COMPARISON OF IBM-2 CALCULATIONS WITH X ( 5 ) CRITICAL POINT SYMMETRY FOR LOW-LYING STATES IN 144-154 Nd

The X(5) would take place when moving continously from the pure U(5) symmetry to the SU(3) symmetry and it implies a definite relations among the level energies and among the E2 transition strengths. It was recently shown that a signature of phase transition is observed in the chain of Sm, Mo and Nd isotopes, where Sm,Mo and Nd display the predicted features of the X(5) symmetry and mark therefore the critical point. However, more detailed studies and experiments are needed to get ideas about this signature. Without entering into detail we have firstly compared the results obtained in our previous study [15] of Nd with that of the limits in X(5) symmetry and then given a clear descripton about the validity of the Hamiltonian parameters used in the study. At the end, we have concluded that some of Nd isotopes display X(5) symmetry features. Key Wordscritical symmetry, interacting boson model, even Nd.

potential for the quadrupole deformation parameter β and a harmonic potential well for the triaxiality deformation parameter γ [8]. The signature of a phase transition between collective vibrator and axially deformed rotor has received considerable attention, in the frame of critical point properties in transitional nuclei [9]. The X(5) would take place when moving continously from the pure U(5) symmetry to the SU(3) symmetry and it implies a definite relations among the level energies and among the E2 transition strengths.
It was recently shown that a signature of phase transition is observed in the chain of Sm [7,10], Mo [11] and Nd [12][13][14] isotopes, where 152 Sm, 104 Mo and 150 Nd display the predicted features of the X(5) symmetry and mark therefore the critical point. However, more detailed studies and experiments are needed to get ideas about this signature. Without entering into detail, which can be found in the Ref. [7], we can give purpose of the present study as follows; (i) To compare the results obtained in Ref. [15] with the limits of X(5) symmetry, (ii) To give a clear descripton about the validity of the Hamiltonian parameters used in this study, (iii)To get a brief conclusion about the relation between 144-154 Nd and X(5) symmetry.
The outline of the remaining part of this paper is as follows; An approximate IBM-2 formulation is given without entering into detail and theoretical background is reviewed in section 2. The calculated R 4/2 , R 0/2 and B(E2) values are compared with the results of some neighboring nuclei and with that of X(5) limits in section 3. The last section contains some concluding remarks.

THEORETICAL BACKGROUND
IBM Hamiltonian takes different forms [16] depending on the regions (SU(5), SU(3), SO(6)) of the traditional IBA triangle. The Hamiltonian that we consider is in the form of [17], where H sd is the standard Hamiltonian of the IBM [18,19], In the IBA-2 model the neutrons' and protons' degrees of freedom are taken into account explicitly. Thus the Hamiltonian [2] can be written as , where n dρ is the neutron (proton) d-boson number operator.
where s + ρ , + d ρm and s ρ , d ρm represent the s and d-boson creation and annihilation operators. The rest of the operators in the equation (3) are defined as and M πv ; .
In this case M πv affects only the position of the non-fully symmetric states relative to the symmetric ones. For this reason M πv is often referred to as the Majorana force.
The electric quadropole (E2) transitions are one of the important factors within the collective nuclear structure. In IBM-2 model, the general linear E2 operator is expressed as [2], In these expression χ ρ is an adimansional coefficient and e ρ is the effective quadrupole charges. Below we show how B(E2;J→J') prescription is implemented in formulation.
The experimental signatures for X(5) behavior are the following [8]. show a characteristic pattern. We will use all of the above points in our search for Nd nuclei of A≤140 displaying behavior similar to the X(5) predictions.
The calculated and experimental energy values are given in table 2 with R 4/2 and R 0/2 ratios. Fig.1 shows the R 4/2 and R 0/2 ratios as a function of neutron number changing from 84 to 94. An harmonic vibrator should have R 4/2 = 2.00, an axially symmetric rotor has R 4/2 =3.33, while X(5) behavior should have R 4/2 =2.91. The serached nuclei in the present study have 2.00 ≤ R 4/2 ≤ 2.96. As it is seen from the table 2 the calculated and experimental energy values are very close to X(5) predictions for 150 Nd, especially. Around N=90, the positions of the excited 0 + states are also close to the X(5) prediction and we note that the spacings in the excited sequence follow the expected behavior.
In table 3, we present the calculated data with available experimental ones for 144-154 Nd. In addition, fig.2 shows the energies of the yrast sequences (normalized to the energy of In fig.3 we present the B(E2;J→J-2) reduced transition strength which is normalized to their respective B(E2; 2 + 1 →0 + 1 ) values and again compare them with the expected behavior for an harmonic vibrator, an axially deformed rotor, and the X(5) prediction. It is clear from table 2 and fig.2 that the Nd nuclei with yrast energies that closely follow the X(5) prediction. However, as can be seen from fig.3, in most of the cases X(5) behavior can be excluded on the basis of the deduced yrast B(E2;J→J-2) values.

3.CONCLUSION
We have searched the validity of our new parameters in IBM-2 formulation and theoretical background is reviewed for 144-154 Nd . The calculated R 4/2 , R 0/2 and B(E2) values are compared with the results of some neighboring nuclei and with that of X(5) limits. At the end, it was seen that some Nd nuclei, especially nuclei around N=90, with yrast energies follow the X(5) prediction closely. But X(5) behavior can be excluded on the basis of the deduced yrast B(E2;J→J-2) values in most of the cases. On the basis of the yrast state energies and yrast intraband transition strenghts, the best candidates were performed to be 148 Nd, 150 Nd and 152 Nd nuclei and the X(5) Picture reproduces the position of the first excited 0 + 2 in the nuclei with N=90.
We suggest that future experiments should focus on more detailed measurements of the excited states in 148 Nd and 152 Nd. Moreover, the detailed information on states above the collective 0 + 2 levels is needed. The present study will be important for understanding the collective excitations in transitional nuclei regarding the applicability of the IBM and the X(5) description.