Numerical Solution of a Sixth-Order Anharmonic Oscillator for Triaxial Deformed Nuclei

Abstract

The Davydov–Chaban Hamiltonian, which describes the quadrupole collective states of triaxial nuclei involving two polar coordinates and three Euler rotation angles, is numerically solved in a basis of Bessel functions of the first kind for a sixth-order anharmonic oscillator potential and a triaxial deformation, respectively. The proposed model is designed to describe a phase transition, as well as coexistence and mixing between an approximately spherical shape and a triaxial deformed one.

1. Introduction

A great interest in solving the Bohr–Mottelson Hamiltonian [1] for different types of potentials in the deformation variable started with the proposal of two solutions called E(5) [2] and X(5) [3]. Both approaches involve an infinite square well potential to describe the critical point of a second-order phase transition between the spherical vibrator [U(5)] [4] and the $\gamma$-unstable system [O(6)] [5,6] of a first-order phase transition between the spherical vibrator [U(5)] and prolate axially rotor symmetric nuclei [SU(3)], respectively [7,8]. This potential represents an approximation of the corresponding potential energy surfaces calculated using coherent state functions [9] for the interacting boson Hamiltonian [10,11,12]. Immediately after, two other critical point symmetries were proposed, Y(5) [13] and Z(5) [14], for the phase transitions between axial and triaxial shapes between prolate and oblate shapes, respectively, followed by their analogs in four dimensions as E(4) [15], X(3) [16] and Z(4) [15]. Because the five-dimensional equation of the Bohr–Mottelson Hamiltonian, after the separation of variables, is reduced to a one-dimensional Schrödinger equation, this allows for an easy adaptation of well-known solutions for potentials, such as harmonic oscillator [4], infinite square well [2,3,6], Kratzer [17], Coulomb [17], Davidson [18], quartic [19], sextic [20] and many other potentials [21,22,23]. While some of these potentials proved to be more appropriate to describe only the critical point, the shape phases or certain regions of the phase transitions, others, such as the sextic potential, showed the ability to cover the entire shape phase transition, including its critical point [24,25,26,27,28,29]. This is possible because the sextic potential, depending on its parameters, can have a single spherical minimum, a single deformed minimum, a flat shape, and simultaneous spherical and deformed minima. It should be noted that the sextic potential cannot be exactly solved in general and, consequently, a quasi-exactly solvable method [30] was applied instead, with the latter allowing for only a certain number of exact solutions to be determined. Nevertheless, as it was demonstrated by many applications [20,24,25,26,27,28,29,31,32,33,34,35], this finite number of exact solutions proved to be enough to describe the most relevant states for the nuclear-shape phase-transition phenomenon. Other phenomena, such as shape coexistence and mixing [36,37,38,39,40,41,42] or anomalous $B\left(E2\right)$ transition rates [43,44], can be described within this model if and only if a general sextic potential is considered (no constraints for parameters) [45]. The equation in this case is numerically solved using a basis of Bessel functions of the first kind [46] involving a method developed in [47]. The advantage of the numerical solution of the sextic potential is that it can describe all these phenomena in a unified way [48]. The shape coexistence and mixing phenomena simply emerge by increasing the height of the maximum, which separates the two minima (phases), in the vicinity of the critical point [45]. The model was developed initially to better describe the critical point of the first-order shape phase transition [46] by simulating two degenerated minima separated by a small barrier (height of the maximum) in agreement with the shape of the potential energy surface predicted by the interacting boson model [3,8,9]. In the next step, the condition for the degeneracy of the two minima was relaxed and the equation was solved for prolate [49] and $\gamma$-unstable cases [50], allowing for a description of phase transitions, coexistence and mixing between an approximately spherical shape and these deformations. Several applications to experimental data, such as ²³⁸Pu, ¹⁵²Nd, ¹⁷⁰Hf [46], ⁷⁶Kr [45,51], ⁷²⁻⁷⁶Se [49], ⁹⁶⁻¹⁰⁰Mo [50], ⁷⁴Kr, ⁷⁴Ge [52], ⁸⁰Ge [53] and ^42,44Ca [54], confirmed the ability of the model to describe these phenomena, but also evidenced the need for its further development to triaxial stable deformation [54]. Some nuclei present a $\gamma$-stable triaxial deformation rather than a $\gamma$-unstable one. Therefore, the main objective of the present work was to propose a new numerical solution of the sextic potential for triaxial nuclei. As a starting point, the Davydov–Chaban Hamiltonian [55] is considered. This choice simplifies the eigenvalue problem compared with the more general Bohr–Mottelson Hamiltonian because the dimensions are reduced from five to four by considering the triaxiality fixed at 30°, which represents the middle of the path between prolate (0°) and oblate (60°) shapes. The Davydov–Chaban Hamiltonian with a sextic potential was previously solved through the quasi-exact method [27] for the shape phase-transition phenomenon, while now by means of the numerical method, the coexistence and mixing phenomena are added to the model.

