Calculation of Energy Levels and The Reduced Transitions Probability For Even-Even Number of Isotones ( N=90)

In this work the Interacting Boson Model IBM-1, DavydovFilippov (D-F) models and the Critical Point Symmetry X(5) have been employed to study the energy levels and the reduced transition probability B(E2) for the (N = 90) transitional nuclei Ba,Ce,Er and Yb in the concepts of theoretical treatment. The best input parameters for the above approaches which lead to the best fit to experimental data are determined. The reduced transition probability B(E2) for the above nuclei have been calculated in a relative and absolute scales by using the most recent available experimental data. ةصلاخلا مت يف هذه مادختسا ةساردلا جذومنأ ةلعافتملا تانوزوبلا لولأا ) IBM-1 ( و جذومنأ فوبيليف و فوديفاد ) D-F ( و جذومنأ رظانت ةجرحلا ةطقنلا E(5) ةقاطلا تايوتسم باسحل ةلزتخملا تلااقتنلاا ةيلامتحاو B(E2) ةيلاقتنلاا ىونلل ) N=90 ( Ba و Ce و Er و Yb متو داجيإ لضفأ تلاماعم لاخدلإا جذامنلل هلاعأ يتلا ىطعت لضفأ ـلو ةقاطلا تايوتسمل ةيرظنلاو ةيلمعلا ميقلا نيب قباطت .B(E2) ميق باسح مت B(E2) ىونلل ةيبسنلا و ةقلطملا هلاعأ تمدختساو يف هذه ةثيدحلا ةيلمعلا تانايبلا تاباسحلا . Calculation of Energy Levels and The Reduced Transitions ... 98 1Introduction The structure of the N = 90 isotones in the vicinity of the Z = 64 has been the focal point of a research groups. A rapid change in deformation occurs at N ≈ 90 nuclei, a long the Nd, Sm, Gd and Dy isotopic chains, as a transition occurs from spherical to axially deformed structure (1-5). The study of phase-shape transitions in nuclei can be done in the interacting boson model (IBM) which reproduces well the data in the transitional Nd-Sm-Gd-Dy region (3,6-8). Recently a new symmetry, called X(5), has been proposed for the critical point of phase-shape transition from spherical to axially deformed nuclei (2-4). The N = 90 isotones Nd and Sm are a good example of the realization of this symmetry (1,3,5,9,10). Additional examples of X(5) behaviors have been suggested in N = 90 isotones Gd and Dy ( 3,11). The aim of the present work is to apply the interacting boson model (IBM-1), the Davydov-Fillipov model and the critical point symmetry X(5) for calculating the values of the energy levels and the reduced electric quadrupole transition probability B(E2) for the other N = 90 transitional nuclei, Ba, Ce, Er and Yb, lying between the SU(5) and SU (3) limits. 2Nuclear Models 2-1 The Interacting Boson Model -1 (IBM-1) The Interacting Boson Model of Arima and Iachello (12-17) ,has widely accepted as a tractable theoretical scheme of correlating, describing and predicting low-energy collective properties of complex nuclei. The most general Hamiltonian subject to the conditions of U(6) symmetry can be written as (15,17). 4 4 4 3 3 3 2 1 ˆ . ˆ ˆ . ˆ ˆ . ˆ ˆ . ˆ ˆ . ˆ ˆ ˆ T T a T T a Q Q a L L a P P a n H o d + + + + + = ε .........(1) where ε is the energy of boson. For the simplest form (which is ε=εd-εs, assuming εs=0). d n ^ is the operator of the d-boson number. a0, a1, a2, a3, a4 parameters represent the strength of the pairing, angular momentum, quadrupole, octupole and hexadecapole interactions respectively. P ^ , ^ L , ^ Q , ^ 3 T and ^ 4 T represents the operators for each interaction respectively. For the transition case between SU(5) and SU(3) limits , the above Hamiltonian reduced to Q Q a L L a n H d ˆ . ˆ ˆ . ˆ ˆ ˆ 2 1 + + = ε .........(2) In order to calculate electromagnetic transition rates, one must specify the transition operators. The (E2) transition operator can be written as (7,15).


