Configuration Mixing in Pt Nuclei within Interacting Boson Model-2

This article provides to descript a consistent of the even-even 182-200 Pt isotopes. This has been achieved using the interacting boson model-2 (IBM-2) and including configuration mixing (IBM-2 CM). Our attention is paid to describe the nuclei shape and to their connecting with shaping coexistence phenomenon. Ten isotopes are studied, ranging from the middle of the neutron shell to very near the doubly closed shell at 208 Pb. The same Hamiltonian is used for all the nuclei studied, with parameters which are constant or smoothly varying. In this study, we showed the transition between more axially symmetric deformed features of light Pt isotopes to γ-unstable and vibrational isotopes (near spherical shape) for 198-200 Pt isotopes.


Introduction
Some nuclei near closed shells appear to have both the vibrational structure expected for a near-spherical shape, and rotational structure, which is typical of deformed nuclei [1]. This phenomenon of shape coexistence involves two configurations of the nucleus which have different numbers of active nucleons. In an IBM description, the two configurations have different boson numbers; N v being the same but N π different, or vice versa. The most common situation involves a difference in N π (N v ) of two bosons between the normal configuration and the so-called intruder configuration, corresponding to a pair excitation across a shell or sub-shell gap [1] [2].
Shape coexistence in atomic nuclei has become a very active field of research during the last decades and clear signals of its existence have been obtained at and near proton or neutron closed shells [1] [2] [3], more in particular in light 2. The Interacting Boson Model-2 (IBM-2)
The quadrupole moment operator, in the IBM-2 model, has the form: where ρ π = or ν , Q ρρ is the quadrupole deformation parameter for pro- The Majorana interaction is represented the term M πν , that accounts for the symmetry energy and shits the states with mixed proton-neutron symmetry state with respect to the fully symmetric ones which affects only the relative location of the states with mixed symmetry, it is with respect to the totally symmetric states. For the reason that little experimental information is familiar states with mixed symmetry that has the form: ( ) (

Electromagnetic Transitions and Quadrupole Moments in IBM-2
Generally, the E2 transition operator of one-body in the IBM-2 is which Q ρ is in the form of Equation (3). For simplicity, the ρ χ has the same value as in the Hamiltonian. Also, one suggests it by the single j-shell microscopy. Generally, the E2 transition results are not sensitive to the choice of e ν and e π , whether e e In IBM-2, the electric quadrupole moment is written as: One can calculate IBM-2 eigen functions and energy eigenvalues are usually achieved numerically by the program code NPBOS [8]. Then, the result of eigenvectors can be calculated transition rates and related properties using the program code NPBTRN [8]. The relationship is between the parameters of Equation (2).

Configuration Mixing in Interacting Boson Model-2 (IBM-2 CM)
Configuration mixing can be treated in the IBM-2 using a technique developed by Duval and Barrett [9]. Separate IBM-2 calculations are achieved for the two configurations and the results are then mixed this is done by the interaction configuration mixing can be remedied in the IBM-2 by a technique method developed by Duval and Barrett [9]. Separate IBM-2 calculations are done for the two configurations and the results are then mixed using the interaction π π π π π π π π α β where the intruder configuration is assumed to involve the proton shell. There are three parameters in the mixing calculation, the mixing strengths α and β in Equation (23), and the pair excitation energy, ∆ , which gives the relative energies of the two unperturbed configurations. The total mixing Hamiltonian is then given by where H 1 (H 2 ) is the IBM-2 Hamiltonian for the first (second) configuration, as given by Equation (11), and an amount fl has been added to the energies of the second configuration.
The mixed wave functions are used to calculate B (E2) values of observed transitions and quadrupole moments. The E2 transition operator is given by [9]: where Q ρ were defined in Equation (3)  For a mathematical simplicity, the neutron boson and proton boson effective charges are often taken to be equal, and the parameters π χ and ν χ (in intruder configuration) are taken to be the same as the Hamiltonian parameters π χ and ν χ respectively. The T(E2) transition operator can then be written as: When two configurations are present, this operator becomes [9]: where ρ π = or ν and i denotes the configuration. The effective charges of the two configurations are not the same, in general.