The plan of this work is the following. In Section 2, the new numerical solution of the Davydov–Chaban Hamiltonian with a sextic potential for triaxial nuclei is proposed. Numerical results and applications using the experimental data of nuclei suspected of manifesting these phenomena are presented in Section 3, while a discussion on the present results and perspectives for further possible applications and development of the new solution is given in Section 4.

2. Hamiltonian Model

The Hamiltonian model, which describes nuclear collective vibrations and rotations around a non-axial deformed shape, has the following expression [55]:

$$H=-{\displaystyle \frac{{ħ}^2}{2B}}\left[{\displaystyle \frac{1}{{\beta }^3}}{\displaystyle \frac{\partial }{\partial \beta }}{\beta }^3{\displaystyle \frac{\partial }{\partial \beta }}-{\displaystyle \frac{1}{4{\beta }^2}}\sum _{k=1}^3{\displaystyle \frac{{\widehat{Q}}_k^2}{{sin}^2\left(\gamma -\frac{2\pi }{3}k\right)}}\right]+V\left(\beta \right),$$

(1)

where ħ and B are the reduced Plank constant and the mass parameter, and $\beta$ and $\gamma$ are the intrinsic deformation coordinates ($\beta$-deviation from sphericity, $\gamma$-axial deformation), while ${\widehat{Q}}_k$ are the projections of the total angular momentum $\widehat{Q}$ in the intrinsic reference frame. The Davydov-Chaban Hamiltonian (1) contains a kinetic term in $\beta$, a rotational term depending on three Euler angles (${\theta }_1$, ${\theta }_2$, ${\theta }_3$) and the $\gamma$ coordinate, respectively a $\gamma$-independent potential. The Hamiltonian was deduced starting from the Bohr–Mottelson Hamiltonian [1,4] by imposing a $\gamma$-triaxial rigidity at ${30}^{\circ }$ for the corresponding classical Hamiltonian. This condition restricts the $\gamma$-vibrations of the nucleus and allows for an exact separation of the variables if the wave function $\Psi (\beta ,{\theta }_1,{\theta }_2,{\theta }_3)=\varphi \left(\beta \right)\psi ({\theta }_1,{\theta }_2,{\theta }_3)$ is considered [15,27,55]:

$$\left[-{\displaystyle \frac{1}{{\beta }^3}}{\displaystyle \frac{d}{d\beta }}{\beta }^3{\displaystyle \frac{d}{d\beta }}+{\displaystyle \frac{\lambda }{{\beta }^2}}+v\left(\beta \right)\right]\varphi \left(\beta \right)=\epsilon \varphi \left(\beta \right),$$

(2)

$$\left[{\displaystyle \frac{1}{4}}\sum _{k=1}^3{\displaystyle \frac{{\widehat{Q}}_k^2}{{sin}^2\left(\gamma -\frac{2\pi }{3}k\right)}}\right]\psi ({\theta }_1,{\theta }_2,{\theta }_3)=\lambda \psi ({\theta }_1,{\theta }_2,{\theta }_3),$$

