Extended Analytical Solutions of the Bohr Hamiltonian with the Sextic Oscillator: Pt-Os Isotopes

Extended analytical solutions of the Bohr Hamiltonian with the sextic oscillator: Pt-Os isotopes S Baid 1,2 , G Lévai 3,∗ and J M Arias 2,4 1 LPHEA, Faculty of Sciences Semlalia, Cadi Ayyad University, Morocco 2 Departamento de FAMN. Facultad de Física, Universidad de Sevilla, 41080 Sevilla, Spain 3 Institute for Nuclear Research (ATOMKI), PO Box 51, H-4001 Debrecen, Hungary 4 Instituto Carlos I de FTC, Universidad de Granada, 18071 Granada, Spain E-mail: [email protected],[email protected] and [email protected] Received 21 October 2022, revised 5 January 2023 Accepted for publication 31 January 2023 Published 21 February 2023 Abstract The sextic oscillator adapted to the Bohr Hamiltonian has been used to describe even Pt and Os isotopes from A=188 to 198 and A=186 to 192, respectively. The purpose of this study was to investigate the possible transition from the γ-unstable to the spherical vibrator shape phases. In this setup the potential appearing in the Bohr Hamiltonian is independent from the γ shape variable, and the physical observables (energy eigenvalues, B(E2)) can be obtained in closed analytical form within the quasi-exactly solvable formalism for the model space containing 30 of the lowest-lying levels. Experimental energy levels have been associated with the theoretical ones. The available electric quadrupole transition data (B(E2), decay preferences) have been taken into account in matching the experimental and theoretical levels. Special attention has been paid to transitions from the first two excited 0 + levels to the 2 1 + and 2 2 +levels, as these indicate the change of shape phases with spherical and deformed potential minimum. The three parameters of the Hamiltonian have been determined by a weighted least square fit procedure. Trends in the location of states belonging to the ground-state, the K π =2 + and two excited K π =0 + bands have been analysed. The trajectory determined by the fitted parameters in the two-dimensional phase space has also been plotted, Journal of Physics G: Nuclear and Particle Physics J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 (28pp)https://doi.org/10.1088/1361-6471/acb78a ∗ Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI. 0954-3899/23/045104+28$33.00 © 2023 The Author(s). Published by IOP Publishing Ltd Printed in the UK 1 and it has been found that all the nuclei are characterized by a deformed potential minimum, except for the heaviest Pt isotope ( 198 Pt), for which the transition to the spherical shape phase is realised. Although the spectroscopic information on the next isotopes of the chains ( 200 Pt and 194 Os)is far less complete, there are indications that these nuclei are also close to or fall within the domain of spherical potential minimum. Keywords: shape phase transitions in nuclei, gamma-unstable nuclei, sextic oscillator potential (Some figures may appear in colour only in the online journal) 1. Introduction The nucleus is a strongly interacting many-fermion system. Its dynamics are thus, governed by the nuclear forces and the Pauli exclusion principle. For the theoretical description of this system, a variety of models has been introduced, each focusing on different aspects. Singleparticle models (like the shell model)describe the nucleus from the point of view of nucleons that move in the average potential generated by the other nucleons. Collective models capture the dynamics of the nucleus in terms of excitations of the nucleus as a whole. One of the most important feature of nuclei is their shape. Its description is carried out in different ways in the individual nuclear models. In the shell model, one can start from a spherical average potential, in which case the nuclear deformation develops due to the occupation of orbits by the protons and neutrons, or, alternatively, the average potential can be chosen non-spherical, which is reflected also in the deformation of the nucleus. In collective nuclear structure models the nuclear shape is considered as an input, and the nuclear excitations are interpreted as oscillations around the equilibrium nuclear shape. The shape associated with the ground-state configuration of the nucleus can be estimated from the location of the nucleus on the chart of nuclides, i.e. from its relative position from closed shells. The equilibrium nuclear shape can change along trajectories of the chart of nuclides, e.g. along isotope chains, so the description of the transition from one shape phase to another one is a notable challenge for all nuclear models. The Bohr Hamiltonian has been developed to describe the quadrupole type excitations of nuclei [1–3]. The key variables are β0 measuring the deviation from the spherical shape and γä[0, π/3]characterizing the deviation from the axially symmetric shape. The orientation of the nucleus is described by the Euler angles, so the problem is defined in a fivedimensional space. The kinetic term depends on all five variables, while the potential is the function of the intrinsic βand γshape variables. The location of the potential minimum defines the nuclear shape: spherical, deformed prolate, deformed oblate, deformed triaxial, etc. An important advance was the introduction of the Interacting Boson Model [4], in which quadrupole nuclear excitations are handled in terms of quadrupole (d)and monopole (s) bosons. The physical operators are constructed from boson-creating and annihilating operators such that the total number of bosons is conserved and the operator exhibits appropriate spherical tensorial (O(3)) character. The model has U(6)group structure, furthermore, it has three limiting symmetries (subgroup chains), each associated with a characteristic nuclear shape: U(5)for a spherical vibrator, O(6)for deformed γ-unstable and SU(3)for axially J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 2 deformed