Electromagnetic Transitions and Structures of even – even 76 − 90 Kr Isotopes within Interacting Boson Model

The Interacting Boson Model is applied to the even Kr 90 76− isotopes. Excitation energies, electromagnetic transition strengths, quadrupole and magnetic dipole moments, and δ (E2/M1) multiple mixing ratios, monopole transitions and mixed symmetry states have been described systematically. It is seen that the properties of low-lying levels in these isotopes, for which the comparison between experiment and IBM-2 calculations is possible, can be satisfactorily characterized by the Interacting Boson Model-2 (IBM-2).

Several some theoretical and experimental studies of even-even Kr isotopes have been carried out: Kaup and Gelberg [1] , have performed systematic analysis of Kr isotopes in IBM-2 , reproduced energy levels. Helleister and Lieb [2] study the energy levels and Electric transition probability and compare with experimental data. Meyer et al., [3] investigated the nuclear structure of the 82 Kr isotope , using in-beam spectroscopy studies and compare the experimental data with the results of IBM-2.
Glannatiempo et al., [4] studied the life-time of the + 2 0 level in the 80 Kr isotope and compare with the calculated value of IBM-2.
Deibaksh et al., [5] have performed the IBM-2 calculation on Kr isotopes , using two-different approaches. The first approach based on the energy of bosons that , ν π ε ε = and the second approach based on the difference the energy of proton boson and energy of neutron boson ν π ε ε ≠ . The results of IBM-2 in a good agreement with experimental data accept for the state + 3 2 . Giannatiempo et al., [6] have studied the symmetry property of the bands in [74][75][76][77][78][79][80][81][82] Kr isotopes by calculating F-spin and the d n component of the wave function of the states of these bands.
Shi Zhu-Yii et al., [7] have studied by using a microscopic sd IBM-2+2q.p. approach, the levels of the ground-band, γ -band and partial two-quasiparticle bands for [72][73][74][75][76][77][78][79][80][81][82][83][84] Kr isotopes are calculated. The data obtained are in good agreement with the experimental results, and successfully reproduce the nuclear shape phase transition of [72][73][74][75][76][77][78][79][80][81][82][83][84] Kr isotopes at zero temperature. The ground-state band is described successfully up to + = 18 π J and E x = 10 MeV. Based on this model, the aligned requisite minimum energy has been deduced. The theoretical calculations indicate that no distinct change of nuclear states is caused by the abruptly broken pair of a boson, and predict that the first backbending of Kr isotopes may be the result of aligning of two quasi-neutrons in orbit 2 / 9 g , which gains the new experimental support of the measurements of g factors in the 78-86 Kr isotopes.
Al-Khudair and Gui-Lu [8] studied the level structure of [76][77][78][79][80][81][82] Kr isotopes within framework of IBM-2 , and performed that the + = 2 π J (one-phonon mixed symmetry state) and + + + = 3 , 2 , 1 π J (two-phonons mixed symmetry states), and have been identified by analyzing the wavefunction of M1 transition. Turkan et al., in 2006 [9] have determined the most appropriate Hamiltonian that is needed for present calculations of nuclei about the 80 ≅ A region by the view of Interacting Boson Model-2 (IBM-2). After obtaining the best Hamiltonian parameters, level energies and B(E2) probabilities of some transitions in [88][89][90] Kr nuclei were estimated. Results are compared with previous experimental and theoretical data and it is observed that they are in good agreement. Turkan et al., in 2009 [10] studied The quadrupole moments of 76,78,80,82,84,88 Kr and 74,76,78,80,82 Se isotopes are investigated in terms of the interacting boson model (IBM), and it was found that a good description of them can also be concluded in this model. Before Electromagnetic Transitions and Structures of even-even 76−90 Kr Isotopes within Interacting Boson Model the quadrupole moments were calculated, the positive-parity states and electromagnetic-transition rates (B(E2)) of even-mass Kr nuclei have also been obtained within the framework of IBM. It was seen that there is a good agreement between the presented results and the previous experimental data. The quadrupole moments of the neighboring Se isotopes were also obtained and it was seen that the results are satisfactorily agree well with the previous experimental data. The aim of this work is to calculate the energy levels and electromagnetic transitions probabilities B(E2) and B(M1), multipole mixing ratios and monopole matrix elements in Kr isotopes, using the IBM-2, mixed symmetry states for these isotopes have been studied in this research, and to compare the results with the experimental data.