Choice of Parameters
The normal configuration for platinum isotopes involves 2 N π = (sometimes denoted as 2π, two proton boson holes), counting from the Z = 82 closed shell.
The neutron configuration for 196 78 118 Pt for example, is 2 N ν = (four neutron boson holes), counting from the N = 126 closed shell. The vibrational spectra can be calculated by diagonalizing the IBM-2 Hamiltonian, (Equation (2)), in the space of two proton and N ν neutron s and d bosons. In order to describe the rotational states, an alternative configuration must be specified and a separate set of IBM-2 calculations made, based on that configuration. The alternate configuration used for the [186][187][188][189][190][191][192][193][194][195][196][197][198][199][200] Pt isotopes involves a two-particle-four-hole excitation in the shell model proton space [1] [9] [10]. This corresponds to two proton boson particles and two proton boson hole in the IBM-2 space. For simplicity, the proton boson particles and hole are treated equivalently, even though the underlying fermion pair degrees of freedom originate in different major shells.
The IBM-2 calculations have been done in model spaces with ( ) 2 2 N π π = and ( ) 4 4 N π π = to describe the vibrational and rotational states, respectively, the two calculations are combined using Equation (11).
In the phenomenological calculations the parameters appearing in the Hamiltonian ((Equation (2) and Equation (11)) in two configurations may in general depend both on proton ( N π ) and neutron ( N ν ) boson number. Guided by the microscopic theory as we have assumed that only 2 4 , π π ε ε and 2 4 , π π κ κ depend on N π ( ) 2 ,4 π π and N ν i.e., ( ) . Thus a set of isotopes, (constant N ν ) have the same value of ν χ , while a set of isotones, (constant N π ), have the same value of π χ . This parameterization allows one to correlate a large number of experimental data. Similarly, when a proton-proton V ππ and neutron-neutron V νν , interaction is added, the coefficients L C are taken as  Table 1 and Table 2).
The values of the parameters used for the present calculations are given in Table 1. The value of the parameter (boson energy) ε for the 4 N π = configuration, 4π ε , is constant for all isotopes and the values of ε for the 2 N π = configuration, 2π ε , are nearly constant. The quadrupole-quadrupole interaction strengths κ trends for ( 2π ) both configurations follow the microscopic predictions [9].
The values of the parameter ν χ used for Pt isotopes are the microscopic predictions from Bijker et al., 1980 [11]. They were reported only for neutron . The values of ν χ for Pt isotopes were determined by extrapolating the microscopic trend to larger neutron number. This was done in Table 1. IBM-2 Hamiltonian parameters for the Pt isotopes (normal configuration 2 N π = ). All energies are in MeV; 2π π χ and 2π ν χ are dimensionless. 0.00 C C C π π π π π π = = = . Journal of Applied Mathematics and Physics a manner consistent with the phenomenologically determined ν χ values which were used for the neighboring platinum and osmium isotones [11]. The same values of ν χ are used for both configurations. The emphasis in this work is on describing overall trends with constant or smoothly varying parameters of the Hamiltonian (Equations. (2), (11)), rather than obtaining the best possible fit to the experimental data for each nucleus. This is done in an effort to find a set of IBM-2Hamiltonian parameters which is appropriate for the entire isotopic chain.
The normal configuration for platinum involves 2 N π = (sometimes ( 2π ) denoted as ( 2π ), two proton boson hole), counting from the Z = 82 closed shell. N ν = neutron s and d bosons. In order to describe the rotational states, an alternative configuration must be specified and a separate set of IBM-2 calculations made, based on that configuration. The alternate configuration used for the Pt isotopes involves a four-particle-four-hole excitation in the shell model proton space. This corresponds to two proton boson particles and two proton boson hole in the IBM-2 space. The two configurations are depicted schematically energy levels in Table 2. For simplicity, the proton boson particles and hole are treated equivalently, even though the underlying fermion pair degrees of freedom originate in different major shells.