rotational nuclei. These symmetries helped the consistent interpretation of observables in a wide range of nuclei. The connection between the IBM Hamiltonian and the Bohr Hamiltonian has also been established [5,6]. A further development was the study of quantum phase transitions in the transitional path from one nuclear shape phase to another. The existence of a second-order phase transition was established between the U(5)(spherical)and the O(6)(deformed γ-unstable)limits, and a first-order one between the U(5)(spherical)and SU(3)(axially deformed)limits [7]. Based on these results, symmetries associated with the critical points of shape transitions were introduced within the Bohr Hamiltonian scheme. Firstly, the E(5)critical point symmetry was proposed for transitions from the spherical vibrator and the γ-unstable phases [8]. In this model, the V(β,γ)potential of the Bohr Hamiltonian is approximated by the infinite square well in the βvariable (the γvariable does not play any role here), which has analytical solutions in terms of Bessel functions. The flat potential shape for the βvariable approximates the situation when the potential minimum V(β=0), associated with the spherical equilibrium shape of the anharmonic vibrator, develops gently through another situation in which a deformed potential minimum V(β>0), associated with the γ-unstable shape, appears without coexistence. The next proposed critical point symmetry was the X(5)[9]occurring during the transition from the spherical vibrator to the axially deformed rotor phase. In this case, the γ shape variable also plays a role, so the formalism becomes more involved, but remains analytically solvable. In this case, the potential energy surface evolves as follows: starting from a spherical minimum, a second minimum higher in energy develops, this minimum goes down in energy until degeneration with the spherical minimum (critical point and shape coexistence)and continues going down in energy to be the lowest minimum. Equivalent critical points have been studied within the IBM scheme (for a nice review see [10]). The introduction of the critical point symmetries triggered a search for nuclei that correspond to the predictions of these models, and also initiated the exploration of potentials that lead to exact solutions in the Bohr Hamiltonian (for a review see [11]). We contributed to these efforts by proposing the sextic oscillator as a potential in the βshape variable [12]. This potential has a flexible shape: depending on the parameters, it can have a spherical minimum, a deformed minimum or both, and the transition between these shape phases can be controlled analytically. It belongs to the class of quasi-exactly solvable (QES)potentials [13,14] meaning that only the lowest few energy levels can be obtained exactly. The electromagnetic transition rates (B(E2),B(E0)) have also been determined in closed form, and benchmark numbers associated with the transition between the shape phases have been calculated [15]. More recently, the range of QES solutions has been extended to wave functions with up to two nodes (previously solutions with up to one node had been considered): this development extended the number of levels to 30, with L π 10 [16]. A systematic description of Ru and Pd isotopes was presented and nuclei close to the transition from the spherical to deformed potential minimum were identified ( 104 Ru, 108 Pd). The sextic oscillator has also been used in the Bohr Hamiltonian with potentials depending on the γshape variable too. Various nuclear shapes have been considered: γ-stable triaxial [17],γ-stable prolate [18],γ-rigid triaxial [19]and γ-rigid prolate [20]nuclei (see [21]for a review). Numerical studies for the sextic oscillator have also been carried out to describe double-well structures [22]. The performance of higher QES solutions of the sextic oscillator has also been studied [23]. Here the solutions had to be determined by numerical diagonalization. The conclusion was that solutions up to two nodes seem to be the best approximation for phase transition studies. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 3 In the present work, we apply the extended sextic oscillator potential QES model to another region of the nuclide chart, where transition between the spherical vibrator and the γunstable shape phases can be expected. This is the chain of even Pt and Os isotopes, located close to the closure of the Z=82 proton shell. Some Pt nuclei have been considered the best examples for O(6)symmetry [24], so it seems worthwhile to study the evolution of the spectroscopic properties along the named isotope chains. Recently, other theoretical works investigate the same region solving the Bohr equation considering energy-dependent parameters and/or deformation-dependent mass [25–28]. The paper is arranged as follows. The essential elements of the sextic oscillator adapted to the Bohr Hamiltonian are presented in section 2. In section 3the experimental levels of Pt and Os nuclei are associated to model states, the model parameters are fitted and the findings are interpreted, while section 4summarizes the results. 2. Review of the model 2.1. The Bohr Hamiltonian The original Bohr collective Hamiltonian is given in terms of five curvilinear coordinates, including the intrinsic deformation variables β,γ, and the three Euler angles (θ 1 ,θ 2 ,θ 3 ). After separating the angular variables, it is written as follows [1,2]: HB Q k V 2 11 sin 3 sin 3 1 4sin ,, 1 k k 2 4 4 2 2 2 22 3 å bb bbb gg gg bgp bg =- ¶ ¶ ¶ ¶+¶ ¶ ¶ ¶ - - + ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥  () () () where Bis the mass parameter and Q k (k=1, 2, 3)are the components of the angular momentum in the intrinsic reference frame. Here we consider the special case of γ-unstable potentials, which depend explicitly only on the βvariable, i.e. V(β,γ)=U(β). For these potentials the β-dependence can be separated into an equation similar to the radial Schrödinger equation by using the substitution ,, , . 