1-Introduction
The structure of the N = 90 isotones in the vicinity of the Z = 64 has been the focal point of a research groups. A rapid change in deformation occurs at N ≈ 90 nuclei, a long the Nd, Sm, Gd and Dy isotopic chains, as a transition occurs from spherical to axially deformed structure (1)(2)(3)(4)(5). The study of phase-shape transitions in nuclei can be done in the interacting boson model (IBM) which reproduces well the data in the transitional Nd-Sm-Gd-Dy region (3,(6)(7)(8). Recently a new symmetry, called X(5), has been proposed for the critical point of phase-shape transition from spherical to axially deformed nuclei (2)(3)(4). The N = 90 isotones 150 Nd and 152 Sm are a good example of the realization of this symmetry (1,3,5,9,10). Additional examples of X(5) behaviors have been suggested in N = 90 isotones 154 Gd and 156 Dy ( 3,11).
The aim of the present work is to apply the interacting boson model (IBM-1), the Davydov-Fillipov model and the critical point symmetry X (5) for calculating the values of the energy levels and the reduced electric quadrupole transition probability B(E2) for the other N = 90 transitional nuclei, 146 Ba, 148 Ce, 158 Er and 160 Yb, lying between the SU(5) and SU (3) limits.

2-1 The Interacting Boson Model -1 (IBM-1)
The Interacting Boson Model of Arima and Iachello (12)(13)(14)(15)(16)(17) ,has widely accepted as a tractable theoretical scheme of correlating, describing and predicting low-energy collective properties of complex nuclei. The most general Hamiltonian subject to the conditions of U(6) symmetry can be written as (15,17 where ε is the energy of boson. For the simplest form (which is ε=ε d -ε s , assuming ε s =0). d n is the operator of the d-boson number. a 0 , a 1 , a 2 , a 3 , a 4 parameters represent the strength of the pairing, angular momentum, quadrupole, octupole and hexadecapole interactions respectively. P ,L ,Q,^3 T and ^4 T represents the operators for each interaction respectively. For the transition case between SU(5) and SU(3) limits , the above Hamiltonian reduced to Q Q a L L a n H d. .
In order to calculate electromagnetic transition rates, one must specify the transition operators. The (E2) transition operator can be written as (7,15).
where α 2 plays the role of the effective boson charge and β 2 is a parameter related to α 2 . The parameter α 2 is related to the reduced transition probability B(E2)as follows: 1-For SU(5) limit (7,15) ) This gives, for the first 2 + state. N α ) 0 B(E2;2

2-2 Davydov and Filippovs (D-F) Model
Rotational levels of even-even nuclei have been treated by  under the assumption that the nuclei possesses equilibrium shapes which are not axially symmetric. The formulae for levels with spins 2 + and 3 + are (18,21,22  γ is the asymmetrical parameter which is determine the deviation of the shapes of the nucleus from axially symmetry. The quantity γ can easily be determined from the ratio of the energies of two levels with angular momenta I=2.  (15) and the relations (13) and (14). The equations for levels with spins 4+, 6+, and 8+ are the roots of the third, fourth and fifth degree respectively (19,21,22). Results of numerical solutions of these equations for several values of γ are given in ref. (19). These equations are not used in the present work; instead the graphical method has been used (see section 3-1-2).
The formulae for the transition probabilities b(E2) are (18,(21)(22)(23) x , can be calculated by using the relations and the wave function coefficient given by (D-F) (19,22). The values of these transition probabilities for several γ values are given by (D-F) (19). In the present work, the graphic method has been used between b(E2) and γ to calculate the above b(E2)(see section 3-2-3). The relation between the reduced Electric transition probability B(E2) and the transition probability b(E2) is (18) Where Q 0 is the intrinsic quadrupole moment.

2-3 Critical Point Symmetry X(5)
Iachello introduced new dynamical symmetries at the critical point of a phase/shape transitions: E(5) for a transition between spherical and deformed γ-soft nuclei (24) and X(5) for transition between spherical and axially deformed nuclei (2,3). His approach was based on analytical solutions of the differential equation for a geometrical (Bohr) Hamiltonian with a flat-bottomed potential in the quadrupole deformation. In the X(5) model an infinite square well potential in β, V(β),is combined with a term V(γ) (3,5): (22) ……………… (23) The potential V(γ) is assumed to be harmonic around γο with 2 ……. (24) Where β is the deformation parameter and γ describing the deviation from axially symmetry.
The present X(5) results for the energies of the ground-state band, β-band and γ-band are shown in Table (1) where the energies are normalized to E2 1 + = 1.0 (3,5).
The IBM-1 has been used in the calculation of the energy spectra. The program PHINT (IBM-code) written in FORTRAN 77 language has been used in the calculations (26). The number of bosons and the best values of the Hamiltonian parameters which gives the best fitting between theoretical and experimental energy levels of the above nuclei are shown in Table (2) .The 0 2 + state (β-band head) of the above nuclei is below the 2 2 + state except for 160 Yb .The term ( a o ∧ P . ∧ P ) has been added to equation (2) in order to raise the 0 2 + state. This term can describe nuclei in which the β-band lie above the γ-band in energy (17).