(3)

where $\epsilon =2BE/{ħ}^2$ and $v\left(\beta \right)=2BV\left(\beta \right)/{ħ}^2$ are the reduced energy and potential, respectively. In Equation (3), $\gamma$ is more like a parameter than a variable. Thus, for $\gamma =\pi /6$, its operator becomes

$${\displaystyle \frac{1}{4}}({\widehat{Q}}_1^2+4{\widehat{Q}}_2^2+4{\widehat{Q}}_3^2)=\widehat{Q}-{\displaystyle \frac{3}{4}}{Q}_1^2,$$

(4)

with the eigenvalue and eigenfunction given by [56]

$${\lambda }_{L,R}=L(L+1)-{\displaystyle \frac{3}{4}}{R}^2,$$

(5)

$${\psi }_{\mu ,R}^{\left(L\right)}\left(\mathrm{\Omega }\right)=\sqrt{{\displaystyle \frac{2L+1}{16{\pi }^2(1+{\delta }_{R,0})}}}\left[D_{\mu ,R}^{\left(L\right)}\left(\mathrm{\Omega }\right)+{(-1)}^L{D}_{\mu ,-R}^{\left(L\right)}\left(\mathrm{\Omega }\right)\right].$$

(6)

Here, L, R and $\mu$ are quantum numbers associated with the total angular momentum and its projections on the intrinsic x’-axis and laboratory z-axis reference frame, while $D_{\mu ,R}^{\left(L\right)}\left(\mathrm{\Omega }\right)$ are the Wigner functions of the Euler angles $\mathrm{\Omega }\equiv ({\theta }_1,{\theta }_2,{\theta }_3)$. An alternative indexing of the wave function can be achieved with the help of the wobbling quantum number $n_w=L-R$ [1,56], which is useful in labeling the collective bands. For example, one has $n_w=0$ for the ground and $\beta$ bands, while $n_w=1$ and $n_w=2$ for the odd and even states of the $\gamma$ band. Furthermore, Equation (2) with $\lambda$ given by Equation (5) is solved for a sextic potential:

$$v\left(\beta \right)=a{\beta }^2+b{\beta }^4+c{\beta }^6,$$

(7)

where a, b and c are independent parameters. Expanding the kinetic term in $\beta$ and introducing a scaled variable $\beta =a^{-{\displaystyle \frac{1}{4}}}\tilde{\beta }$ in Equation (2), one obtains

$$\left[-{\displaystyle \frac{d^2}{d{\tilde{\beta }}^2}}-{\displaystyle \frac{3}{\tilde{\beta }}}{\displaystyle \frac{d}{d\tilde{\beta }}}+{\displaystyle \frac{L(L+1)-\frac{3}{4}{R}^2}{{\tilde{\beta }}^2}}+{\tilde{\beta }}^2+A{\tilde{\beta }}^4+B{\tilde{\beta }}^6\right]F\left(\tilde{\beta }\right)=\tilde{\epsilon }F\left(\tilde{\beta }\right),$$

(8)

where $A=b{a}^{-{\displaystyle \frac{3}{2}}}$, $B=c{a}^{-2}$ and $\tilde{\epsilon }=a^{-{\displaystyle \frac{1}{2}}}\epsilon$. Equation (8) is numerically diagonalized in a basis of Bessel functions of the first kind, which, in turn, are solutions of the same equation but for an infinite square well potential [15]:

$$\begin{array}{c}\hfill v\left(\tilde{\beta }\right)=\{\begin{array}{cc}0,& \tilde{\beta }\le {\tilde{\beta }}_w,\\ \infty ,& \tilde{\beta }>{\tilde{\beta }}_w.\end{array}\end{array}$$

(9)

