Energy Levels and Electromagnetic Transition of Some Even-Even Xe Isotopes

In this work, the energy levels electromagnetic transition B(E2) and B(M1), branching ratios, mixing ratios and electric quadrupole moment of even-even Xe isotopes have been investigated using Interacting Boson Model (IBM-1). The results were compared with some previous experimental and theoretical values, it was seen that the obtained theoretical results are in agreement with the experimental data.


Introduction
The interacting boson approximation represents a significant step towards 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 recently found application to problems in atomic, molecular, and high-energy physics (Firestone, 1996, Pan & Draayer, 1998).The IBM-1 is a valuable interactive model developed by Iachello andArima (Iachello &Arima, 1974, Arima &Iachello, 1975).It has been successful in describing the collective nuclear structure by prediction of low lying states and description of electromagnetic transition rates in the medium mass nuclei.IBM-1 defines a six-dimensional space described in terms of the unitary group, U(6).Different reductions of U(6) give three dynamical symmetry limits known as harmonic Oscillator, deformed rotator and asymmetric deformed rotor which are labeled by U(5), SU(3) and O(6), respectively (Kumar et al., 2010, Cejnar et al., 2010).
Xenon isotopes belong to a very interesting but complex region of the periodic Table known as the transition region.The Xe isotopes can exhibit excitation spectra close to the O(6) symmetry.After some theoretical investigations it was concluded that the xenon isotopes should lie in a transitional region from U(5)-to an O(6)-like structure as the neutron number decreases from the closed shell N=82 (Casten & Von Brentano, 1985).The chain of 120-134 Xe isotopes are interesting because of the existence of transitional nuclei where the nuclear structure changes from rotational to vibrational shapes.Many authors studied this area of isotopes experimentally and theoretically.IBM-1 model has been used in calculating the energy of the positive parity low-lying levels of Xe series of isotopes (Dilling et al., 2002).For the neutron number 66≤ N ≤72, the energy ratio E4 + /E2 + is almost 2.5 already pointing to γ-soft shapes.In recent years many works have been done on the structure of Xenon isotopes; Kusakari and M. Sugawara (Kusakari & Sugawara, 1984) calculated the energy levels, the back-bending in the yrast bands , reduced transition probabilities for the positive-parity states of 122-130 Xe within the framework of the interacting boson model.Xing-Wang Pan et al (Pan et al., 1996) and Al-Jubbori (Al-Jubbori et al., 2016) described the low-lying energy levels for even-even and even-odd nuclei 126-132 Xe and 131-137 Ba by a unified analytical expression with two (three) adjustable parameters of the fermion model.Wang Bao-lin (Bao-lin, 1997) calculated the triaxial deformation parameters β and γ in the O(6)-like nuclei by comparing quadrupole moments between the interacting boson model IBM and the collective model studied.A.D. Efimov et al (Efimov et al., 2002) studied B(E2; 0 → 2 ) values in the intruder bands of 112,114,116,118 Sn within the framework of the interacting boson model (IBM1) and comparison with the ground state bands in the even-mass Xe isotopes to find a similarity not only for the energy spaces, but for the B (E2) values as well.B. Saha et al (Saha et al., 2004) calculated the B(E2) and B(M1) values for the isotope of 124 Xe within the framework of the interacting boson model (IBM1).
Nureddin Turkan (Turkan, 2007) described the quadrupole collective states of the medium-heavy nuclei within the framework of the interacting boson model (IBM) and calculated the energy levels and B(E2) values for the even-even 122-134 Xe isotopes.The results were compared with the previous experimental and theoretical data and it has been observed that they are in good agreement.Ismail Maras et al (Maras et al., 2010) studied the ground state, quasi beta and quasi gamma band energies for the 114,116,118,120 Xe isotopes using the interacting boson model (IBM) and calculated the energy levels using PHINT and NPBOS program codes.M. A. Jafarizadeh et al (Jafarizadeh et al., 2013) studied the properties of 114-134 Xe isotopes in the U(5)↔SO( 6) transitional region of IBM and calculated the energy levels and B(E2) transition rates.Zhang Da-Li and Ding Bin-Gang (Da-Li & Bin-Gang, 2013) studied the structure evolution of the 124-134 Xe isotopic chain in the framework of the proton-neutron interacting model (IBM2), they have been calculated the B(E2) transition branching ratios, and the M1 excitations.L. Coquard et al (Coquard et al., 2011) described the low-lying collective states in 126 Xe and calculated the B(E2) values within the framework of the IBM which show a good agreement with the measured values.L Prochniak (Próchniak, 2015) studied the collective properties of the even-even 118−144 Xe isotopes within a model employing the general Bohr Hamiltonian, calculated the low energy spectra and B(E2) transition probabilities and compared the results with the experimental values.
The aim of the present work is study the energy levels, B(E2) and B(M1) values and explore the description of E2/M1 mixing ratios using IBM-1 for 120-128 Xe isotopes.Furthermore, calculate the value of electric quadrupole (Q J ) of these isotopes within the framework of the IBM-1 and comparing the results with the most recent experimental data and previous studies.