The Model
In the IBM-2 the structure of the collective states in even-even nuclei is calculated by considering a system of interacting neutron (ν) and proton (π) boson s( The Majorana term M πν shifts 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, we did not attempt to fit the parameters appearing in eq. (3) , but rather took constant values for all Kr isotopes. : The quadrupole moment Q ρ is in the form of equation (2), for simplicity, the χ ρ has the same value as in the Hamiltonian. This is also suggested by the single j-shell microscopy, π e and υ e are proton and neutron boson effective charges respectively. In general, the E2 transition results are not sensitive to the choice of e ν and e π , whether e ν = e π or not.
The reduced electric quadrupole transition probability B (E2) is given by: The M1transition operator is given : where ) ( π υ L L is the neutron and (proton) angular momentum operator where π g and υ g are the effective boson (proton, neutron) gyromagnetic -factors.
The reduced magnetic dipole transition probability B(M1) is given by: In the IBM-2 , the monopole transition (E0) operator is given by [18] which is related to the transition matrix ) 0 (E ρ by the expression [12]: where R 0 is the nuclear radius constant (R 0 =1.25*10 -15 m).
The Monopole transition probability is defined by :

Results and Discussion Hamiltonian Interaction Parameters
Since the Hamiltonian contain many parameters it is unpractical and not very meaningful to vary all parameters freely. Instead it is convenient to use the behavior of the parameters predicted by a microscopic point of view as a zeroth-order approximation. In a simple shell-model picture based upon degenerate single nucleon levels [13] the expected dependence of Here ) 0 ( ρ κ and ) 0 ( ρ χ are constants, and ρ Ω is the pair degeneracy of the shell. We see that while ρ κ has always the same sign, ρ χ changes sign in the middle of the shell. In realistic cases the estimates of eq.1 are expected to be valid only approximately. In our approach we have imposed somewhat weaker constrains on the parameters: (i) it is assumed that within a series of isotones (isotones) ) ( π ν χ χ does not vary at all; (ii) the parameters κ ε , and ν χ are assumed to be smooth functions of ( ) ν N .
Concerning the sign of ν χ and π χ a complication arises. From very simple microscopic consideration it follows that the s , χ (which also determine to a large extent the sign of the quadrupole moment of the first excited state + 1 2 are negative in the region where the valence shell is less than half filled (particle-boson) and positive in the region where the valance shell is more than half filled (hole-boson). Quantitatively, such a behavior was confirmed in other phenomenological calculations with IBM-2. For example in a study of the Ba isotopes with 72 < N < 80 good fit to the energy levels was obtained with 90 . 0 ≈ ν χ [14,15]. Since in the naïve It seen that parameters are constant or vary smoothly: within a series of isotopes π χ does not vary, the variation in ε is very small and there is a slight decrease of the value of κ for the lighter Kr isotopes. The change in character of the spectra through a series of isotopes is essentially due to two effects: (i) the decrease of the value of ν χ , and (ii) the decrease of the number of neutron bosons ν N . We note that the behaviors of ν χ κ ε , , and π χ is I qualitative agreement with microscopic considerations. It was found that both ν 0 C and ν 2 C vary for the isotopes. Such a behavior agree with the trend found in other regions [16]. The positive value of 2 ξ guarantees that no low-lying anti-symmetric multiplets occur for which there is no experimental evidence.  Using the parameters in table 1, the estimated energy levels are shown in the figures, along with experimental energy levels. As can be seen, the agreement between experiment and theory is quite good and the general features are reproduced well. We observe the discrepancy between theory and experiment for high spin states. But one must be careful in comparing theory with experiment, since all calculated states have a collective nature, whereas some of the experimental states may have a particle-like structure. Behavior of the ratio of the energies of the first + 1 4 and + 1 2 states are good criteria for the shape transition [17]. The value of R 4/2 ratio has the limiting value 2.0 for a quadrupole vibrator, 2.5 for a non-axial gamma-soft rotor and 3.33 for an ideally symmetric rotor. R 4/2 remain nearly constant at increase with neutron number. The estimated values change from isotope to another (see table 2 ), this meaning that their structure seems to be varying from axial gamma soft to quadrupole vibrator Since Kr nucleus has a rather vibrational-like character, taking into account of the dynamic symmetry location of the even-even Kr nuclei at the IBM phase Casten triangle where their parameter sets are at the transition region and closer to U (5) character and we used the multiple expansion form of the Hamiltonian for our approximation. The shape transition predicted by this study is consistent with the spectroscopic data for these nuclei. Kr are typical examples of isotopes that exhibit a smooth phase transition from vibrational nuclei SU (5) to soft triaxial rotors O(6). [9]  isotopes and we remark that the energy of the + 1 3 state is predicted systematically too high. This is a consequence of the presence of a Majarona term πν M in the Hamiltonian (eq. 3). We have chosen the parameters of the Majarona force in such a way that it pushes up states which are not completely symmetric with respect to proton and neutron bosons, since there is no experimental evidence for such states. However, experimental information becomes available about these states with mixed symmetry, this situation could possibly be improved. In the present case it would have been possible to further higher its energy by constant the value of where B(E2) is the reduced transition probability, π N and ν N are the boson numbers of proton and neutron respectively, is the total boson number. The difference between the effective charge and the charge of the single nucleon is referred to as the polarization charge. The value of effective charge may depend somewhat on the orbit of the nucleon. In particular, the polarization effect decreases when the binding energy of the nucleon becomes small.   Experimental data are taken from [8,18] It is well known that absolute gamma ray transition probabilities offer the possibility of a very sensitive test of nuclear models and the majority of the information on the nature of the ground state has come from studies of the energy level spacing. The transition probability values of the exited state in the ground state band constitute another source of nuclear information. Yrast levels of eveneven nuclei (I i = 2,4,6,.....) usually decay by E2 transition to the lower lying yrast level with In