For this case, Equation (8) reduces to

$$\left[-{\displaystyle \frac{d^2}{d{\tilde{\beta }}^2}}-{\displaystyle \frac{3}{\tilde{\beta }}}{\displaystyle \frac{d}{d\tilde{\beta }}}+{\displaystyle \frac{L(L+1)-\frac{3}{4}{R}^2}{{\tilde{\beta }}^2}}\right]f\left(\tilde{\beta }\right)\equiv H_{ISW}f\left(\tilde{\beta }\right)=\omega f\left(\tilde{\beta }\right),$$

(10)

where $H_{ISW}$ and $\omega$ denote the Hamiltonian and eigenvalue for the infinite square well potential. Furthermore, the function changes to $f\left(\tilde{\beta }\right)={\tilde{\beta }}^{-1}J\left(\tilde{\beta }\right)$, leading to

$$\left[-{\displaystyle \frac{d^2}{d{\tilde{\beta }}^2}}-{\displaystyle \frac{1}{\tilde{\beta }}}{\displaystyle \frac{d}{d\tilde{\beta }}}+{\displaystyle \frac{L(L+1)-\frac{3}{4}{R}^2+1}{{\tilde{\beta }}^2}}\right]J\left(\tilde{\beta }\right)=\omega J\left(\tilde{\beta }\right),$$

(11)

which is changed to the Bessel form

$$\left[{\displaystyle \frac{d^2}{d{z}^2}}+{\displaystyle \frac{1}{z}}{\displaystyle \frac{d}{dz}}+\left(k^2-{\displaystyle \frac{{\nu }^2}{z^2}}\right)\right]J\left(z\right)=0,$$

(12)

by adopting the notations

$${\nu }^2=\sqrt{L(L+1)-{\displaystyle \frac{3}{4}}{R}^2+1},k^2=\omega ,$$

(13)

using the change of variable $z=k\tilde{\beta }$. The boundary conditions at ${\beta }_w$ give the eigenvalues and the eigenfunctions for the infinite square well potential:

$${\omega }_{n,\nu }={\left(k_{n,\nu }\right)}^2={\left({\displaystyle \frac{z_{n,\nu }}{{\tilde{\beta }}_w}}\right)}^2,$$

(14)

$$f_{\nu ,n}\left(\tilde{\beta }\right)={\displaystyle \frac{\sqrt{2}}{{\tilde{\beta }}_w}}{\displaystyle \frac{{\tilde{\beta }}^{-1}{J}_{\nu ,n}\left(\frac{z_{\nu ,n}}{{\tilde{\beta }}_w}\tilde{\beta }\right)}{J_{\nu +1}\left(z_{\nu ,n}\right)}},$$

(15)

where $z_{\nu ,n}$ is the n^th zero of the Bessel function $J_{\nu }\left(z\right)$ of index $\nu$. Furthermore, the wave function for the sextic potential is written in terms of these functions as

$$F_{\nu ,\xi }\left(\tilde{\beta }\right)=\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{f}_{\nu ,n}\left(\tilde{\beta }\right)\equiv \sum _{n=1}^{n_{Max}}{A}_n^{\xi }|\nu n\rangle ,$$

(16)

where $n_{Max}$ is the dimension of the basis truncation, while $\xi =n_{\beta }+1$ is related to the $\beta$ vibration quantum number $n_{\beta }=0,1,\dots ,n_{Max}-1$. For numerical applications using the experimental data, $n_{Max}=20$ is enough to obtain a convergence accuracy of ${10}^{-7}$. This is calculated according to Equation (18) from Ref. [46]. Introducing the wave function (16) in Equation (8), the following result is obtained:

$$\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{H}_{ISW}|\nu n\rangle +\sum _{n=1}^{n_{Max}}{A}_n^{\xi }\left({\tilde{\beta }}^2+A{\tilde{\beta }}^4+B{\tilde{\beta }}^6\right)|\nu n\rangle =\tilde{\epsilon }\sum _{n=1}^{n_{Max}}{A}_n^{\xi }|\nu n\rangle .$$

(17)