2 ii 2 bgq b fb gq Y =F - () ()() () The β-differential equation is then u d d 12 .3 2 22 fb b tt bbfb fb-+ ++ += ⎡ ⎣ ⎢⎤ ⎦ ⎥  ()()() () () () () Here τis the seniority quantum number, which determines the allowed Langular momentum values (see later). The reduced energies and potentials are defined as E B2 2 =   and uU B2 2 bb=  () ( ) , respectively. Note that the deformation variable βis dimensionless, and so are u(β)and ò. The mass parameter Bhas physical dimension kg m 2 and its chosen value sets the energy scale. 2.2. The sextic oscillator potential adapted to the Bohr Hamiltonian Similarly to [12,15,16], we chose the potential u(β)as the sextic oscillator, which is a member of the QES potential family [13]. A characteristic feature of these potentials is that J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 4 only their lowest few bound-state solutions can be obtained in closed analytical form. In general, their solutions are written in terms of an infinite power series, which can be reduced to an Mth order polynomial form for certain combination of the parameters appearing in the formulas. In the case of the sextic oscillator the coefficients of the power series are obtained from a three-term recurrence relation. This can be terminated by a suitable choice of the parameters, in which case the coefficients of the polynomial solutions are obtained by finding the roots of an (M+1)th order algebraic equation. In general, the roots have to be calculated numerically, however, up to M=2(i.e. the cubic algebraic equation)they can be determined in closed form. It has to be noted that the conditions for the termination of the power series imply constraints on the potential parameters, so the QES solutions can be obtained only for special subsets of the general sextic oscillator. It may also be noted that the QES solutions of the sextic oscillator can also be obtained by a suitable transformation from the bi-confluent Heun equation: see [29]on the relation of the two approaches. Here we review the essential ingredients of our model; any further details can be found in [12,15,16]. The polynomial QES solutions of the sextic oscillator can be written in the form ab Pexp 42 ,4 M2422 f bb b b b~-- t+ ⎡ ⎣⎤ ⎦ () ( ) () () where P (M) (β 2 )is an Mth order polynomial in β 2 . Note that since βis dimensionless, so are a and b. Note also that normalizable solutions can only be obtained for a0. Substituting equation (4)into equation (3)one finds that the form of the sextic potential used in the Bohr Hamiltonian is [12,15,16] u b ac ab a u42 , 5 22426 0 bbbb=- + + + pp () ( ) () where aand bare real parameters appearing in the exponential factor of equation (4), while c π is a constant that is determined by τand Mas cM 2 27 2.6t=+ + p () The u 0 pconstants (defined later on)are used to match the τ-even and τ-odd parts of the spectrum: see [16]for the details on the calculations. Note that the a=0 choice recovers the harmonic oscillator. The shape of the potential equation (5)is determined by the signs of the coefficient of the quadratic, quartic and sextic terms The latter, a 2 is always positive, so the potential increases infinitely with increasing β. The sign of the coefficient of the quadratic term determines the potential close to the origin. For b 2 >4ac π , the potential has a local minimum at β=0, while for b 2 <4ac π it has a local maximum. For b 2 =4ac π the quadratic term vanishes, so the potential is flat near the origin, and the coefficient of the quartic term determines what kind of extremum it has there. This coefficient, 2ab, has the same sign as b. In other words, the (a,b) parameter space of the model is divided by the parabola a=b 2 /(4c π )into three domains: I. Above the parabola the potential has a local maximum at the origin, and has a minimum for β>0, irrespective of the sign of b; II. Below the parabola and for b>0 all the coefficients are positive, so the potential has only a minimum at the origin. (for a=0 the potential reduces to the harmonic oscillator); III. Below the parabola and for b<0 the potential has a local minimum at the origin, another minimum at β>0 and a local maximum in between. It can be proven that the minimum at β>0 is always deeper [16]. Equation (6)implies that for a>0, c π is different for even and odd values of τ. This also means that the potential is slightly different for states with even and odd value of τ. The difference appears in the coefficient of the quadratic term in β, i.e. close to the origin and it is not too significant in general. This J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 5 duality of the potential in the parity of τis similar to parity-dependent potentials, which are used in various branches of physics. However, it raises the question of how to adjust the two spectra to each other. In [12,15,16]this problem was handled by introducing the u 0 pconstant in the potential equation (4). In particular, the minima of the two potentials were prescribed to coincide. This can be reached by the following choice of the u 0 +and u0 -used in the τ-even and τ-odd potential forms, respectively: u0, 7 0 = + () u ba ba ab a b a ba ab a aba ba 15 17 2;if15 17 2 ; if 15 17 0; if 17 ,8 0 2022 02 04042 06061 2 202042 061 2 1 2 1 2 bb bb bb bbb = --- +-+- < -- - - << > - +- +- +- --+ ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ()()()() [( ) ( ) ] [( ) ( ) ] ( ) ( )() () () ( ) ( ) () () where abb ac 1 3212. 9 021 2 b=-+ pp [( )] () See [12,15,16]for the detailed calculations. In the first applications of the sextic oscillator in the Bohr Hamiltonian [12,15]solutions with M=0 and 1 were considered, while the extension up to M=2 was worked out in [16]. This required the determination of the roots of a cubic algebraic equation in closed form. The normalised solutions are written as Ndg ab 1exp 42 .10 224 42 f bb bb bb=++ -- aataa + ⎡ ⎣⎤ ⎦ () ( ) ( ) Here αrepresents the quantum numbers (ξ,τ).