Energy Spectra
The configuration mixing calculations are done using the computer code NPMIX [10], which calculates the energy eigenvalues and eigenfunctions. The computer codes NPBEMX and BEMIX [10] are subsequently used to calculate matrix elements for transition rates and other properties. The mixing between the two configurations ( 2π ) and ( 4π ) is apparent in the experimental data shown in Figures 1-3. Consider, for example, the 1 4 + , 2 0 + and 2 4 + , 2 0 + states in [186][187][188][189][190][191][192][193][194][195][196] Pt isotopes. In the latter, the vibrational (2) state is lower in energy than the rotational ( 4π ) 1 4 + and 2 4 + states. The states are close in energy and mutually repel, with the vibrational 4 + state being lower in energy and the rotational 4 + state higher in energy than would be expected without configuration mixing.
In the second configuration ( 4π ) configuration interaction parameters In the IBM-2-CM approach, the lightest Pt isotopes, are deformed ( 172-178 Pt).
In 186 Pt isotope a prolate shape and a γ-soft minimum coexist, but a well-deformed prolate minimum quickly develops in 180 Pt isotope, becoming the most pronounced prolate minimum at the mid-shell, i.e., in 182 Pt isotope with the prolate shape remaining well pronounced up to 186 Pt isotope. Moving towards heavier mass Pt isotopes, γ-flat energy surfaces start to develop. For 188 Pt isotope, a much extended energy surface develops in the γ-direction (deviation from symmetry axis), becoming completely γ-unstable when reaching [190][191][192][193][194][195][196] Pt isotopes, the 198-200 Pt isotopes tends to spherical shape because approach to magic number. We show in Figure 1 the evolution of the energy of the ground state band ( 1 2 + , 1 4 + , 1 6 + , and 1 8 + ) increases toward the middle of the major shell with the number of the valence neutrons and remain almost constant for 180 186 A ≤ ≤ isotopes. Although these tendencies are well reproduced, the rotational properties or rotational features are somewhat enhanced in the calculated levels for 180 184 A ≤ ≤ , which are slightly agreement in energy with the experimental data. From both results, the IBM-2 CM and the experimental data, we observed the fingerprints for structural evolution with a jump between 186 Pt and 188 Pt isotopes, also the shape of the nucleus changed from prolate ( 186 Pt isotope) to oblate ( 188 Pt isotope) deformation, this case is called phase transition. The 188 Pt isotope takes the oblate shape because the yrast states gradually departure to neutron closed shell.
One can also find signatures for a shape or phase transition in the systematics of the quasi-β-band levels shown in Figure 2. In 180-186 Pt isotopes configuration mixing, the 2 0 + baldhead and the state 3 2 + nearly constant in both IBM-2 data and experimental data. The two levels are pushed up rather significantly from Journal of Applied Mathematics and Physics  - [21]. 186 Pt isotope to 188 Pt isotope. Consistently with the systematics in the ground-state band. The calculated 2 0 + and 3 2 + states are higher than, but still follow, the experimental data. The 1 3 + state lies close to the 2 4 + state. However, this calculation predicts that this trend persists even for [188][189][190][191][192][193][194][195][196][197][198] Pt isotopes, whereas the energy spacing between the experimental 1 3 + and 2 4 + states for these isotopes is larger. Similar deviation occurs for high spin states. This means that our calculations suggest the feature characteristic of the O (6) symmetry (γ-unstable).
The γ-band states shown in Figure 3, we can see the good agreement between , and thus appear to be in between the O(6) symmetry and triaxial rotor symmetry. The deviation of the γ -band structure seems to be nothing more than a consequence of an algebraic nature of the IBM-2, and indeed has also been found in existing phenomenologically calculations of IBM-2.
A characteristic feature of the γ-unstable limit of the IBM-2 is a bunching of γ-band states according to 2 + , (3 + , 4 + ), (5 + , 6 + ), ..., that is, 3 + and 4 + are close in energy, etc. This even-even staggering is observed in certain O(6) nuclei but not in all and in some it is, in fact, replaced by the opposite bunching (2 + , 3 + ), (4 + , 5 + ), ... which is typical of a rigid triaxial rotor [22]. From these qualitative observations it is clear that the even-even γ-band staggering is governed by the γ-degree of freedom (i.e., triaxiality) as it changes character in the transition from a γ-soft vibrator to a rigid triaxial rotor. The energy surface becomes rather flat, evolving towards a spherical minimum at 200 Pt and beyond. The possibility of triaxial deformation was not considered (tends to spherical shape).