Furthermore, the left side of this equation is multiplied by $\langle \nu m|$ to obtain

$$\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{\left({\displaystyle \frac{z_{\nu ,n}}{{\tilde{\beta }}_w}}\right)}^2{\delta }_{mn}+\sum _{n=1}^{n_{Max}}{A}_n^{\xi }\langle \nu m|\left({\tilde{\beta }}^2+A{\tilde{\beta }}^4+B{\tilde{\beta }}^6\right)|\nu n\rangle =\tilde{\epsilon }\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{\delta }_{mn},$$

(18)

while the matrix elements for the sextic potential are calculated as follows:

$$\begin{array}{ccc}\hfill \langle \nu m|{\tilde{\beta }}^{2i}|\nu n\rangle & =& {\int }_0^{{\tilde{\beta }}_w}{f}_{\nu ,m}\left(\tilde{\beta }\right){\tilde{\beta }}^{2i}{f}_{\nu ,n}\left(\tilde{\beta }\right){\tilde{\beta }}^{3}d\tilde{\beta },i=1,2,3\hfill \\ & =& {\displaystyle \frac{2}{{\tilde{\beta }}_w^2}}{\displaystyle \frac{1}{J_{\nu +1}\left(z_{\nu ,m}\right)J_{\nu +1}\left(z_{\nu ,n}\right)}}{\int }_0^{{\tilde{\beta }}_w}{J}_{\nu }\left({\displaystyle \frac{z_{\nu ,m}}{{\tilde{\beta }}_w}}\tilde{\beta }\right)J_{\nu }\left({\displaystyle \frac{z_{\nu ,n}}{{\tilde{\beta }}_w}}\tilde{\beta }\right){\tilde{\beta }}^{2i+1}d\tilde{\beta }.\hfill \end{array}$$

(19)

By changing the variable using $x=\tilde{\beta }/{\tilde{\beta }}_w$, the parameter ${\beta }_w$ is moved in front of the integral as a scaling parameter:

$$\begin{array}{ccc}\hfill \langle \nu m\left|{\tilde{\beta }}^{2i}\right|\nu n\rangle & =& {\displaystyle \frac{2{\tilde{\beta }}_w^{2i}}{J_{\nu +1}\left(z_{\nu ,m}\right)J_{\nu +1}\left(z_{\nu ,n}\right)}}{\int }_0^1{J}_{\nu }\left(z_{\nu ,m}x\right)J_{\nu }\left(z_{\nu ,n}x\right)x^{2i+1}dx\hfill \\ & =& {\tilde{\beta }}_w^{2i}{I}_{mn}^{(\nu ,i)}.\hfill \end{array}$$

(20)

This latter aspect simplifies the fitting procedure of the experimental data because these integrals, denoted by $I_{mn}^{(\nu ,i)}$, can be calculated only once a subroutine is independent of any parameter. By using the above result in Equation (18), one can write it as follows:

$$\sum _{n=1}^{n_{Max}}{A}_n^{\xi }\left[{\left({\displaystyle \frac{z_{\nu ,n}}{{\tilde{\beta }}_w}}\right)}^2{\delta }_{mn}+\left({\tilde{\beta }}_w^2{I}_{mn}^{(\nu ,2)}+A{\tilde{\beta }}_w^4{I}_{mn}^{(\nu ,4)}+B{\tilde{\beta }}_w^6{I}_{mn}^{(\nu ,6)}\right)\right]=\tilde{\epsilon }\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{\delta }_{mn},$$

(21)

or, in a compact form as:

$$\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{H}_{mn}=\tilde{\epsilon }\sum _{n=1}^{n_{Max}}{A}_n^{\xi }{\delta }_{mn},$$

(22)

where $H_{mn}$ is the general matrix element. This is further used to find the eigenvalue $\tilde{\epsilon }$ and then the total energy of the nuclear system:

$$E_{\xi ,L,R}({\tilde{\beta }}_w,A,B)=\sqrt{a}{\displaystyle \frac{{ħ}^2}{2B}}{\tilde{\epsilon }}_{\xi ,L,R}({\tilde{\beta }}_w,A,B).$$