ξ=1, 2, and 3 is the traditional notation referring to the number of nodes nof the β-wave function as ξ=n+1. The coefficients d α and g α can be determined in the knowledge of the roots of the cubic algebraic equation, while the normalization constants N α can be calculated exactly in terms of confluent hypergeometric functions [12,16]. The energy eigenvalues are determined for τ-even and τ-odd states separately. According to equation (6), the τ+2M=4 choice that allows the combinations τ=0, M=2; τ=2, M=1 and τ=4, M=0 corresponds to c + =15/4, while τ+2M=5 includes the combinations τ=1, M=2; τ=3, M=1 and τ=5, M=0 corresponding to c − =17/4[12](this difference in c + and c − also defines the difference of the τ-even and τ-odd potentials as 2aβ 2 ). The energy eigenvalues are displayed in table 1. The various i M l () quantities appearing there are obtained from solution of the (M+1)th order algebraic equations. Introducing s=(2τ+5)/4, the quantities appearing in table 1are as follows [16]: bb a bb a 2 2 18 , 2 2 22 , 11 12 12 1 2 1 2 l l = + = +   () () () () ()  J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 6 s s 5 4 , 5 4 for 0 7 4 , 7 4 for 1, 12 ii ii 22 22 lt lt =L = = =L = = () () () () () () ()  where sbr sbr sbr ab r rrs b as 42cos 3, 42cos 33 , 42cos 33 , cos 64 , 16 324 1 , 13 1 2 2 2 3 2 3 2 1 2 f pf pf f L=- L=+ - L=+ + =- º= + + ⎛ ⎝⎞ ⎠ ⎛ ⎝⎞ ⎠ ⎛ ⎝⎞ ⎠ ⎡ ⎣⎤ ⎦ () () () () () ( ( )) ( ) () () () and the prescription rbsgn sgn=-() () has to be satisfied. The (ξ,τ)multiplets contain one or more physical states with given Langular momentum value. The rules determining the allowed values follow from the O(5)⊃O(3)decomposition: construct τ=3n Δ +ν, with n Δ =0, 1,.... Then the allowed L-values are L=ν,ν+1,K, 2ν−2, 2ν(note that the value L=2ν−1 is missing). Table 1displays which Lvalues are allowed for the given (ξ,τ)multiplets. In the original version of the model [12,15], the states Table 1. The energy eigenvalues of the sextic oscillator adapted to the Bohr Hamiltonian [16]. The constant terms u0 pin equations (7)and (8)that depend on the even or odd value of τare subtracted for simplicity. The i M l () and i M l ˜ () quantities are displayed in equations (11)and (12). The fifth column displays the angular momenta Lcontained in the given (ξ,τ)multiplet. ξτM u ,0 - xt p  L 10 2 b b ;0, ;0, 2 2 1 2 l l > < ⎧ ⎨ ⎩ () () 0 12 1 b 9 1 l+- () 2, 4 14 0 b 5 3 2 l+ () 2, 4, 5, 6, 8 20 2 b 5 3 2 l+ () 0 22 1 b 9 1 l+ + () 2, 4 30 2 bb b 5 ;0, ;0, 1 2 2 2 l l +> < ⎧ ⎨ ⎩ () () 0 11 2 bb b 7 ;0, ;0, 2 2 1 2 l l +> < ⎧ ⎨ ⎩ ˜ ˜ () () 2 13 1 b11 1 l+- ˜() 0, 3, 4, 6 15 0 15b2, 4, 5, 6, 7, 8, 10 21 2 b 7 3 2 l+˜() 2 23 1 b11 1 l+ + ˜ () 0, 3, 4, 6 31 2 bb b 7 ;0, ;0, 1 2 2 2 l l +> < ⎧ ⎨ ⎩ ˜ ˜ () () 2 J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 7 belonging to the same multiplet were considered degenerate in energy. During the application of the extended model (up to M=2)[12]it was found that the physical states do not follow this degeneracy, rather typically, the levels with higher Lappear higher in the spectrum. For this reason, in [16]the Hamiltonian was extended by a phenomenological rotational term cL ·L. The justification of this rotational term has been presented in [12], referring to certain algebraic models of nuclear structure. The Casimir invariants of algebras associated to these models, e.g. the IBM are typically written as the sum of scalar products of the type A (j) ·A (j) . The most well-known example is the SU(3)algebra, in which case the Casimir invariant is expressed in terms of the L·Land the Q·Qterms corresponding to j=1 and j=2, respectively. Introducing the physically relevant quadrupole–quadrupole interaction naturally introduces a rotational splitting in the spectrum. In the case of O(6)and O(5)algebras, which play an important role in the present model, the L·Lterm also appears in the Casimir operators together with other scalar products of the type A (j) ·A (j) [4], so a rotational splitting can appear naturally. With the rotational term, the number of the model parameters increased to three: a,band c. In what follows, the energy scale will be redefined such that E=0 corresponds to the ground-state energy. The rescaled energy eigenvalues will be denoted as E .14 , , 1,0 =- xt xt  () The electric quadrupole transitions are calculated using the first-order transition operator [3,8,30] TtD D Dcos 1 2sin . 15 iii E2 ,0 2 ,2 2 ,2 2 bqg q qg=++ mmm- ⎡ ⎣ ⎢⎤ ⎦ ⎥ () ( () ()) ( ) () () () () The integrals in βnecessary for its matrix elements can be calculated exactly in terms of confluent hypergeometric functions [16]. This first-order transition operator leads to selection rules Δτ=±1, which also means that the matrix elements for the electric quadrupole moments vanish. This selection rule can be relaxed by adding a second-order term to (15): TTtD D Dcos 2 1 2sin 2 . 16 iii E2 E2 2 ,0 2 ,2 2 ,2 2 bqg q qg=+ ¢-++ mmm ¢ - ⎡ ⎣ ⎢⎤ ⎦ ⎥ () () ( () ()) () () () () () () () Note that the first-order operator in equation (15)is an O(6)generator, so implies strict selection rules for τ, which is an O(5)quantum number. In contrast to this, the second-order transition operator (16)is no more an O(6)generator, so transitions with Δτ=0, ±1, ±2 are allowed. It should be noted that the Jacobian in the (β,γ)intrinsic variables is sin 3 4 bg∣() ∣ . 3. Application for Pt and Os isotopes In a previous study [16], the model was applied to even-mass Ru and Pd nuclei, which are located near the closure of the Z=50 shell. In analogy with that work, here nuclei near the closure of the Z=82 shell are discussed: the chain of even-mass Pt (A=188 to 198)and Os (A=186 to 192)isotopes. The heaviest Pt isotopes also approach the N=126 shell closure. These regions are expected to accommodate nuclei with spherical vibrator and γ-unstable [30] character, and may present examples for the transition between these shape phases. The selection criterion was that the given isotope should have rich enough experimental spectrum up to about E=2500 keV and L π =10 + . Data were taken from NNDC compilation. 