American Journal of Modern Energy
Volume 1, Issue 1, June 2015, Pages: 17-24

Structure Evolution in Odd-Even Eu- 155 Nucleus within IBFM-2

Saad N. Abood1, Mohamed Bechir Ben Hamida2, Laith A. Najim3

1Physics Department, College of Science AL-Nahrain, University Baghdad, Baghdad, Iraq

2Laboratory of Ionized Backgrounds and Reagents Studies (LEMIR), High School of Sciences and Technology of Hammam Sousse (ESSTHS), University of Sousse, Sousse, Tunisia

3Physics Department College of Science, Mosul University, Mosul, Iraq

Email address:

(S. N. Abood)
(M. B. B. Hamida)
(L. A. Najim)

To cite this article:

Saad N. Abood, Mohamed Bechir Ben Hamida, Laith A. Najim. Structure Evolution in Odd-Even Eu- 155 Nucleus within IBFM-2. Frontiers in Heterocyclic Chemistry. Vol. 1, No. 1, 2015, pp. 17-24. doi: 10.11648/j.ajme.20150101.12


Abstract: In this work, We investigate the energy levels and the electromagnetic transition probabilities B(E2) and B(M1) for Eu-155 within framework of IBFM-2. The results are in reasonably agreement with the available experimental values. The result of an IBFM-2 multilevel calculation with the 3s1/2, 2d3/2, 2d5/2, and 1g7/2 single particle orbits is reported for the positive parity states of the odd-mass Eu-155 isotopes. Also, an IBM-2 calculation is presented for the low-lying states in the even-even Sm-154 core nucleus.

Keywords: Interacting Boson Model, Interacting Boson-Fermion Model, Energy Levels, B(E2), B(M1) and Mixing Ratios


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 high-energy physics [1]. The application of this model to deformed nuclei is currently a subject of considerable interest and controversy.

The interacting boson model-1 (IBM-1) [2] and its extension to the odd-A nuclei, the interacting boson-fermion model (IBFM-1) [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 proton-neutron pairs and one thus is led to consider only proton-proton and neutron-neutron pairs. The corresponding model is the interacting boson model-2 (IBM-2) [4, 5]. The introduction of fermions in this models leads to the interacting boson-fermion model-2 (IBFM-2). In addition to a more direct connection with the spherical shell model, the interacting boson-fermion model-2 (IBFM-2) has features that cannot be obtained in the interacting boson-fermion model-1 (IBFM-1). Here, we apply the IBFM-2 model to account for 155Eu isotope.

Detailed work has been done on the structure of europium nuclei in recent years; Bhattacharya et al., [6] studied on level structure, single-nucleon-transfer spectroscopic factors, electromagnetic transition strengths, and relative gamma-ray 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 155Eu from the (n, ) reaction. Lo Bianco et al, .[9] studied gamma-ray transitions in 147Eu and analyzed in terms of the interacting boson-fermion 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 →Jf ) and -ray E2/M1 mixing ratios for selected transitions of some isotopes. Akaya et al, . [11] studied on the gamma-gamma angular correlation and eK  directional correlation methods and -ray E2/M1 mixing ratios of 154Eu were investigated. Yazar et al., [12] studied some electromagnetic transition properties of 153-155Eu Isotopes within IBFM-2. The aim of the present work is to do a systematic study of the 155Eu isotope within the IBFM-2 model.

It is generally believed that such positive parity spin states can be explained in particle-core coupled type models. 155Eu has 63 protons and 92 neutrons, it is thus appropriate to describe 155Eu in the IBFM-2 by the coupling of a single fermion to the 154Sm even-even core. Over the major shell N = 50, there are four available positive parity single-particle levels, the 1g7/2, 2d5/2, 2d3/2, and 3s1/2. For the boson core, the IBM-2 basis states are used. To describe the positive-parity states, however, it is necessary to consider the inclusion of all four negative-parity 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 155Eu isotope. The results of the IBFM-2 multilevel calculations for 155Eu are presented for the energy levels and the transitions probabilities, which are compared with the corresponding experimental data.

2. The Interacting Boson Model and Even-Even Core

The interacting boson model [15] provides a unified description of collective nuclear states in terms of a system of interacting bosons. The 155Eu isotopes have 63 protons 92 neutrons, which fill the orbits above major shell closure at N = 50, characterized by 13 particle-like proton states. It is thus appropriate to describe 155Eu in the IBFM-2 model by coupling of a single fermion (proton) to the 154Sm even-even nuclear core.

The interacting boson model (IBM) has become widely accepted as a tractable theoretical scheme of correlating, describing, and predicting low-energy collective properties of complex nuclei. In this model it was assumed that low-lying collective states of even-even 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 (s-boson) and another with J = 2 (d-boson). In the original form of the model known as IBM-1, proton, and neutron-boson degrees of freedom are not distinguished. The model has an inherent group structure, associated with it. In the IBM-2 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 d-boson energy,  is the strength of the quadrupole interaction between neutron and proton bosons.