(23)

The total energy depends on three quantum numbers $(\xi ,L,R)$; two free parameters $(A,B)$; a scaling parameter $\sqrt{a}{\displaystyle \frac{{ħ}^2}{2B}}$; and a special parameter $\tilde{\beta }$, which is fixed so that the infinite square well potential intersects the tail of the sextic potential at a very high spin state. The scaling parameter can be removed if the energies are normalized to the energy of the first exited state of the ground band:

$$\begin{array}{ccc}\hfill R_{\xi ,L,R}({\tilde{\beta }}_w,A,B)& =& {\displaystyle \frac{E_{\xi ,L,R}({\tilde{\beta }}_w,A,B)-E_{1,0,0}({\tilde{\beta }}_w,A,B)}{E_{1,2,2}({\tilde{\beta }}_w,A,B)-E_{1,0,0}({\tilde{\beta }}_w,A,B)}}\hfill \\ & =& {\displaystyle \frac{{\epsilon }_{\xi ,L,R}({\tilde{\beta }}_w,A,B)-{\epsilon }_{1,0,0}({\tilde{\beta }}_w,A,B)}{{\epsilon }_{1,2,2}({\tilde{\beta }}_w,A,B)-{\epsilon }_{1,0,0}({\tilde{\beta }}_w,A,B)}}\hfill \end{array}$$

(24)

The eigenvector components $A_n^{\xi }$, given by Equation (22), determine the wave function (16), which is used to calculate other physical quantities related to the experimental data. Such an example is the probability density distribution of the $\tilde{\beta }$ deformation:

$${\rho }_{\xi ,L,R}={\left[F_{\xi ,L,R}\left(\tilde{\beta }\right)\right]}^2{\tilde{\beta }}^3,$$

(25)

which is analyzed in relation to the shape of the effective sextic potential:

$$v_{eff}\left(\tilde{\beta }\right)={\displaystyle \frac{L(L+1)-\frac{3}{4}(R^2+1)}{{\tilde{\beta }}^2}}+{\tilde{\beta }}^2+A{\tilde{\beta }}^4+B{\tilde{\beta }}^6.$$

(26)

The effective potential (26) is obtained by writing Equation (8) in a Schrödinger form through the change of function $F\left(\tilde{\beta }\right)={\tilde{\beta }}^{-{\displaystyle \frac{3}{2}}}\phi \left(\tilde{\beta }\right)$:

$$\left[-{\displaystyle \frac{d^2}{d{\tilde{\beta }}^2}}+{\displaystyle \frac{L(L+1)-\frac{3}{4}(R^2+1)}{{\tilde{\beta }}^2}}+{\tilde{\beta }}^2+A{\tilde{\beta }}^4+B{\tilde{\beta }}^6\right]\phi \left(\tilde{\beta }\right)=\tilde{\epsilon }\phi \left(\tilde{\beta }\right).$$

(27)

Another important quantity, which is considered a signature for the presence of the shape coexistence and mixing phenomena [57,58], is the electric monopole transition (E0) between the first excited $0_2^{+}$ and the ground state $0_1^{+}$:

$${\rho }^2(E0;0_2^{+}\to 0_1^{+})={\left({\displaystyle \frac{3Z}{4\pi }}\right)}^2{\beta }_M^4\langle F_{2,0,0}\left(\tilde{\beta }\right)|{\tilde{\beta }}^2|F_{1,0,0}\left(\tilde{\beta }\right)\rangle ,$$