5 5 http://www.nndc.bnl.gov/nndc/ensdf/. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 8 All calculations have been done, as mentioned above, including in the Hamiltonian an extra term cL 2 so as to break the L-degeneracy within a τmultiplet: HucLL d d 12 ,17 2 22 fb b tt bb=- + ++ ++ ⎡ ⎣ ⎢⎤ ⎦ ⎥ ()()() () · ( ) where uU B2 2 bb=  () ( ) is given by equation (5). Thus, one has three parameters: (a,b) defining the potential and ccontrolling the L-splitting. Using the reduced energy quantities, c is dimensionless, similar to aand b. In what follows, we set the energy scale such that ÿ 2 /(2B)=1 keV, i.e. B=500ÿ 2 MeV −1 . With this choice, the physical energy values measured in keV coincide with the calculated dimensionless energy values. 3.1. State and band assignment Model states were assigned to the experimental levels based on their energy and, if available, the B(E2)values of their electric quadrupole transitions. The three model parameters (a,band c)were fitted to the experimental spectrum using a two-step procedure, which was also applied in [16]. In the first step the lowest two even-spin states, and the lowest odd-spin states were identified with model states. These were the (ξ,τ)L π =(1, 3)0 + ,(2, 0)0 + ,(1, 1)2 + , (1, 2)2 + ,(1, 2)4 + ,(1, 3)4 + ,(1, 3)6 + ,(1, 4)6 + ,(1, 4)8 + ,(1, 5)8 + ,(1, 5)10 + , and the (1, 3)3 + , (1, 4)5 + ,(1, 5)7 + levels (note that there is only one 10 + state in the model spectrum, while 1 + and 9 + states are missing). The relative order of the levels with the same Lis unambiguous: states with higher τare expected to lie higher. The only exception occurs for the L π =0 + states. Here both (ξ,τ)L π =(1, 3)0 + and (2, 0)0 + states can be the first excited level, depending on the parameters. However, the electric quadrupole decays of the two states are different, which helps their assignment [16]. The (1, 3)0 + level decays to the (1, 2)2 + level, which is expected to be the second excited 2 + state, while the (2, 0)0 + level decays to the (1, 1)2 + level, which is typically the first excited one. This procedure was applied to the six Pt and four Os isotopes selected here. There were only a handful of levels for which no experimental candidate could be identified: the corresponding model states all belong to the (1, 5)multiplet (one 8 + , one 10 + , and three 7 + states). After this first round of the level assignment, the parameters were fitted to the experimental spectrum. Following the procedure applied in [16]various weights were used for the individual levels. In particular, in order to avoid the overrepresentation of less wellknown higher-lying states, the weight of unity (w=1)was distributed evenly among the levels belonging to the same (ξ,τ)multiplets. In particular, the members of the (1, 2),(1, 3), (1, 4)and (1, 5)multiplets carried the weight 1/2, 1/4, 1/5 and 1/7, respectively, while the states that stood alone in their multiplet (e.g. (1, 1)2 + ,(2, 0)0 + , etc)had weight w=1. Furthermore, when the experimental spin-parity of the level was ambiguous, then the corresponding weight was halved. As discussed previously, the bandhead state of the first two excited K π =0 + bands can be assigned either to the (1, 3)or the (2, 0)quantum numbers. Experimental data for the B(E2) values for the transitions from the second and third 0 + states to the first two 2 + states is displayed in table 2. It is seen that the 0 2 +state prefers decaying to the 2 2 +state in most cases. Relatively strong transitions have been observed for 194 Pt, 196 Pt, 190 Os and 192 Os. The B(E2) value is smaller for 188 Os, but the preference holds in that case too. In terms of the model, this indicates that the 0 2 +states can be assigned to the model state with (ξ,τ)=(1, 3). Based on the position of the first excited 0 + state, the same assignment seems reasonable for the remaining nuclei, with the exception of 198 Pt. In this case, this state prefers decaying into the first excited 2 + level, which supports its assignment to the model state with quantum numbers (2, 0)0 + . J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 9 the pairs [(1, 3)3 + ,(1, 3)4 + ],[(1, 4)5 + ,(1, 4)6 + ]and [(1, 5)7 + ,(1, 5)8 + )]is determined by the phenomenologic rotational term, cL ·L, the cparameter of which comes out relatively small from the fitting procedure. It has to be mentioned though, that the general structure of the K π =2 + band is changing along the Pt chain, as it can be seen in figure 2. Starting from the light side of the chain, a typical rotational spacing can be observed for most nuclei. Then the splitting between the 2 + and 3 + states begins to increase, while that between the 3 + and 4 + states is shrinking, and for 198 Pt the latter two states form an almost degenerate doublet. Figure 3. Members of the first two excited K π =0 + bands for the Pt isotopes. Lines indicate the theoretical levels (ξ,τ)L=(1, 3)0, (1, 4)2 for the first band, and (2, 0)0, (2, 1)2, (2, 2)4 and (2,3)6 for the second one, while the corresponding experimental states are represented by various symbols of the same colour. The indices ξ=1 and ξ=2 denote the two bands. Figure 4. The same as figure 1for the Os isotopes. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 16 Similar trend is seen for the 5 + members of this band and its neighbors. This trend is less visible in the Os chain (see figure 5). It is seen in figures 3and 6that the members of the band built on the bandhead state identified with the (2, 0)0 + model state exhibit a uniform decreasing trend in energy as the mass number is increasing, while the members of the bands built on the (1, 3)0 + state stay roughly at the same energy throughout the chains. This leads to the phenomenon discussed before, i.e. that the (2, 0)0 + state gets below the (1, 3)0 + state in 198 Pt. This crossing does not occur in the Os chain, although the two 0 + bandhead states get close to each other in 192 Os. Figures 7–13 present the most well-known part of the spectra, together with the available B(E2)values given in W.u. The theoretical calculations have been done using the linear Figure 5. The same as figure 2for the Os isotopes. Figure 6. The same as figure 3for the Os isotopes. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 17 quadrupole operator, equation (15). The parameter tin equation (15)was fixed to match the experimental B E2; 2 0 11  ++ () . A geometric estimate of the tparameter is ZR 3 4 0 2p() , where R 0 =1.2 A 1/3 fm. This estimated value is around 9 e b and 8.6 e b for the Pt and Os isotopes, respectively. Except for the two light Os isotopes, the estimated value is within 10 percent to the calculated one. Table 5summarizes the experimental information on the electric quadrupole transitions between states discussed here. There are only a few known B(E2)values in the three light Pt Figure 7. The experimental and theoretical energy (in MeV)spectrum of 194 Pt with the E2 transitions. The linear E2 operator, equation (15), has been used with t=8.58 e b, in order to match the transition B E2; 2 0 11  ++ () . Transitions predicted to be forbidden in the model are displayed in red in the experimental (left)panel. Two forbidden transitions with B(E2)0.36 W.u. were omitted for the sake of clarity. Figure 8. The same as figure 7for 196 Pt. The linear E2 operator, equation (15), has been used with t=8.37 e b, in order to match the transition B E2; 2 0 11  ++ () . Seven forbidden transitions with B(E2)0.56 W.u. were omitted for the sake of clarity. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 18 isotopes, while the data set is more abundant in 194 Pt, 196 Pt and in the Os isotopes. It is reasonable to discuss separately the transitions with Δτ=±1 that are allowed using the linear E2 transition operator (15)and those that are forbidden. The average strength of the allowed transitions is typically one or two orders of magnitude larger. However, there are some relatively strong transitions with Δτ=±2 that the model predicts to be forbidden. This is the case for three of the four Os isotopes with the E2 transition from the 4 3 +state to the 2 2 +state. The strength of these transitions is in the range of 7–10 W.u., which reaches 10 to 17 percent of the strength of the transition from the 2 1 + state to the ground-state in the given nucleus. In Figure 9. The same as figure 7for 198 Pt. The linear E2 operator, equation (15), has been used with t=9.24 e b, in order to match the transition B E2; 2 0 11  ++ () . Two forbidden transitions with B(E2)0.05 W.u. were omitted for the sake of clarity. Figure 10. The same as figure 7for 186 Os. The linear E2 operator, equation (15), has been used with t=11.48 e b, in order to match the transition B E2; 2 0 11  ++ () . Two forbidden transitions with B(E2)1.2 W.u. were omitted for the sake of clarity. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 19 Figure 11. The same as figure 7for 188 Os. The linear E2 operator, equation (15), has been used with t=10.35 e b, in order to match the transition B E2; 2 0 11  ++ () . Three forbidden transitions with B(E2)1.31 W.u. were omitted for the sake of clarity. The transition indicated with green arrow occurs within the (ξ,τ)=(1, 3)multiplet and is forbidden due to the Δτ=0 selection rule. Figure 12. The same as figure 7for 190 Os. The linear E2 operator, equation (15), has been used with t=9.53 e b, in order to match the transition B E2; 2 0 11  ++ () . Four forbidden transitions with B(E2)0.8 W.u. were omitted for the sake of clarity. The two transitions indicated with green arrow occurs within the (ξ,τ)=(1, 3)multiplet and are forbidden due to the Δτ=0 selection rule. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 20 the present model the 4 3 +and the 2 2 +states are interpreted as model states with quantum numbers (ξ,τ)=(1, 4)and (1, 2), so the Δτ=±1 selection rule forbids the E2 transition between them when using the β-linear operator, equation (15). This indicates, probably, a non-negligible mixing with other 4 + and 2 + states. In 186 Os the strongest theoretically forbidden transition is that from the 2 2 +to the ground-state. Figure 13. The same as figure 7for 192 Os. The linear E2 operator, equation (15), has been used with t=8.32 e b, in order to match the transition B E2; 2 0 11  ++ () . Three forbidden transitions with B(E2)0.41 W.u. were omitted for the sake of clarity. Table 5. Statistics of the experimentally observed B(E2)values (in W.u.). Transitions between states that are assigned to model states are considered. The number and the mean B(E2)values are displayed in the second and third column for transitions with Δτ=±1 allowed within the model. The same data are displayed in the fourth and fifth column for forbidden transitions (corresponding mainly to Δτ=±2), supplemented by the largest B(E2)value, which is predicted to be zero in the model, as well as with the initial and final states of the corresponding transition. The three transitions within the same (ξ,τ)multiplet are not included. Nucleus Allowed E2 transitions Forbidden E2 transitions Number 〈B(E2)〉Number 〈B(E2)〉Largest B(E2) L ip L f p 188 Pt 5 133.0 0 190 Pt 1 56.1 0 192 Pt 6 77.5 2 0.62 0.68 194 Pt 11 45.3 4 3.89 14.4 03 + 2 2 + 196 Pt 11 35.6 8 0.58 2.8 198 Pt 6 32.0 3 0.23 0.60 186 Os 13 87.6 7 3.0 9.4 2 2 + 0 1 + 188 Os 11 69.8 6 3.0 10.2 4 3 + 2 2 + 190 Os 16 59.3 7 2.46 7.3 4 3 + 2 2 + 192 Os 16 51.0 6 3.0 10.6 4 3 + 2 2 + J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 21 Another example is the transition from the E x =1547 keV 0 + state to the 2 2 +state in 194 Pt. This transition is comparable in strength to that to the 2 1 + state (see also table 2). In the present approach this 0 + state has been