BETWEEN IBA-I AND IBA-II MODEL AND ELECTROMAGNETIC TRANSITIONS OF SOME ERBIUM ISOTOPES

Since the lowest levels are symmetric under the interchange of neutrons and protons from calculations in the interacting boson approximation IBA-2 model, IBA-1 model space, in which neutron and proton degrees of freedom are not distinguished can be considered as a subspace of the IBA-2 model space. Using the microscopic background of the IBA-2 model, a correspondence can be established between IBA-1 and IBA-2 model space. Since the space of the IBA-1 model can be regarded as a subspace of the IBA-2 model there is a unique way to “Project” the operators of the IBA-2 model onto those of IBA-1. This projection can be carried out by using the Fspin formalism. In the IBA-2, the lowest states are indeed fully symmetric, the calculations with the help of this projection, we explore the energy levels and the electric quadrupole transition probabilities B(E2;Ii → If) and γ-ray E2/M1 mixing ratios for selected transitions of Er .Owing to admixtures of non-fully-symmetric states in IBA-2, we renormalized the parameters (ε) and (κ).This is the first time we show that this projection can be applied to some heavier isotopes and the results obtained for Er isotopes are reasonably in good agreement with the previous experimental values.


1-INTRODUCTION
Detailed work has been done on the structure of erbium nucleus in recent years; R.F.Casten et all [1] studied on (n,γ) reaction for 168 Er and obtained a number of new levels for the first time, I. Alfter et all [2] determined M1/E2 multipole mixing ratios of erbium isotopes by experiment, L.M. Chen [3] studied the negative parity high-spin states of even-odd erbium nucleus with mass number 159<A<165 within the framework of the interacting boson fermion model, B.R. Barrett et all [4] calculated the multipole mixing ratios of 168 Er with the framework of the interacting boson approximation, R.S. Gau et all [5] calculated the energies of excited states and the values of B(E2) of 155-165  Er by using the interacting boson fermion model.
The interacting boson approximation represents a significant step forward in our understanding of nuclear structure.It offers a simple Hamiltonian, capable of describing collective nuclear properties across a wide range of nuclei, and is founded on rather general algebraic group theoretical techniques which have also found recent application to problems in atomic, molecular, and high-energy physics [6][7].The application of this model to deformed nuclei is currently a subject of considerable interest and controversy [8].
In the first version, IBA-1, no distinction is made between neutron and proton degrees of freedom.An unsatisfactory aspect of this model is that there is no clear connection with a microscopic structure of nucleus.The microscopic theory strongly suggests that it is important to treat the neutron and proton degrees of freedom independently.This has led to the introduction of the second, generalized, version of the IBA-model, IBA-2.In the second version, the neutron and proton degrees of freedom are treated explicitly.In this model the nucleus is described explicitly in terms of neutron (s ν ,d ν ) and proton (s π ,d π ) bosons.From calculations in the IBA-2 model it appears that the lowest levels are symmetric under the interchange of neutrons and protons.This symmetry is most easily discussed in terms of a variable called F-spins [9].In the case of boson systems F-spin plays a role similar to that of isospin in the case of fermion systems.
The relation between the IBA-1 and IBA-2 model is obtained by identifying states of the former to the fully symmetric i.e. maximal F-spin states of the latter model.Since the space of the IBA-1 model can be regarded as a subspace of the IBA-2 model there is a unique way to "Project" the operators of the IBA-2 model onto those of IBA-1.This projection can be carried out by using the F-spin formalism [10].
From these considerations it follows that IBA-1 and IBA-2 model can be related to each others and the states of the IBA-1 model can be identified with the fully symmetric states in the IBA-2 model.It is the purpose of this work to study these relations and applied to 166-168 Er isotopes.
The Project approximation used in this study has been extensively described by Olaf Scholten for the neodymium, samarium and gadolinium isotopes [10].We shall present here only the results of calculation and refer the reader to that work of the Project approximation for details.In section 2 we study the positive parity spectra of the 166-168 Er isotopes.In the same section E2 and M1 transition probabilities and electric quadrupole transition probabilities B(E2;I i → I f ) are analyzed.Finally, the work is summarized in section 3.