(28)

where Z and ${\beta }_M$ are the charge number and a scaling parameter, with the latter being fixed for the experimental data of the quadrupole electromagnetic transition between the first excited $2^{+}$ and ground state $0_1^{+}$. The quadrupole electromagnetic transitions (E2) are calculated using the following transition operator [14]:

$$T_{\mu }^{\left(E2\right)}=t\beta \left[D_{\mu ,0}^{\left(2\right)}\left(\mathrm{\Omega }\right)cos\left(\gamma -{\displaystyle \frac{2\pi }{3}}\right)+{\displaystyle \frac{1}{\sqrt{2}}}\left(D_{\mu ,2}^{\left(2\right)}\left(\mathrm{\Omega }\right)+D_{\mu ,-2}^{\left(2\right)}\left(\mathrm{\Omega }\right)\right)sin\left(\gamma -{\displaystyle \frac{2\pi }{3}}\right)\right],$$

(29)

where [55]

$$t={\displaystyle \frac{3R^{2}Ze}{4\pi }}{\beta }_M$$

(30)

is a scaling parameter that depends on the nuclear radius $R=R_0{A}^{1/3}$, where A is the atomic mass and $R_0=1.2$ fm (1 fm = ${10}^{-15}$ m); charge number Z; elementary charge e; and ${\beta }_M$. The parameter ${\beta }_M$, which also appears in Equation (28), is related to the scaled variable $\tilde{\beta }$ and the quadrupole deformation variable ${\beta }_2$ by the simple formula ${\beta }_2={\beta }_M\tilde{\beta }$. Taking $\gamma =\pi /6$ in Equation (29), one obtains a simplified form for the transition operator [14]:

$$T_{\mu }^{\left(E2\right)}=-{\displaystyle \frac{1}{\sqrt{2}}}t\beta \left(D_{\mu ,2}^{\left(2\right)}\left(\mathrm{\Omega }\right)+D_{\mu ,-2}^{\left(2\right)}\left(\mathrm{\Omega }\right)\right)$$

(31)

The $B\left(E2\right)$ electromagnetic transitions are then given by [14]

$$\begin{array}{ccc}\hfill B(E2;{\xi }_i{L}_i{R}_i\to {\xi }_f{L}_f{R}_f)& =& {\displaystyle \frac{5}{16\pi }}{\displaystyle \frac{t^2}{2}}{\displaystyle \frac{1}{(1+{\delta }_{R_i,0})(1+{\delta }_{R_f,0})}}{I}_{{\xi }_i{L}_i{R}_i,{\xi }_f{L}_f{R}_f}\hfill \\ & & \times {\left[C_{R_{i}2R_f}^{L_{i}2L_f}+C_{R_i-2R_f}^{L_{i}2L_f}+{(-1)}^{L_i}{C}_{-R_{i}2R_f}^{L_{i}2L_f}\right]}^2,\hfill \end{array}$$

(32)

where the Cs are the Clebsch–Gordan coefficients that represent the contribution of the matrix element over the Euler angles, while

$$I_{{\xi }_i{L}_i{R}_i,{\xi }_f{L}_f{R}_f}=\langle F_{{\xi }_i,L_i,R_i}\left(\tilde{\beta }\right)|\tilde{\beta }{|F_{{\xi }_f,L_f,R_f}\left(\tilde{\beta }\right)\rangle }^2$$

(33)

are the matrix elements over the $\beta$ variable. The latter can be expressed as a function of $\tilde{\beta }$ and finally of x, given by Equation (20) by a renormalization of the scaling parameter t by a factor that depends on the scaling parameter a.

The present model is able to describe triaxial deformed nuclei without $\gamma$-vibrations, and more importantly, due to the sextic potential consideration in the $\beta$ variable, it can cover the shape phase transition, shape coexistence and mixing phenomena between an approximately spherical shape and a triaxially deformed one. Moreover, unusually small $B\left(E2\right)$ transitions can also be explained within the proposed model. These aspects are addressed in more detail in the next two sections.