Electric Transition Probability B(E2)
Calculations of electromagnetic properties give us a good test of the nuclear models prediction. The electromagnetic matrix elements between eigenstates were calculated using the programs NPBTRN and NPBEMX for IBM-2 and IBM-2 CM model respectively.
From Equations (7), we note that an E2 transition mainly depends on identifying proton and neutron bosons effective charges e π and e ν . These isotopes lying in region between SU(3) limit (rotational nuclei) and O(6) limit (γ-soft nuclei), therefore, the relationship between ( e π , e ν ) and the reduced transition probability B(E2) for rotational limit SU(3) and γ-soft limit O (6)  is the experimental reduced transition probability from the first excited states ( 1 2 + ) to the ground state ( 1 0 + ), N is the total number of bosons.
The relations (26) and (27), was used to estimate the effective boson charges for proton and neutron bosons ( e π , e ν ). In these calculations, we use the following criteria to determine the effective charges.
0.173 e.b e π = is a constant throughout the whole isotopic chain and the e ν changes with neutron number. This is true if the neutron (proton) interaction does not depend on the proton (neutron) configurations. The values of e π and e ν are determined by fitting to the five ( )  Pt and 196 Pt isotopes from the first configuration ( 2 N π = ). They are given in Table 3. For the configuration mixing, the effective charges for bosons is evaluated in the same manner in normal configuration, For simplicity, The ratio of the two quadrupole effective charges, 4 2 e e in Equation (25) is taken to be the same quadrupole interaction strengths, as the ratio of the corresponding 4 2 π π κ κ , for each isotope (see Equation (12) and Table 1, Table 2). This is reasonable, since the effective charge and the strength of the quadrupole interaction are both proportional to the mean square proton radius. Thus, the only new parameter needed to determine the reduced transition rates is 2 e . The value for the Pt isotopes was determined by fitting of ( ) In Table 4  values, which are of the same order of magnitude and display a typical decrease towards the middle of the shell.
As a consequence of possible M1 admixture the ( ) quantity is rather difficult to measure. There is no experimental data to compare the values  Table 4. This quantity is rather small since this transition is forbidden in all three symmetries of the IBM-2.
As a consequence of possible M1 admixture the ( ) are small values in sometimes because these transitions between different bands (cross over transitions) and the selection rules which determine these transition.
In Table 5, the quadrupol moment, qualitatively, for first excited state Q + mean a negative intrinsic quadrupole moment (for ground state 0 Q ). For the beta band, a negative ( ) 1 2 Q + means a negative intrinsic quadrupole moment 0 Q . The negative 0 Q implies that the nucleus has an oblate shape; the positive intrinsic quadrupole moment 0 Q means that the nucleus has a prolate shape. The overall the IBM-2 and IBM-2 CM results is a good agreement with the experimental data.

Hamiltonian (Equation
Concerning electric transition probability B(E2) values, we find that in all calculations the overall trend is reproduced reasonably very well in some transitions, but notice some discrepancies present case of the decay of the 2 0 + states in heavier Pt isotopes, in general, better than the values calculated by Bjjker et al., [11].