Method
The IBM-1 of Arima and Iachello (Arima & Iachello, 1976) 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 number N of bosons.Each boson could occupy two levels one with angular momentum (L = 0) (s-boson) and another, usually with higher energy, with (L = 2) (d-boson) (Sharrad et al., 2012).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.The IBM-1 Hamiltonian can be expressed as (Casten & Cizewski, 1978, Abrahams et al., 1981, Iachello & Arima, 1987).
This Hamiltonian contains two terms of one body interactions, (ε s and ε d ), and seven terms of two-body interactions where ε s and ε d are the single-boson energies, and C L , v L and u L describe the two boson interactions.However, it turns out that for a fixed boson number N, only one of the one-body terms and five of the two body are terms independent, as it can be seen by noting N = n s +n d .

Results
The obtained results can be discussed separately for energy levels, transition probabilities B(E2) and quadrupole moment Q J , B(M1) values and mixing ratio.

Energy levels
The γ-unstable limit of IBM-1 has been applied for 120-126 Xe nuclei due to the values of the experimental energy ratios (E2 :E4 :E6 :E8 =1:2 .5:4.5:7).Therefore, these nuclei have γ-unstable dynamical symmetry O( 6) with respect to IBM-1.The adopted Hamiltonian is expressing as (Casten & Warner, 1988, Iachello, 2001): The Xenon isotopes have a number of proton bosons 2, and number of neutron bosons varies from 7 to 10.The parameters value used in the present work are presented in Table 1.

Xe
The calculations of the g-bands, β-bands and γ-bands are compared with the experimental data (http://www.nndc.bnl.gov/chart/getENSDFDatasets.jsp,Kitao et al.,2002, Tamura, 2007, Katakura & Wu, 2008, Katakura & Kitao, 2002) for all isotopes under study, and as it given in Table 2.In this Table, one can see a agreement between experimental data and the IBM-1 calculations.Levels with '*' correspond to cases for which the spin and/or parity of the corresponding states are not well established experimentally.
A measure of the deviation of a charge distribution from a spherical shape is the electric quadrupole moment of the distribution.Then the quadrupole moment (Q J ) is an important property for nuclei and can be determined if the nucleus spherical (Q = 0), deformed oblate (Q<0) or prolate (Q>0) shapes.The electric quadrupole moments of the nuclei can be derived from the transition rate B(E2,J i → J f ) values according to Eq. ( 5) (Iachello & Arima, 1987, Kassim & Sharrad, 2014): where: J is the total angular momentum.The calculations of the electric quadrupole moment Q J within the framework of IBM-1 is shown in Table ( 5) for 120-126 Xe nuclei.This Table shows that the Q2 has a negative value for all interested nuclei.The calculated Q4 has been positive value.b- (Kitao et al.,2002) c- ( Efimov et al., 2002) d- ( Tamura, 2007) e- ( Katakura & Wu, 2008) f- ( Coquard, 2010) g- ( Katakura & Kitao, 2002)

