1. Introduction

The structure of even-even Pt isotopes with neutron number has provided a challenge to the theoretical models. These nuclei show a few important features, such as the systematism of 2⁺ states and its connection with the systematism of the first excited 0⁺ with respect to the 2⁺ states. These features have been investigated by algebraic models like the IBM-2, which distinguishes the proton-pairs from neutron-pairs, including mixing the two different boson configurations [1-7]^[1]. The interacting boson approximation has been rather successful at describing the collective properties of several mediums and heavy nuclei. In this model, the low energy state of even –even nuclei is described in terms of interaction between s(L=0) and d(L=2 ) bosons. Through adjusting a small number of parameter, it produces the majority of the low-ling state of such nuclei ^{[1, 2]}. The early version of the interacting boson model called IBM-1, where no distinction is made between protons (s_π, d_π), neutron bosons (s_ν,d_ν), and the number of bosons taken to be the number of nucleons outside the closed shell divided by two. The second version of the Interacting Boson Model ( IBM-2 ), distinguishes between protons (s_π,d_π) and neutron bosons(s_ν,d_ν). While the first version has a symmetry limitation, like; dipole magnetic transition, mixed symmetry states and electric monopole transitions, the second version has no limitation ^{[3, 4]}.

The aim of the present investigation about platinum A=192-202 nuclei which lies in the region between U(5) and O(6) is to explore deeply in the structure of these isotopes and not just in the energy levels. We made calculations for the spectra, E2, M1 and E0 transition probabilities plus the mixing ratios.

2. The Interacting Proton and Neutron Model

The IBM-2 Hamiltonian is written as ^{[5, 6]};

(1)

Where is the energy difference between s and d boson, is the number of d bosons, where ρ corresponds to π (proton) or ν ( neutron) bosons, the second term denotes the quadrupole – quadrupole interaction between proton and neutron with strength κ, where the quadruple operator is define as ^[7].

(2)

The term V_ππ and V_νν which corresponds to the interaction between like – boson, are sometimes included in order to improve the fit to experimental energy spectra and they are expressed as ^[8].

The terms V_ππ and V_νν which correspond to the mutual interaction are sometimes - like boson- included in order to improve the fit to experimental energy spectra and they are expressed as.

(3)

The Majorana term contains three parameters and can be written as;

(4)

where the Majorana term plays a great role in studying the mixed symmetry states of some excited energy levels.

3. IBM-2 Parameters

The ^192-202Pt isotopes have Z=78, the numbers of boson proton N_π = 2 and neutron boson N_v vary from 6 to 1 boson hole. The parameters used for the best fitting are listed in Table 1. We used the χ_π = -0.88, ζ_1,3 = -0.083 and CL_π(0,2,4)=0 for all isotopes and the observed spectra are closed to experimental data ^[9].

4. Energy Spectra

The energy spectra of ^192-202Pt isotopes are produced by using the parameters in Table 1. But before going into details of our main prospective, one should decide to which limit of the IBM-1 these nuclei belong. The ratio is the key of this problem, where this ratio is equal to 2, 2.5 and 3.3 for U(5), O(6) and SU(3) respectively. Figure 1 shows the ratio for ^192-202Pt isotopes as a function of neutron number which indicates that theses isotopes belong more closely to O(6) symmetry group, specially the case of ¹⁹⁶Pt. The same results are also predicted by the IBM-2 results.

Table 1. IBM-2 Parameter for even-even ^192-202Pt isotopes

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Figure 1. The experimental ratio in comparison with the IBM-2 output and the IBM-1 limits

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5. Electromagnetic Transition

In IBM-2, the E2 transition operator is given by ^{[10, 11]}.

(5)

where e_π and e_ν are boson effective charges, depending on the boson number N_ρ, these parameters are free and can take any value to fit the experimental data. In the present work, the effective charge of proton e_π = 0.1 (eb) and neutron e_ν = 0.17 (eb) (derived from experimental data of . The calculated reduced electric quadrupole probabilities B(E2)of ^192-202Pt isotopes are shown in the Table 2. The calculated values are acceptable in comparison to the available experimental data and they look to have a good systematism.

