Structure Evolution in OddEven Eu 155 Nucleus within IBFM2
Saad N. Abood^{1}, Mohamed Bechir Ben Hamida^{2}, Laith A. Najim^{3}
^{1}Physics Department, College of Science ALNahrain, University Baghdad, Baghdad, Iraq
^{2}Laboratory of Ionized Backgrounds and Reagents Studies (LEMIR), High School of Sciences and Technology of Hammam Sousse (ESSTHS), University of Sousse, Sousse, Tunisia
^{3}Physics Department College of Science, Mosul University, Mosul, Iraq
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To cite this article:
Saad N. Abood, Mohamed Bechir Ben Hamida, Laith A. Najim. Structure Evolution in OddEven Eu 155 Nucleus within IBFM2. Frontiers in Heterocyclic Chemistry. Vol. 1, No. 1, 2015, pp. 1724. doi: 10.11648/j.ajme.20150101.12
1. Introduction
The interacting boson model 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 highenergy physics [1]. The application of this model to deformed nuclei is currently a subject of considerable interest and controversy.
The interacting boson model1 (IBM1) [2] and its extension to the oddA nuclei, the interacting bosonfermion model (IBFM1) [3], have proved to be able to give a successful description of widely varying classes of nuclei situated away from closed shell configurations.
In heavy nuclei, the neutron excess prevents the formation of correlated protonneutron pairs and one thus is led to consider only protonproton and neutronneutron pairs. The corresponding model is the interacting boson model2 (IBM2) [4, 5]. The introduction of fermions in this models leads to the interacting bosonfermion model2 (IBFM2). In addition to a more direct connection with the spherical shell model, the interacting bosonfermion model2 (IBFM2) has features that cannot be obtained in the interacting bosonfermion model1 (IBFM1). Here, we apply the IBFM2 model to account for ^{155}Eu isotope.
Detailed work has been done on the structure of europium nuclei in recent years; Bhattacharya et al., [6] studied on level structure, singlenucleontransfer spectroscopic factors, electromagnetic transition strengths, and relative gammaray branching. Guchhait et al. .[7] determined the level energies, spectroscopic factor, and E2 transition strengths. Prokofjev et al, .[8] studied on the ray and conversion electron spectra of ^{155}Eu from the (n, ) reaction. Lo Bianco et al, .[9] studied gammaray transitions in ^{147}Eu and analyzed in terms of the interacting bosonfermion model. There are also theoretical studies of particular isotopes with different models. Yazar et al., [10] explored the energy levels and the electric quadrupole transition probabilities B(E2; Ji →J_{f} ) and ray E2/M1 mixing ratios for selected transitions of some isotopes. Akaya et al, . [11] studied on the gammagamma angular correlation and e_{K} − directional correlation methods and ray E2/M1 mixing ratios of ^{154}Eu were investigated. Yazar et al., [12] studied some electromagnetic transition properties of ^{153155}Eu Isotopes within IBFM2. The aim of the present work is to do a systematic study of the ^{155}Eu isotope within the IBFM2 model.
It is generally believed that such positive parity spin states can be explained in particlecore coupled type models. ^{155}Eu has 63 protons and 92 neutrons, it is thus appropriate to describe ^{155}Eu in the IBFM2 by the coupling of a single fermion to the ^{154}Sm eveneven core. Over the major shell N = 50, there are four available positive parity singleparticle levels, the 1g_{7/2}, 2d_{5/2}, 2d_{3/2}, and 3s_{1/2}. For the boson core, the IBM2 basis states are used. To describe the positiveparity states, however, it is necessary to consider the inclusion of all four negativeparity single particle levels. The inclusion of multilevel possibilities into the IBFM has been analyzed by Scholten [13], who developed a formalism based on the BCS equations. The single particle energies were calculated using the relations given by [14]. Here, we apply the IBFM model to account for the ^{155}Eu isotope. The results of the IBFM2 multilevel calculations for ^{155}Eu are presented for the energy levels and the transitions probabilities, which are compared with the corresponding experimental data.