Magnetic Transition Probability
The B(M1) reduced transition probabilities were calculated using eq.6, and the boson gyromagnetic factors  table 4. A good agreement between the theory and the available experimental data is achieved. As can be seen from the table yields to a simple prediction that M1 matrix elements values for gamma to ground and transitions should be equal for the same initial and final spin. Also the size of gamma to ground matrix elements seems to decrease as the mass number increases.
The results shows that the transitions between low-lying collective states are relatively weak. This is because of the increase of 2-There are evidences that M1 small mode exists in all spectra. 3-one cannot make decisive conclusions related to the agreement between theoretical and experimental data from the above table due to the lack of experimental data. However both experiments and theory predicts small M1 component which is due to symmetry and forbiddances of band crossing gamma transitions. 4-The γ γ → M1 matrix elements are larger than the g → γ M1 matrix elements by a factor of 2 to 3. Again, this agree qualitatively with the perturbation expressions derived in [22] . 5-The size of the g → γ M1 matrix elements seems to decrease with increasing mass. Specially, a change in g → γ M1 strengths occurs when the gamma band crosses the beta band.
The M1 properties of collective nuclei are certainly very sensitive to various, even small, components in the wave functions either of collective or non-collective character. In the Kr 90 88− isotopes it was shown that the inclusion of excitations across the major shell and two quasi-particle states is important. One excepts that also for Kr 90 88− isotopes (which are near to closed shell for neutron) similar effects come into play. As above analysis suggests they can manifest in considerable renormalization of IBM-2 boson g-factors from their slandered values. The magnetic dipole moment for first excited state is given by It is clear that the two effects contribute to the dependence of the magnetic moments on proton and neutron number: the dependence of π g and ν g on proton and neutron number and the variation of the matrix elements of the operator ) ( ν π L L with π N and ν N . As will be better shown below , the former effect is the related to the shell structure of the orbits, while the latter is related to the average number of proton and neutron boson taking part in the collective motion.
Experimental data are taken from [8 ,18] These are compared with experimental and theoretical results in table 5, where one can see good agreement with estimated and experimental values. The variations in sign of the E2/M1 mixing ratios from one nucleus to another for the same class of transitions, and within a given nucleus for transitions from different spin states, suggest that a microscopic approach is needed to explain the data theoretically. For such reason, the sign of the mixing ratio is not taken into consideration. Sign convention of mixing ratios has been explained in detail by Lange et al. [23] These results exhibit disagreement in some cases, with one case showing disagreement in sign. However, it is a ratio between very small quantities and any change in the dominator that will have a great influence on the ratio. The large calculated value for The E0 transition occurs between two states of the same spin and parity by transferring the energy and zero units of angular momentum, and it has no competing gamma ray. The E0 transition is present when there is a change in the surface of the nucleus. For example, in nuclear models where the surface is assumed fixed, E0 transitions are strictly forbidden, such as in shell and IBM models. Electric monopole transitions are completely under the penetration effect of atomic electrons on the nucleus, and can occur not only in + + → 0 0 transition but also, in competition with gamma multipole transition, and depending on transition selection rules that may compete in any ∆J = 0 decay such as a 2 + → 2 + or any I i = I f states in the scheme. When the transition energy greater than 2moc 2 , monopole pair production is also possible.
The E0 reduced transition probability is given in eq. 9. The parameters in equation 9, can be predicted from the isotope shift [20] (see table 7), since such data are not available for Kr isotopes, we calculate these parameters by fitting procedure into two experimental values of isotopic shifts (equation (2-59)). The parameters which were subsequently used to evaluate the ) 0 (E ρ -values were; β 0π = 0.062 eb , β 0ν = −0.021 eb and γ 0v =0.032 fm 2 . From the table 6, in general there is no experimental data to compare with the IBM-2 calculations.
The monopole matrix element is important for nuclear structure and the model predictions due to their sensitivity for the nuclear shape. We conclude that more experimental work is needed to clarify the band structure and investigate an acceptable degree of agreement between the predictions of the models and the experimental data.
We also find good agreement between the calculated and experimental values for isotope shifts for all mercury isotopes (table 7) but the isomer shift result for Kr isotopes is in poor agreement with the experimental value.