In order to calculate M1 transition probability, one should estimate the effective factors for proton and neutron. It is found that in this case Sambataro relation ^{[12, 13]} is useful and written as;

Table 2. The B(E2) transition of ^192-202Pt. isotopes, units (eb)²

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Table 3. The B(M1) transition of ^192-202Pt. isotopes, units (µ_N)²

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(6)

where , are -factors of nuclear proton and neutron respectively. The total – factor associated with magnetic momentum, is and in this work we use the experimental value of magnetic momentum for the 2⁺₁ state to estimate the g factor, where = 0.860 (µ_N).It is found that the estimated values are =0.315(µ_N) and = 0.398(µ_N), and () = 0.083(µ_N). The M1 operator is obtained by making l = 1 in the single boson operator of the IBM-2 and it can be written as^[14];

(7)

The calculated values for B(M1) is acceptable to some extent comparing with the available experiments values which are usually very few in nuclear data sheet. The B(M1) transition of ^180-190Pt. isotopesis calculated and presented in Table 3.

After calculating the matrix elements of B(E2) and B(M1) of gamma transition it is possible to compare the strength of E2 and M1 transition in terms of the multipole mixing ratio (δ) which written as ^[15];

(8)

is the transition energy between the two states units (MeV) and Δ(E2/M1) is the ratio between reduced matrix element for E2 and M1 transition which is expressed in the form ^[16];

(9)

In Table 4 we present the values of the mixing ratio for the selected platinum isotopes in comparison with the available experimental values. It is seen that both the magnitude and sign of are correctly obtained for most of the selected isotopes. The agreement between experimental values and those of IBM-2 looks acceptable in comparison withthe available experimental data.

Table 4. Mixing ratios( eb/µ_N )for ^192-200Pt

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Monopole transitions is caused by anelectromagnetic interaction between the nuclear charge and the atomicelectron penetrating the nucleus. The E0 transition occurs between two states of the same parity and spin by transferring the energy and zerounit of angular momentum. The transitions can occur not only in 0⁺→ 0⁺ transition but also, in competition with gammamultipole transition and depending on transition selection rules maycompete in an ∆I =0 decay such as a 2⁺ → 2⁺. For transitions energies greater than 2m₀c², monopole pairproduction is also possible. The reduced transitions probability written as ^[17]

(10)

Where e in the electronic charge, R is the nuclear radius and ρ²(E0) is the matrix element transition is given by;

(11)

where are the parameters estimated by using the fitting in isomer shift, which is a measure of the difference between the excited state and the ground state in a given nucleus: ^[18]. In this work, the values of these parameters are (0.076, -0.026) fm² for proton and neutron respectively. To find the strength of E0 transition related to E2 transition,one must calculate the X(E0/E2) ratio which can be written as;

(12)

Where, I_f = I_f ' for, I_i = I_f ≠ 0, and I_f ' =2 for, I_i = I_f = 0.

Table 5. The monopole transition matrix elementρ(E0) and X(E0/E2) for ^192-202Pt

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This ratio shows the strength of the transition between and , where it is noted that the IBM-2 calculated values are not entirely consistent with the experimental values available, and the reason is due to the strength of the transition between E2 and E0, as well as the fact that the difficulty of defining a unified parameters for () give us the theoretical values in (IBM-2) closer to the experimental ones. The experimental values available for ρ(E0) and ( are also very few. Table 5 shows the monopole transition matrix elements ρ(E0) and X(E0/E2) for ^192-202Pt. isotopes.

6. Conclusion

In this paper we have calculated the microscopic properties of ^{192- 202}Pt isotopes in details. We have presented results for energy spectra and energy level ratio to indicatethe shape phase transition in this chain of isotopes by using the IBM-2. It is found that most of these isotopes belong to _unstable limit of IBM-1 and one of them is a perfect O(6) nucleus (¹⁹⁶Pt). On the other hand, the E2 and M1 reduced transition probabilities are calculated and compared with the available experimental data. Moreover, the calculations are extended to δ(E2/M1) and X(E0/E2) ratios and showed acceptable agreement with the available experimental data, specially the sign of the mixing ratio.

Acknowledgments.

The authors of this work would like to express many thanks to the Department of Physics, the College of Sciences, and the University of Babylon for their support.

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