2. The Interacting Boson Model and EvenEven Core
The interacting boson model [15] provides a unified description of collective nuclear states in terms of a system of interacting bosons. The ^{155}Eu isotopes have 63 protons 92 neutrons, which fill the orbits above major shell closure at N = 50, characterized by 13 particlelike proton states. It is thus appropriate to describe ^{155}Eu in the IBFM2 model by coupling of a single fermion (proton) to the ^{154}Sm eveneven nuclear core.
The interacting boson model (IBM) has become widely accepted as a tractable theoretical scheme of correlating, describing, and predicting lowenergy collective properties of complex nuclei. In this model it was assumed that lowlying collective states of eveneven nuclei could be described as states of a given (fixed) number N of bosons. Each boson could occupy two levels one with angular momentum J = 0 (sboson) and another with J = 2 (dboson). In the original form of the model known as IBM1, proton, and neutronboson degrees of freedom are not distinguished. The model has an inherent group structure, associated with it. In the IBM2 model the neutrons and protons degrees of freedom are taken into account explicitly. Thus the Hamiltonian [16, 17] can be written as,
(1)
(2)
Here is the dboson energy, is the strength of the quadrupole interaction between neutron and proton bosons.
In the IBM2 model, the quadrupole moment operator is given by:
(3)
where or , is the quadrupole deformation parameter for neutrons and protons . Where the terms and are the neutronneutron and protonproton dboson interactions only and given by:
(4)
The last term is the Majorana interaction, shits the states with mixed protonneutron symmetry with respect to the totally symmetric ones. Since little experimental information is known about such states with mixed symmetry, which has the form:
(5)
The general onebody E2 transition operator in the IBM2 is
(6)
(7)
Where is in the form of Eq.(3). For simplicity, the has the same value as in the Hamiltonian. This is also suggested by the single jshell microscopy. In general, the E2 transition results are not sensitive to the choice of and , whether = or not. Thus, the reduced electric quadrupole transition rates between states are given by:
(8)
The electric quadrupole moment in IBM2 is given:
(9)
In the IBM2, the M1 transition operator up to the onebody term (l =1) is
(10)
Where and . The and are the boson gfactors (gyromagnatic factors ( in unit that depends on the nuclear configuration. They should be different for different nuclei.
(11)
The magnetic dipole moment operator is given by:
(12)
The reduced magnetic dipole transition rates between states are given by:
(13)
3. The Interacting BosonFermion Model
In the IBFM, oddA nuclei are described by the coupling of the odd fermionic quasiparticle to a collective boson core. The total Hamiltonian can be written as the sum of three parts:
H = HB + HF + VBF (14)
where H_{B} is the usual IBM2 Hamiltonian [1617] for the eveneven core, H_{F} is the fermion Hamiltonian containing only onebody terms and V_{BF} is the bosonfermion interaction that describes the interaction between the odd quasinucleon and the eveneven core nucleus. H_{F} is the fermion Hamiltonian containing only onebody terms and V_{BF} is the bosonfermion interaction that describes the interaction between the odd quasinucleon and the eveneven core nucleus. V_{BF} is dominated by three terms: a monopole interaction characterized by the parameter A_{0} which plays a minor role in actual calculations; the most important arise from the quadrupole interaction [18] characterized by and the exchange of the quasiparticle with one of the two fermions forming a boson [19] characterized by . H_{F} is the fermion Hamiltonian containing only onebody terms and
(15)
where the are the quasiparticle energies and is the creation (annihilation) operator for the quasiparticle in the eigen state . The bosonfermion interaction V_{BF} that describes the interaction between the odd quasinucleon and the eveneven core nucleus contains, in general, many different terms and is rather complicated, but has been shown to be dominated by the following three terms:
(16)
Where the core boson quadrupole operator is given by the equation (3), and is a parameter shown by microscopic theory to lie between and −. V_{BF} is dominated by three terms: a monopole interaction characterized by the parameter A_{0} which plays a minor role in actual calculations, the most important arise from the quadrupole interaction [20, 21] characterized by, and the exchange of the quasiparticle with one of the two fermions forming a boson [22] characterized by A_{0}.