3-1-2 D-F Model
The energy levels of even-even 146 Ba, 148 Ce, 158 Er and 160 Yb nuclei possessing spins 2 and 3 have been calculated by using equations (8), (9) and (10) .The results for numerical solutions of the equations for levels with spins 4,6 and 8 for several values of γ have been taken from (19) and replotted as shown in Figure(2) These plots are analyzed by using the MATLAB program to obtain the values of energy levels for 146 Ba, 148 Ce, 158 Er and 160 Yb nuclei from knowing their γvalues. The energy levels and γvalues the above nuclei are listed in Table (3).The constant (A) has the same energy dimension.

3-1-3 Critical Point Symmetry X(5)
A nucleus with an R 4/2 value near 2.91 in a known spherical to axially deformed transition region is immediately of interest as a prospective X(5) nucleus (2,3,5,11). The 146 Ba, 148 Ce, 158 Er and 160 Yb nuclei are constitute an example of such spherical to axially deformed transition region. We used the critical point symmetry X(5) to calculate the energy levels of the above nuclei. The energy levels from X(5) [see Table (1)] are fitted to the first experimental excited state with spin 2 (E2 1 + ) of the above nuclei and determined the other energy levels by using the conversion constant. This constant was calculated by dividing the (E2 1 + ) exp from ref. (25) by the corresponding value obtained from critical point symmetry (E2 1 ) theo .

3-2-2 IBM-1.
The reduced transition probability B(E2) calculations were carried out by using the computer program FBEM (IBMT-code) (26). The parameters used in this program namely E2SD and E2DD were determined for each 146  The parameter α 2 is calculated from the experimental value of B(E2;2 1 + →0 1 + ) using equations (5) and (7) for SU(5) and SU(3) limits to get two values. The parameter β 2 is calculated from α 2 for SU(5) limit by using the relation (16) By taking the average of the two values for each α 2 and β 2 parameters and slightly change the average values in order to get a good agreement between theoretical and experimental B(E2) values. The values of E2SD and E2DD for the above nuclei are listed in Table (2).

4-1-Energy level
The energy values of the low-lying positive parity states of 146 Ba, 148 Ce, 158 Er and 160 Yb nuclei calculated by the IBM-1, D-F models and critical point symmetry X(5) are compared with experimental values (27)(28)(29)(30) as shown in Table( 5). It can be seen from this table that the most of present results from IBM-1 calculations are in good agreement with the experimental energy levels values within the associated errors which are found to be less than 24% for all levels. It is obvious that the IBM-1 calculations give better values than those of D-F predictions, especially, for 4 1 + , 4 2 + , 6 1 + and 8 1 + states, in addition there is good agreement between experimental data and the theoretical energy levels calculated by X(5) limit, especially, for 2 1 + , 4 1 + , 6 1 + and 8 1 + states. The significant apparent discrepancy is that for 2 3 + state of 146 Ba , 2 2 +, 2 3 +, and 3 1 + states of 148 Ce, 0 2 + ,2 2 + , 2 3 + , 4 2 + and 3 1 + states of 158 Er and 160 Yb nuclei. Two basic predictions of the X(5) model are that R 4/2 = 2.91 ( or 2.71 ≤ R 4/2 ≤ 3.11) and E(0 2 + )/E(2 1 + ) = 5.67 (3,5) . The R 4/2 equal to 2.83,2.87,2.74 and 2.63 and E(0 2 + )/E(2 1 + ) equal to 5.81,4.87,4.2 and 4.47 for 146 Ba, 148 Ce, 158 Er and 160 Yb nuclei respectively. It is evident from Table (5) and the above ratios that the 146 Ba and 148 Ce nuclei have a structure near to the X(5) symmetry. One can be observe from Table (5) that the D-F model is not able to reproduce the 0 2 + , 0 3 + and 2 3 + states.

4-2 Reduced Transition Probabilities
The absolute experimental Reduced transition probabilities B(E2) and the relative B(E2) values for 146 Ba, 148 Ce, 158 Er and 160 Yb nuclei have been calculated from the available experimental data (27)(28)(29)(30) and compared with those predicted by the IBM-1, D-F models and critical point symmetry X(5) is shown in Table (

1-
The results from IBM-1 calculations are in good agreement with the experimental values between the SU(5) and SU(3) limits. .

2-
The theoretical energy values for 6 1 + and 8 1 + states obtained from D-F model are higher than the experimental data. The D-F model is not able to reproduce the 0 2 + , 0 3 + and 2 3 + states.