interpreted as the (ξ,τ)=(2, 0)bandhead state, which is not expected to decay to the 2 2 +state with quantum numbers (1, 2). The other 0 + model state with quantum numbers (1, 3)can decay to this state, however, it has been assigned to the first excited 0 + state at E x =1267 keV on grounds of its decay pattern (see table 2). In the remaining Pt nuclei ( 192 Pt, 196 Pt and 198 Pt)the selection rules seem to be realized with better accuracy. It is notable that in many transitions with significant B(E2)value that the model predicts to be forbidden, the 2 2 +state appears as the initial or final state. This seems to indicate that this state has non-negligible mixing with other 2 + states. It is worthwhile to discuss transitions within the (ξ,τ)multiplets separately. These correspond to Δτ=0, and as such, they are forbidden if the linear E2 transition operator is used. It is found that only three such transitions occur, and only in 188 Os and 190 Os. The strongest one is (1, 3)4 + to (1, 3)3 + with B(E2)=54 +24/−19 W.u. in 190 Os. Such a strong transition within the K π =2 + band would be characteristic of a rotational nucleus, indicating triaxial configuration. The two other examples (1, 3)6 + to (1, 3)4 + in 190 Os and (1, 3)4 + to (1, 3)6 + in 188 Os with B(E2)=6(4)and 15(6)W.u., respectively, are weaker and correspond to transitions between the K π =2 + and the ground-state bands. An overall observation regarding the B(E2)values is that the model generally predicts stronger transitions than what is obtained from the experiment. It has to be mentioned here that the theoretical B(E2)values are scaled by the requirement that the B E2; 2 0 11  ++ () is reproduced. This may lead to a general overestimation of the transition strengths. However, the fact that the tparameter of the linear E2 transition operator (15)falls within a relatively narrow range may be considered as the sign of consistency. In-band transitions in the groundstate bands are generally reproduced well: with a few exceptions, the calculated B(E2)values are close to the upper limit of the experimental data. Transitions within the K π =2 + band are also reasonably reproduced: here the strength of transitions from the higher states tend to be overestimated. Interband transitions between the K π =2 + and the ground-state band are typically also overestimated by a factor of two or three. Here there is a clear difference between the Pt and Os isotopes. According to the prediction of the model, the strength of the transitions to the 2 1 + state from the 4 1 + and 2 2 +states is equal (this applies also to the O(6)limit of the IBM). This expectation is fulfilled for the Pt isotopes, but it is not for the Os isotopes. In this case, the mean values of the former transition are several times stronger. The largest difference (∼6to8)occurs on the light end of the isotope chain, and it gradually diminishes (to 3 and less than 2)for 190 Os and 192 Os (it may also be noted that the errors of the 2 2 +to 2 1 + transitions are comparable to the mean values, so the experimental ratios might be closer to the theoretical ones). This seems to indicate that the model performs better for the heavy Os isotopes, located near the closure of the N=126 neutron shell. This finding seems to be confirmed by microscopic calculations. In [31]the structural evolution in neutron-rich Os and W isotopes was investigated using IBM and Hartree–Fock–Bogoliubov calculations with the Gogny-D1S energy density functional (EDF). It was found that the obtained energy surfaces for Os isotopes are rather flat in the γdegree of freedom: in fact, for 192 Os it was calculated to be γ-independent. Similar features can be seen in, 6 which displays plots concerning the deformation evolution of nuclei based on microscopic calculations. The lighter Os isotopes exhibit features close to a rotational character. Transitions from the excited 0 + bandhead states and the 2 + states built on them are also overestimated by the model in general. Here the 6 https://wwwphynu.cea.fr/science_en_ligne/carte_potentiels_microscopiques/carte_potentiel_nucleaire_ eng.htm. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 22 calculated values exceed the experimental ones by a factor of two to five. Generally, the results are rather similar to those obtained for the Ru and Pd isotopes [16]. It has to be noted that the B(E2)values were calculated using the first-order transition operator (15). As it has been mentioned previously, the electric quadrupole moments calculated with this operator vanish. The experimentally known quadrupole moments of the 2 1 + state are the following: 5192 Pt: Q=+0.55(21)eb, 194 Pt: Q=+0.48(14)eb, 196 Pt: Q=+0.62 (8)eb, 198 Pt: Q=+0.42(12)eb, 186 Os: Q=−1.63(4)eb, 188 Os: Q=−1.46(4)eb, 190 Os: Q=−1.18(3)eb, 192 Os: Q=−0.96(3)e b. In summary, the ground-state configuration of the Pt isotopes has oblate shape, while that of the Os isotopes is prolate. The electric quadrupole momenta are non-zero, however, they are clearly smaller in magnitude that the corresponding values of the neighboring nuclei (W and Hf nuclei located closer to the mid-shell region typically have Q∼−2 e b, while 184 Os has Q=−2.7 e b). A more realistic reproduction of the electric quadrupole momenta would be possible using the more general E2 operator (16) that includes a quadratic term too. This operator would also relax the selection rules by allowing transitions with Δτ=±2 and 0 too. However, we are not considering this option here. As a summary of the present subsection, it may be noted that the performance of the sextic oscillator as a γ-independent potential in the Bohr Hamiltonian improves with increasing mass number in both chains. Indications for this are the trends in the splitting between even- and odd-spin levels in the K π =2 + bands (for Pt nuclei)and the trends in the quality of fitting, Dand in the ratios B EBE2; 4 2 2; 2 2 11 21  ++ ++ ()() (for the Os nuclei). The lighter members of the chains, closer to the middle of the neutron shell might have influence from other shape configurations (e.g. rotational)too. Figure 14. The location of the Pt and Os nuclei in the (a,b)phase space based on the fitted aand bparameter values. Open circles indicate the tentative position of the 200 Pt and 194 Os nuclei. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 23 3.3. Consequences regarding the phase space and potential shapes Figure 14 displays the trajectory defined by the fitted aand bparameters in the (a,b)phase space for the Pt and Os isotopes. Most locations are above the critical parabola and close to the a-axis on its left, i.e. the bparameters are typically small and negative. This parameter combination corresponds to negative coefficients of the quadratic and quartic terms of the sextic oscillator potential (5). This leads to a potential with a deformed minimum (|β|>0). The only exception is 198 Pt, which is located below the critical parabola, close to the b-axis with b>0. For this parameter combination all three terms of the sextic oscillator have positive coefficients, leading to a spherical minimum (β=0). Furthermore, the small aalso means that this nucleus is close to the harmonic oscillator limit. These trajectories are different from those obtained for the Ru and Pd isotopes using the same model [16]. There the b parameter was found to be positive for each isotope, furthermore, most of the points were located below the critical parabola, corresponding to a spherical minimum. The Ru isotopes followed a trajectory from the right (b∼300)to the left (b∼100)as the mass number increased from A=98 to 108, crossing the critical parabola near A=104. The parameter b also decreased (from 300 to 200)for the Pd isotopes, as the mass number increased from 102 to 110, but the trajectory moved roughly parallel with the critical parabola, staying below it. The potentials generated from the fitted parameter values aand bare displayed in figures 15 and 16. It is worth noting that for these γ-unstable systems the potential as a function of βis symmetric with respect to β=0, that is for each nucleus minima of equal depth are found for ±|β|because the potential only contains even powers in βand it does not depend on the γdegree of freedom. This means that a prolate system with β 0 >0 and γ 0 =0 has the same energy as an oblate system with β 0 >0 and γ=π/3. But this last situation is equivalent to β=−β 0 and γ=0. Since we are not plotting γ,infigures 15 and 16 we include only positive βvalues, the figures are a projection of the energy surface on the γ=0 plane for Pt isotopes (oblate shapes)and on the γ=π/3 for Os isotopes (prolate shapes). A symbol has been plotted in the actual minimum for each nucleus. Figure 15. The potentials u(β)calculated from the parameters aand bdetermined from the fitting procedure for Pt nuclei. The zero of the energy scale is chosen at the groundstate energy for all isotopes. Filled circles indicate the location of the potential minima. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 24 It is seen in figure 15 that the potential associated with 198 Pt differs from the other potentials in that it is the only one corresponding to a spherical minimum ( 0 min b= ). This is also the only example for which the first excited 0 + state is interpreted as the E 2,0 , rather than the E 1,3 theoretical level. All the rest of the Pt isotopes are calculated to be deformed with moderate β-deformations. The transition from the deformed minimum to the spherical one occurring in the Pt chain may be interpreted as the consequence of approaching the N=126 closed neutron shell, where nuclei with the spherical (or U(5)in IBM)symmetry are expected (note that a similar trend was found in the Ru chain [16], which approaches the N=82 shell closure for the light end of the chain). The investigation of 200 Pt is thus reasonable to decide the validity of this interpretation. For this, the properties of the first two 0 + states and the E2 decays from them are crucial. There is a state with tentative 0 + spin-parity assignment at 1118 keV and another one with firm 0 + at 1583 keV, but there are no B(E2)data available for 200 Pt. However, based on the findings on 198 Pt, one may expect that the first state is the model state with (2, 0)quantum numbers, while the second one is that with (1, 3). Considering heavier Pt isotopes, the energy spectrum of 200 Pt is far less well-known than those of the other Pt isotopes discussed here, so a parameter fit can be carried out only with fewer states. Applying the fitting procedure the ten lowest-lying positive-parity states one obtains a=5249, b=179 and c=11.5. These values place this nucleus above the critical parabola in figure 14, i.e. into the domain with deformed potential minimum (β>0). Most probably this is the consequence of the fact that the first excited 0 + state is substantially higher (1118 keV)than the corresponding state in 198 Pt (914 keV). Concerning Os, figure 16, all studied isotopes are calculated to be deformed (oblate, due to the know quadrupole moments)with moderate β-deformations. It should be worthwhile to investigate experimentally the next Os isotope too, i.e. 194 Os, to see whether the transition to the spherical domain can be observed. The indication for this would be the presence of two excited 0 + states relatively low in energy. The spectrum of 192 Os is close to this arrangement (956 and 1206 keV). However, the corresponding states in 194 Os are located at 697 and 1540 keV, which is a rather different pattern. The possibility of an unobserved 0 + state cannot be excluded, as most experimental states have tentative spin-parity assignment. There is a state, for example, at 1141 keV, which decays only into the 2 2 +state. This state has a tentative spin- Figure 16. The same as figure 15 for the Os nuclei. J. Phys. G: Nucl. Part. Phys. 50 (2023)045104 S Baid et al 25