are boson operators with and denotes normal ordering whereby contributions that arise from commuting the operators are neglected. The first term in V_{BF} is a monopole interaction which plays a minor role in actual calculations and the dominant term are the second and third, which arise from the quadrupole interaction. The third term represents the exchange of the quasiparticle with one of the two fermions forming a boson; Talmi [19] has shown that this exchange force is a consequence of the Pauli principle for the quadrupole interaction between protons and neutrons. The remaining parameters in Equation (17) can be related to the BCS occupation probabilities u_{j} , v_{j} of the singleparticle orbits:
(17)
(18)
Where are single particle matrix elements of the quadrupole operator and
(19)
are the structure coefficients of the d boson deduced from microscopic considerations, with being the energy of a pair relative to an pair [23].
The BCS occupation probability and the quasiparticle energy of each single particle orbital can be obtained by solving the gap equations:
(20)
(21)
where E_{j} is the single particle energy calculated from the relations in [20], is the Fermi level energy, and is the pairing gap energy, which was chosen to be 12A^{−1/2} MeV [24]. That leaves the strengths A_{0}, , and as free parameters which are varied to give the best fit to the excitation energies.
The total number of bosons and fermions is then: and
4. Results and Discussion
4.1. Interacting Boson Model 2
4.1.1. Energy Levels
The isotopes chosen in this work are A=154 due to the presents of experimental data for the energy levels. We have , (12 protons outside the closed shell 50), and = 5 for Sm^{154}, measured from the closed shell at 82. While the parameters ,as well as the Majorana parameters with k =1,2,3, were treated as free parameters and their values were estimated by fitting with the experimental values. The procedure was made by selecting the traditional value of the parameters and allowing one parameter to vary while keeping the others constant until the best fit with the experimental obtained. This was carried out until one overall fit was obtained. The best values for the Hamiltonian parameters are given in table 1.
The IBM2 Hamiltonian is nonlinear 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. The least square fit procedure was used to find the best fit to the three lowest bands of the ^{154}Sm isotopes under consideration.
Values of the interaction parameters for the ^{154}Sm isotopes in the IBM–2 Hamiltonian (for^{154}Sm, in terms of code NPBOS notation are given in the table (1)).
Table 1. IBM2 Hamiltonian parameters , all parameters in MeV units except and are dimensionless.
CLπ(L=0,2,4)  CLν(L=0,2,4)  ξ 3  ξ 1=ξ2  χπ  χν  K  Ɛ  Isotope 
0.5, 0.4, 0.8  0.5, 0.4, 0.8  0.1  0.12  1.2  0.8  0.039  0.34  Sm154 
Concentration was made on the to make a reasonable fit to experimental data. A sample of experimental and theoretical values of energy levels are taken in Fig.1. As one can see an overall a good agreement was obtained for the gamma and beta bands for ^{154}Sm. Figure (1), show a comparison between experimental and theoretical energy levels of the ground band in ^{154}Sm isotope, the agreement is very good for the 2_{1} and 4_{1} states.
The ratio for ^{154}Sm equal 3.256 for experimental date [25] and 3.191 for IBM2. These values for the ration put the nucleus in transitional region from gamma soft to rotational shape.
4.1.2. Electric Transition Probability
In IBM2, the E2, transition operator is given by the equation (10), and are boson effective charges depending on the boson number or and they can take any value to fit the experimental results . The method explained in reference [26]. The effective charges calculated by this method for ^{154}Sm isotopes were and .Table 3 given the electric transition probability.
Theand values increased as neutron number increases toward the middle of the shell as the value of has small value because contain mixtures of M1. The value of is small because this transition is forbidden (from quasibeta band to ground state band). The values of IBM2 in a good agreement with available experimental data [25].