2-THEORY AND METHOD OF CALCULATION
In IBA-2 the neutron and proton degrees of freedom are treated explicitly.This has the advantage of being closer to a microscopic theory.The matrices that have to be diagonalized are, however, much larger.One can regard the IBA-1 model space, in which neutron and proton degrees of freedom are not distinguished, as a subspace of the IBA-2 hamiltonian one can thus Project out its IBA-1 pieces [10].In the present work the relevant terms in the IBA-2 Hamiltonian Where the dot denotes a scalar product.The first term represents the single-boson energies for proton and neutron bosons and n dρ is the number of d-bosons where ρ corresponds to π(proton) or ν(neutron) bosons.The second term denotes the main part of the boson-boson interaction, i.e the quadrupole-quadrupole interaction between neutron and proton bosons with strength κ.The quadrupol operator is where χ ρ determines the structure of the quadrupole operator and is determined empirically.The square brackets in (2) denote angular momentum coupling.
The terms V ππ and V νν , correspond to interactions between like-bosons.They are of the form The isotopes 162−170 Er have N π = 7, and N ν varies from 6 to 9, while the parameters κ, χ ν , χ π and ε were treated as free parameters and their values were estimated by fitting to the measured level energies.This procedure was made by selecting the "traditional" values of the parameters and then allowing one parameter to vary while keeping the others constant until a best fit was obtained.This was carried out iteratively until an overall fit was achieved.The best fit values for the Hamiltonian parameters are given in Table 1.In the present work IBA-2 Hamiltonian parameters are normalized with the help of IBA-1 model Hamiltonian.In the IBA-2 calculation the lowest states are indeed fully symmetric; the calculation with the help of this projection gave good results for the excitation energies.Because of the admixtures of non-fully-symmetric states in IBA-2 model space, the projection gave some difficulties and we had to renormalize the parameters (ε) and (κ).The IBA-2 Hamiltonian is non-linear in the parameters.To obtain the values of the parameters which give the best fit we have to calculate for each energy level the difference between its experimental and calculated values.Then we have to sum over the squares of all these differences and to find a local minimum to this summation.Therefore, in particular a minimum where the ε π and ε ν parameters are equal to the experimental values in the appropriate semi-magic nuclei.The least square fit procedure was used to find the best fit to the three lowest bands (g.s., γ-state and βstate bands) of the erbium isotopes under consideration.The best fit obtained for these isotopes is shown in fig.2a-2b.The numerical diagonalization of Hamiltonian has been carried out by using the PHINT code [11].The values of the main parameters of the Hamiltonian are given in Table-1.
The calculated excitation energies for 166-168 Er isotopes as well as the experimental ones are compared in Figure 2a-2b.The general agreement between experiment and model is quite good.A successful nuclear model must yield a good description not only of the energy spectrum of the nucleus but also of its electromagnetic properties.The most important electromagnetic features are the E2 transitions.The B(E2) values were calculated by using the E2 operator, (4) Where the Q π and Q ν operators are defined in eq. ( 2) and e π and e ν are the "effective charges" for the proton bosons and the neutron bosons.For simplicity the "effective charges" e π and e ν were taken as equal (e=0.120eb.Some calculated B(E2) values from the ground state band and B(E2) ratios are given in table 2.
Since erbium nucleus has a rather rotational character, taking into account of the dynamic symmetry location of the even-even erbium nuclei at the IBM phase triangle where their parameter sets are at the O(6)-SU(3) transition region and closer to SU(3) rotational character and possessing good rotational states, we used the multiple expansion form of the Hamiltonian for our approximation.In order to find the value of the effective charge we have fitted the calculated absolute strengths B(E2) of the transitions within the ground state band to the experimental ones.The best agreement is obtained with the value e π = e ν = e = 0.120 eb, as shown in table-2.The B(E2) values depend quite sensitively on the wave functions, which suggest that the wave functions obtained in this work are reliable.E2:Ml multipole mixing ratios; Arima and Iachello in their original interacting boson approximation (IBA-1) gave the M1 operator in the restricted case of SU(5) dynamic symmetry [13] and as well as the general case [10].However, even when starting with the general operator, they derived the E2/M1 mixing ratio by neglecting the term which break the SU(3) symmetry [14].It follows that the reduced mixing ratio is given by the same simple formula for both SU(5) and SU(3) symmetries.The formula contains only one parameter and the initial and final spins.Warner [15] has developed an IBA description of the E2/M1 mixing ratio and his point of departure was essentially the same as that of Scholten et al [16].To present time, several systematic studies [17,18] have been performed within the framework of the IBA.

RESULTS AND DISCUSSION
The strongly deformed even-even erbium isotopes have been described by IBA-2 Hamiltonian.In these calculations no truncation has been put in the huge neutron-proton boson spaces.This is the first time that IBA-2 Hamiltonian parameters are obtained by the projection that we have developed by using the F-spin formalism from the operator of IBA-2 model over the operator of IBA-1 model space.The single d-boson energies were determined from the experimental data -the 0-2 spacing in the appropriate semimagic nuclei, where this data is known.It was found that although Hamiltonian yields a good description of the energy levels of the 166-168 Er isotopes.
For totally symmetric states, the description of the nuclear properties is approximately equal in going from IBA-1 to IBA-2.However IBA-2 model which distinguished neutrons and protons has a clear microscopic connection with the spherical shell model while the IBA-1 has not.The present work demonstrates that IBA-2 Hamiltonian parameters based on IBA-1 model gave good results for the excitation energies and the electric quadrupole transition probability B(E2;I i →I f ) of 166- 168 Er isotopes.For the non-fully-symmetric states, we renormalized the parameters (ε) and (κ) and obtained good results.In the present calculations we have shown the ability of the projection in correlating different properties in erbium isotope in terms of a few parameters.
We have also examined the mixing ratio ( ) of transitions linking the γ and ground state bands.The transitions which link low spin states and were obtained in the present work are in good agreement and show a little bit irregularities.We find that the transitions which link low-spin states and which were obtained in the present work are largely consistent with this requirement although some may be considered to show irregularities.
In the treatments of the IBA-2 Hamiltonian mentioned above few IBA-2 interactions were used.In the IBA-2 model there are additional interactions with (or without) microscopic basis.It is possible that by adding some of interactions to our IBA-2 Hamiltonians, the wave functions will be altered such that the agreement with the mixing ratios could be improved.A deeper understanding of the microscopic basis of the IBA-2 model will certainly help to find the interactions that must be included in the IBA-2 Hamiltonians in order to provide better description of the strongly deformed even-even nuclei.

Fig. 1 .
Fig. 1.The parameters ε d and κ employed for the IBA-2 calculations for erbium isotopes with even neutron numbers 94 up to 102.

Fig. 2a Fig. 2b .
Fig.2aThe three lowest rotational bands in spectra of the 166 Er.In each band the experimental data are plotted on the left and calculated values on the right