In the IBM-2 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 neutron-neutron and proton-proton d-boson interactions only and given by:

(4)

The last term  is the Majorana interaction, shits the states with mixed proton-neutron 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 one-body E2 transition operator in the IBM-2 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 j-shell 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 IBM-2 is given:

(9)

In the IBM-2, the M1 transition operator up to the one-body term (l =1) is

(10)

Where  and . The  and  are the boson g-factors (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 Boson-Fermion Model

In the IBFM, odd-A nuclei are described by the coupling of the odd fermionic quasi-particle to a collective boson core. The total Hamiltonian can be written as the sum of three parts:

H = HB + HF + VBF               (14)

where HB is the usual IBM-2 Hamiltonian [16-17] for the even-even core, HF is the fermion Hamiltonian containing only one-body terms and VBF is the boson-fermion interaction that describes the interaction between the odd quasi-nucleon and the even-even core nucleus. HF is the fermion Hamiltonian containing only one-body terms and VBF is the boson-fermion interaction that describes the interaction between the odd quasi-nucleon and the even-even core nucleus. VBF is dominated by three terms: a monopole interaction characterized by the parameter A0 which plays a minor role in actual calculations; the most important arise from the quadrupole interaction [18] characterized by  and the exchange of the quasi-particle with one of the two fermions forming a boson [19] characterized by . HF is the fermion Hamiltonian containing only one-body terms and

(15)

where the  are the quasiparticle energies and  is the creation (annihilation) operator for the quasiparticle in the eigen state . The boson-fermion interaction VBF that describes the interaction between the odd quasi-nucleon and the even-even 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 −. VBF is dominated by three terms: a monopole interaction characterized by the parameter A0 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 A0.

  are boson operators with  and denotes normal ordering whereby contributions that arise from commuting the operators are neglected. The first term in VBF 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 uj , vj of the single-particle 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 quasi-particle energy  of each single particle orbital can be obtained by solving the gap equations:

(20)

(21)

where Ej 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 A0, , 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 Sm154, 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 IBM-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. The least square fit procedure was used to find the best fit to the three lowest bands of the 154Sm isotopes under consideration.

Values of the interaction parameters for the 154Sm isotopes in the IBM–2 Hamiltonian (for154Sm, in terms of code NPBOS notation are given in the table (1)).

Table 1. IBM-2 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 154Sm. Figure (1), show a comparison between experimental and theoretical energy levels of the ground band in 154Sm isotope, the agreement is very good for the 21 and 41 states.

The ratio  for 154Sm equal 3.256 for experimental date [25] and 3.191 for IBM-2. These values for the ration put the nucleus in transitional region from gamma soft to rotational shape.

Figure 1. Comparison of calculated IBFM-2 energy levels for positive parity with experimental data of 154Sm [25].

4.1.2. Electric Transition Probability

In IBM-2, 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 154Sm 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 IBM-2 in a good agreement with available experimental data [25].

The quadrupole moment is given in equation (9) for first excited state in 154Sm 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.

Table 2. Electric Transition probability B(E2) for 154Sm isotopes in e2b2 units.

21→01 41→21 61→41 22→21
Exp. IBM-2 Exp. IBM-2 Exp. IBM-2 Exp. IBM-2
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 154Sm 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 (IBM-2 and experimental data in sign and magnitude.

Table 3. Mixing Ratios for 154Sm (Exp. data are taken from Ref. [27]).

21→21 42→41 23→21 31→21
Exp. IBM-2 Exp. IBM-2 Exp. IBM-2 Exp. IBM-2

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 IBM-2 are due to the F-spin vector character of  state in 154Sm. The is still sizable in 154Sm (increased with increased neutron number) because the transition from ground state to mixed symmetry state  in IBM-2 .

Table 4. Mixing Ratios for 154Sm (Exp. data are taken from Ref. [15,21]).

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 IBFM-2 eq. (5) was diagonal's by means of the computer program ODDA [29] in which the IBFM-2 parameters are identified as: A0  and  . The parameters for the 154Sm core are derived in the present work and given in table 1, while the quasi-particle energies and occupation probabilities used in this work are given in Table 6. the boson-fermion monopole interaction was omitted (A0 = 0.0), there are only two ( and ) free varying boson-fermion interaction parameters for the even-odd 155Eu isotope.

Table 5. Boson- Fermion interaction in 155Eu isotopes for positive- parity state.

nucleon symbol

Proton

0.170 0.210 0.0
neutron

0.172 0.220 0.0

The BCS parameters for the multilevel calculations of 155Eu are given in table 6.

Table 6. BCS parameters for the multilevel calculations of 155Eu.

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 IBFM-2 energy levels calculation is used to fit experimental energy levels with the boson-fermion parameters which is given in table (1) for 155Eu nucleus. The monopole interaction parameter A0 is set to zero. The dependence of VBF 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 155Eu 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.

Figure 2. Comparison of calculated IBFM-2 energy levels for positive parity with experimental data of 155Eu [25].

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 odd-even 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 fm2.

In the table 7 the values of the E2 transitions for 155Eu 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 155Eu 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.

Table 7. Electric Transition Probability B(E2) in e2.b2 (Exp. data are taken from Ref. [30]).

3/21→5/21 3/22→7/21 5/22→5/21 5/22→7/21
Exp. IBFM-2 Exp. IBFM-2 Exp. IBFM-2 Exp. IBFM-2
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 155Eu 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 155Eu nucleus.

Table 8. Mixing ratio of 155Eu.

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 even-even 154Sm nucleus for ground, beta gamma bands are reproduced well. The energy spectra of the odd-even 155Eu nucleus can be reproduced quite well with the help of only two (and ) freely varying boson-fermion interaction parameters. The monopole interaction (A0) 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 154Sm 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 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.


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