The quadrupole moment is given in equation (9) for first excited state in ^{154}Sm isotopes are very well described. As mentioned above, the calculated values of (= 1.765 e.b) indicated this nucleus has prolate shape in first excited states.
21→01  41→21  61→41  22→21  
Exp.  IBM2  Exp.  IBM2  Exp.  IBM2  Exp.  IBM2 
0.922(40)  0.913  1.186(39)  1.231  1.374(47)  1.393  0.012  0.014 
4.1.3. Magnetic Transition Probability
After calculated the E2 matrix elements we lock after the M1 matrix elements as in equation (13) .The direct measurement of B(M1) matrix elements is difficult normally, so the M1 strength of gamma transition may be expressed in terms of the multipole mixing ratio which can be written as [27]
(22)
Having fitted E2 matrix elements, one can then use them with to obtain M1 matrix elements and then the mixing ratio , compare them with the prediction of the model using the operator (eq.9). The have to be estimated, if they are not been measured in the case of ^{154}Sm isotope. The g factors may be estimated from experimental magnetic moment of the state (μ=2g). In phenomenological studies and are treated as parameter and kept constant for a whole isotope chain. The total g factor defined by Sambataro et. al., [28] as:
(23)
Many relations could be obtained for a certain mass region and then the average values for this region could be calculated. One of the experimental B(M1) and the relation above been used to find that .The estimated values of the parameter are , these were used to calculate the mixing ratio. The ratios were calculated for some selected transitions and listed with the available experimental data in table 3. A good agreement between theoretical results (IBM2 and experimental data in sign and magnitude.
21→21  42→41  23→21  31→21  
Exp.  IBM2  Exp.  IBM2  Exp.  IBM2  Exp.  IBM2 
 34  1.1  0.055 
 0. 20  7.5  5.22 
The magnetic dipole transition probability is given in table 4, there is no experimental data to compare the theoretical results. The value is small that is, implying some collective effects, The large B(M1) values in IBM2 are due to the Fspin vector character of state in ^{154}Sm. The is still sizable in ^{154}Sm (increased with increased neutron number) because the transition from ground state to mixed symmetry state in IBM2 .
21→21  23→21  01→11  31→21 
0.007  0.020  1.460  0.262 
4.2. Interacting Boson Fermion Model 2
4.2.1. Energy Levels
The Hamiltonian of IBFM2 eq. (5) was diagonal's by means of the computer program ODDA [29] in which the IBFM2 parameters are identified as: A_{0} and . The parameters for the ^{154}Sm core are derived in the present work and given in table 1, while the quasiparticle energies and occupation probabilities used in this work are given in Table 6. the bosonfermion monopole interaction was omitted (A_{0} = 0.0), there are only two ( and ) free varying bosonfermion interaction parameters for the evenodd ^{155}Eu isotope.
nucleon  symbol 



Proton 
 0.170  0.210  0.0 
neutron 
 0.172  0.220  0.0 
The BCS parameters for the multilevel calculations of ^{155}Eu are given in table 6.
 2d5/2  1g7/2  3s1/2  2d3/2 
 1.279  0.955  2.198  2.099 
 0.810  0.438  0.050  0.052 
The IBFM2 energy levels calculation is used to fit experimental energy levels with the bosonfermion parameters which is given in table (1) for ^{155}Eu nucleus. The monopole interaction parameter A_{0} is set to zero. The dependence of V_{BF} on the specificity of each nucleus is counted for in the occupation probabilities appearing in the exchange term and in the quadrupole term . The best agreement with experiment for the level calculations of ^{155}Eu nucleus is found by slightly varying the occupation probability to 2j to allow a better fit with the experiment (see fig. 2). The present choice of parameters gives also a good agreement with experimental data.
4.2.2. Electric Transition Probability
The calculation of electromagnetic transitions gives a good test of the nuclear model wave functions. In this section we discuss the calculation of the E2 transition strengths and results with the available experimental data. In general, the electromagnetic transition operators can be written as a sum of two terms, the first of which acts only on the boson part of the wave function and second only on the fermion part.