Reduced transition probabilities B(M1) and E2/M1 mixing ratios
Similarly, from Eq. ( 3), the M1 operator would be just β1[ × ](1).For calculating M1 transitions, the IBM-1 rule must be extended to second-order in the U(6) generators (Abrahams et al.,1981, Casten & Warner, 1988).The most general second-order M1 generator can then be written as (Casten & Warner, 1988, Arima & Iachello, 1979): where is the effective boson g factor and can be written as (Arima & Iachello, 1979): = Z/A, (8) where Z and A are the atomic and mass numbers, respectively.Indeed from Eq. ( 7), N is the number of bosons, is angular momentum, ( 2) is general matrix element of the E2 transition and is d-boson number operators.The first term of Eq. ( 7) is diagonal and does not contribute to transitions.The last terms yield the M1 matrix element can be written as (Casten & Warner, 1988, Warner, 1981): where ̀ and denote additional quantum numbers.In the first term the spin factors given separately in Refs.(Abrahams et al.,1981, Casten & Warner, 1988) for the cases L → L±1 and L → L, and it has been combined into the single factor ƒ ƒ , given by: The second term of Eq. ( 9) only contributes to transitions between states of the same spin, since the corresponding operator in Eq. ( 7) is diagonal in .In the O(6) symmetry of the IBM-1, the operator contributes to both diagonal and off diagonal matrix elements.The matrix elements between the representations σ = N and σ = N −2 can be written as (Iachello & Arima, 1987,Casten & Warner, 1988): For L±1→ L transitions, Eq. ( 9) leads to a particularly simple expression for the reduced 2/ 1 mixing ratio, namely The reduced mixing ratio is related to the quantity normally measured, δ( 2/ 1) by (Casten & Warner, 1988): ( 2/ 1) = 0.835[ /( 1)] ( 2/ 1) (13) where is in MeV and ( 2/ 1) is in eb/µN.The spin dependence of Eq. ( 10) has already been derived in the framework of the geometrical model by Grechukhin (Grechukhin, 1963) in an analogous way by expressing the relevant part of the M1 operator in terms of the quadrupole coordinates of the nuclear surface.In the IBM-1 model, the value of the constant B in Eq. ( 9) and the validity of the spin dependence can be investigated by looking at empirical values of [ ( 2/ 1) ƒ ƒ −1 .To L → L transitions, the inclusion of both s and d bosons in the IBM-1 formalism gives rise to the additional contribution in Eq. ( 12) from the operator, and the L → L, M1 matrix element thus, in principal, depends on the relative sizes and Signs of the two terms.

Participant Flow
For experimental and quasi-experimental designs, there must be a description of the flow of participants (human, animal, or units such as classrooms or hospital wards) through the study.Present the total number of units recruited into the study and the number of participants assigned to each group.Provide the number of participants who did not complete the experiment or crossed over to other conditions and explain why.Note the number of participants used in the primary analyses.(This number might differ from the number who completed the study because participants might not show up for or complete the final measurement.)

Conclusion
The energy levels and electromagnetic transitions of moderately deformed even-even Xe nuclei have been studied in the present work.It can be described in terms of the O(6) limit in the framework of the IBM.The B(E2) , B(M1) transition probabilities, the E2/M1 mixing ratios, branching ratios, and electric quadrupole moment Q J for the even-even 120-126 Xe isotopes with neutron numbers between 66 and 72 are calculated using the IBM.The calculated B(E2) and B(M1) values all in good agreements with experimental data.

Table 1 .
Adopted values for the parameters used for IBM-1 calculations.All parameters are given in MeV, except N. ).

Table 5 .
The electric quadrupole moment Q J in eb unit.

Table 6 .
The parameters of T(M1) used in the present work.All parameters are given in (µ N ), except N.