Mixed Symmetry States
One of the advantage of the IBM-2 is ability of reproducing the mixed symmetry states. These states are created by a mixture of the wave function of protons and neutrons that are observed in most even-even-even nuclei. This mixed symmetry states (MSS) has been observed in many nuclei. In more vibrational and In all Kr isotopes that the second + 3 states to be the lowest π I + = 3 mixed symmetry states with two phonon excitation. The low-lying levels with angular momentum greater than + 3 with a large mixed symmetry states component are predicted in this work.
The energy fit to several levels is very sensitive to the parameters in the Majorana term which also strongly influence the magnitude and sign of the multipole mixing ratios of many transitions. In particular we find that the calculated energies of a number of states are affected in a very similar way and these might be considered to have a mixed-symmetry origin, or contain substantial mixed-symmetry components. Those with a mixed-symmetry origin have no counterpart in IBM-1. The energy dependence of the + 2 2 and + 4 2 levels is consistent with the mixed-symmetry character of the + 3 2 level being shared with neighboring states [24,25].
The influence of the parameters on these states is shown in table 1. The 2 ξ term strongly affects the energies of all of the levels considered to have a mixed-symmetry character or to contain mixed-symmetry components . In obtaining this plot the 1 ξ and 3 ξ terms were maintained at their best-fit values. The mixing ratio data, discussed in the above section have a strong dependence on Most experimentally observed low-spin levels, apart from + 1 states below.5 MeV; have their counterpart in the IBM-2 level spectrum although the energy match is not good in every case. It also appears that we may identify the members of the family of mixed-symmetry states corresponding to the [N-1,1] representation [12]. The small E2/M1 mixing ratios are consistent with this interpretation but level lifetimes are required for a firmer identification.
In Kr isotopes, all hitherto discovered MSS have been reviewed in [9]. It has been shown that the lowest lying MSS is the one quadrupole phonon MSS labeled as +  2 states indicate that substantial E0 components occur in these decays from mixed-symmetry states. The E0 matrix element describing such decay is proportional to π β 0 and ν β 0 , although the β values are small, their sign difference results in the E0 matrix being greatest.

Conclusion
We have presented results of calculations of the properties of the 76-90 Kr isotopes found in many cases good agreement between our calculations and experiment. However, we have also found that 76-90 Kr isotopes, that there are several 0 + states whose properties cannot be reproduced by our calculations. These intruder states are presumably associated with other with the other low-lying degrees of freedom. Since they appears to be present also in other mass regions, it is clear that they must be explicitly included in the calculations if one wants to describe all observed low-lying states.