derivative to zero, and the location of the critical point is obtaind when E becomes flat at  = 0 or equating the second order derivative at  = 0 to zero.This yield to .
Therefore the equilibriumvalue is Therefore the relation betweenthe parameters to give the critical point is  =  − 1  0 −  − (4 − 8).Thus the most general PES of the critical point in the U(5)-O(6)phase transition term is: The analysis of the two dynamical symmetry limits of the IBM provides a good test to the PES's presented in the above formalism.
(i) For the U( 5) limit ( 0 = 0,  2 = 0), the equilibrium shape of the nucleus is always spherical, this yields  ,  =  + That is The energy functional is γ independent and has a minimum at  = 0.

The U(5)-O(6)Shape transition
The transition between the spherical and -unstable shapes can be studied by considering five cases.
Case 1: (  0 =  1 =  3 = 0) The critical point appear when  2 = 0, yielding   = − 2 (4 − 8) Introducing the control parameter  1 ,such that Then the critical point is located at   = 4−8 5−8 For large N limit If we eliminate the contribution of the one body terms, the coefficients  2 and  4 becomes The critical point appearswhen  2 = 0, yielding   =  0 ( − 1) Introducing the control parameter  ′ such that Then the critical point is located at In this case the PES has a flat behavior (~ 4 )for small ,an inflection point is at  = 1 and approaches a constant for large .The global minimum at  = 0 is not well localized and the PES exhibits considerable instability in  resembling a squarewell potential for 0 ≤  ≤ 1.

Application to Xenon Isotopic chain
The xenon isotopic chain along the mass region~120 − 130 represent excellent example for studying O(6)-U(5) shape phase transition which give a good test of the proposed nuclear IBM .We will study the second order shape phase transition between the U( 5 where N is the number of the experimental fitting points and ∆  . are the experimental errors.The adopted best parameters are listed in Table [1].These model parameters give a satisfactory description of the experimental data [31]and theoretical calculations [32]In Figure (7), the corresponding PES's plotted for this isotopicchain of nuclei which evolve from -unstable nuclei to spherical vibrator when moving from the lighter 122 Xe to heavier 132 Xe isotopes.To get the characteristic of collectivity in our Xe isotopic chain, the behavior of the energyratios  /2 = (   )/(2 1 + ) have been calculated for 122 Xe and 132 Xe and compared to those of the predicted dynamical symmetries for the O(6) and U(5) limits of the IBM , Specially the ratio  4/2 is a good criterion for the shape transition.The value  4/2 has a limiting value 2 for quadrupole vibrator and 2.5 for a non-axial  soft rotor.As it is seen from the Figure ( 8)  4/2 decreasing gradually from 2.5 for 122 Xe to 2.25 for 130 Xe.This means that this structure varies from -soft rotor along harmonic vibrator.So the energy spectrum of the 122-132 Xe nuclei can be studied between the rotational and vibrational limits.

Conclusion
The paper is focused on the U(5)-O(6)second order shape phase transition of the IBM.The considered Hamiltonian has been written in multipole form and studied in some different cases in order to produce shape phase transition.The PES's have been calculated as expectation values of the Hamiltonian operator within the intrinsic coherent states.The PES's in each case are investigated and analyzed and the equilibrium deformation and critical points are determined.The values of the PES's for doubly even Xe isotopic chain are studied systematically.The model parameters are adjusted by fitting the excitation yrast energies with the calculated ones by performing a computer search program in order to minimize the rootmean-square(rms) deviation between the experimental excitation energies and calculated ones.The phase diagram for Xe nuclei exhibits second order shape phase transition from spherical U(5) to -unstable O(6) when moving from heavier isotope 132 Xe(with boson number =4)to lighter ones 122 Xe (with boson number N=9)

Figure( 2 )
Figure(2) Calculated PES's as a function deformation parameter  U(5)-O(6) shape transition for case one at three values of control parameter  =   ,   ,   the critical point is at   =   .

Figure( 3 )Figure( 4 )
Figure(3)the same as in Figure(2) but for the case 2 at the control parameter  ′ = ., . and . .The critical point is at  ′ = . .The PES is illustrated in Figure(3) for three values of  ′ The critical points connecting U(5) and O(6)in the above two cases are however different and can be viewed as two different lines in nuclear shape phase diagram, with the respective critical point lying in two different points in Casten triangle.In both cases the PES's display a spherical minimum in  = 0 for larger than the critical value   , while having a deformed minimum for values of  smaller than the critical value.At the critical point, the PES in both cases occurs in leading order a  4 behavior, but differs for the higher order terms.Case 3:Modified O(6) to produce transition Putting  =  2 = 0 in the original Hamiltonian and adding the term ( + 4), then the parameters of the PES's become  2 = −4  − 1  0 +    4 =  (30)  0 =  0  − 1 +   + 4  In Figure (4),we show the PES's corresponding to modified O(6) limit, with chosen parameters to produce a shape transition at N=7.The parameters are  = 580  ,  0 = 12   = 65  and  = 4,7,13.Case 4: modified U(5) to produce transition Putting  0 =  2 = 0 in the original Hamiltonian and adding the term ( 4  4 . 4 −    )where  4 is the hexadecapole oprator  4 = [ † ×  ] (4)

Figure( 6 )
Figure(6) The same as in Figure(5) but when   =  with the parameters   = (. − .),   = (. + ), and   = .( − ) ) and O(6) by analyzing the PES's of the Xenon isotopic chain122-132 Xe.The structure parameter of the quadrupole operator is taken to be zero .For each nucleus parameters of the PES's A2 and A4 which are linear combination of the original parameters of the Hamiltonian have been adjusted by fitting the experimental excitation energies of the yrast band to the calculated ones using a computer search program to minimize  such that

Figure( 8 )
Figure(8)The evolution of yrast energy ratios  / = ()/() as a function of angular momentum I for the nuclei 122 Xe and 130 Xe and comparison with the prediction of U(5) and O(6) dynamical symmetry limits.