Transition operators can be written in the same way as in eq. (3). There are now four terms describing proton and neutron bosons and fermions,
(24)
The boson terms are given in equation (3). The fermion terms can, to the lowest order, be written as:
(25)
Particularly important in oddeven nuclei are the transition operators which induce E2 and Ml transitions. It is customary in the operators to separate the dependence on the angular momenta and from the coefficients that determine the strengths of the transitions. This is done by introducing effective charges and moments. For E2 transitions, one has:
(26)
where now the single particle indices are written explicitly. The quantities and are the fermion effective charges. The free values of these charges are 1 and 0 respectively, in units of the electron charge. Shell model calculations indicate that and . Following, the boson part is written as
(27)
A superscript B has been added to in order to distinguish it from the fermion charges. The units of are different from those of since the radial integral is already included in eq. (18). The boson effective charges have the same units as the product
(28)
That is the units are e fm^{2}.
In the table 7 the values of the E2 transitions for ^{155}Eu with the experimental data, The transition is a good agreement with experimental data. The main discrepancies occur in the case of the B(E2) involving the depopulation of excited states of ^{155}Eu at about 0.213 MeV. This apparent breakdown of the present model has two probable cases:
1 The configuration space used in the present calculation is not large enough. It may
be better to include protons and neutrons as active nucleons.
2 A satisfactory comparison with the experiments is quite difficult due to the large errors on the experimental values, moreover the theoretical B(E2) values for that the transition seem to be systematically too high. This can be explained by the fact that many small components of the initial and final wave functions contribute coherently to the value of this reduced E2 transition probability.
In general, the calculated values agree with the experimental data reasonably well. The B(E2) values depend quite sensitively on the wave functions, which suggest that the wave functions obtained in this work are reliable. The model may be applied to many other nuclei and its many other nuclear properties.
3/21→5/21  3/22→7/21  5/22→5/21  5/22→7/21  
Exp.  IBFM2  Exp.  IBFM2  Exp.  IBFM2  Exp.  IBFM2 
0.0036  0.0053  0.0023  0.0033    0.0732    0.00346 
4.2.3. Magnetic Transition Probability
The mixing ratios for some selected transitions in the ^{155}Eu nucleus is calculated from the useful eq. (10) of mixing ratio as above and with the help of B (E2) and B (MI) values, which are obtained from ODDA program [29] ; the results are given in table (8). In general, the calculated mixing ratio of ^{155}Eu nucleus.
3/21→5/21  3/22→7/21  5/22→5/21  5/22→7/21 
0.02  0.0043  0.006  2.0 
5. Conclusions
We can summarize the main results and conclusions of this study as follows. Energy level for eveneven ^{154}Sm nucleus for ground, beta gamma bands are reproduced well. The energy spectra of the oddeven ^{155}Eu nucleus can be reproduced quite well with the help of only two (and ) freely varying bosonfermion interaction parameters. The monopole interaction (A_{0}) plays a minor role in the actual calculations. The most important effects arise from the quadrupole interaction () and the exchange of the quasiparticle with one of the two fermions forming a boson interaction ().
A satisfactory comparison with the experiments is quite difficult due to the errors in the experimental values; moreover the theoretical B (E2) values for the transition seem to be systematically too small. This can be explained by the fact that many small components of the initial and final wave functions contribute coherently to the value of this reduced E2 transition probability. In general, the calculated electromagnetic properties of the ^{154}Sm nucleus do not differ significantly from those calculated in experimental work. The calculated values in this study show that the transitions connect the levels with the same parity and the E2 transitions are predominant. The later includes transitions originating from the beta and gamma bands, which supports the idea that the beta and bands may be quadrupole excitations of the perturbed ground state, but the existence of M1 indicates that the beta and gamma bands cannot be pure quadrupole excitations of the ground state band.
We have also examined the mixing ratio (E2/M1) of transitions linking the ground state bands. We find that the transitions which link lowspin states and which were obtained in the present work are largely consistent with this requirement, although some may be considered to show irregularities.
References