3. Numerical Results

A first application of the model was made for the experimental data of the ^192,194,196Pt isotopes [59,60,61] and compared in

Table 1. The experimental energy spectrum of the ground, $\beta$ and $\gamma$ bands of the ^192,194,196Pt isotopes [59,60,61] were compared with the corresponding values calculated with the present numerical solution (NS) with respect to the quasi-exact solutions QE I ($k=4$) [27] and QE II ($k=10$) [46], where k gives the number of the exact solutions within the quasi-exactly solvable model [27,28,29]. For ¹⁹²Pt, ¹⁹⁴Pt and ¹⁹⁶Pt, the values of the fitted parameters $(A,B)$ were $(-100,127.2)$, $(-93.81,100)$, $(-50,68.5)$, respectively, and ${\tilde{\beta }}_w=(1.47,1.15,1.50)$, respectively.

	192Pt				194Pt				196Pt
${\mathit{R}}_{\mathit{\xi },\mathit{L},\mathit{R}}$	Exp.	QE I	QE II	NS	Exp.	QE I	QE II	NS	Exp.	QE I	QE II	NS
R1,4,4	2.479	2.439	2.412	2.278	2.470	2.415	2.414	2.317	2.465	2.513	2.418	2.231
R1,6,6	4.314	3.787	3.821	3.797	4.298	3.835	3.869	3.916	4.290	3.709	3.691	3.664
R1,8,8	6.377	5.773	5.780	5.531	6.392	5.880	5.879	5.772	6.333	5.579	5.511	5.279
R1,10,10	8.624	7.350	7.453	7.461	8.672	7.573	7.629	7.873	8.558	6.914	6.970	7.060
R1,2,0	1.935	1.653	1.660	1.796	1.894	1.661	1.669	1.817	1.936	1.646	1.637	1.770
R1,3,2	2.910	2.303	2.330	2.509	2.809	2.332	2.355	2.558	2.852	2.249	2.264	2.450
R1,4,2	3.795	4.229	4.218	4.198	3.743	4.268	4.267	4.343	3.636	4.179	4.090	4.040
R1,5,4	4.682	4.342	4.360	4.395	4.563	4.402	4.422	4.553	4.526	4.243	4.197	4.223
R1,6,4	5.905	6.358	6.440	6.604					5.644	6.041	6.069	6.271
R1,7,6	6.677	6.065	6.171	6.428
R1,8,6	8.186	9.163	9.221	9.100					7.730	8.564	8.573	8.563
R2,0,0	3.776	3.397	3.510	3.317	3.858	3.706	3.727	3.615	3.192	2.954	3.031	3.002
R2,2,2	4.547	4.995	5.134	5.146	4.603	5.409	5.407	5.643	3.828	4.308	4.476	4.697
rms		0.614	0.601	0.605		0.543	0.525	0.525		0.682	0.686	0.714

with the results obtained with the quasi-exact solution (QE) of the Bohr Hamiltonian with a sextic potential [27,28]. As one can see, even if there are some differences in the calculated energies of the two models, especially for the heads of the ground and $\gamma$ bands, the values of the root mean square (rms) quantities were almost the same. This was not a surprise when taking into account that they have the same Hamiltonian and potential, but only different solving methods. Actually, the result represents a test and a validation of the present model which, compared with the quasi-exact model dedicated only to the shape phase-transition phenomena, can cover new phenomena, such as shape coexistence and mixing.

Thus, the model could describe both shape phase transitions and shape coexistence with mixing phenomena by involving the special properties of the sextic potential. For example, Figure 1 illustrates the ability of the model to describe a phase transition from an approximately spherical shape to a deformed one crossing the region of the critical point. The sextic potential had an approximately spherical minimum for ⁹⁸Ru, a flat shape for ^102,104Ru (critical point region) and a deformed minimum for ¹⁰⁸Ru. The corresponding probability density distribution presented a single peak for the $0_1^{+}$ and $2_1^{+}$ states ($n_{\beta }=0$, no node) and two peaks for $0_2^{+}$ ($n_{\beta }=1$, one node). The peaks were centered around the minima, with a larger width at the critical point. The picture was dramatically changed when the barrier (maximum) that separated the spherical and deformed minima was introduced and increased step by step as in Figure 2. In panel (a) of Figure 2, the effective sextic potential has two degenerated minima, with the ground state being slightly below the barrier. One can observe in panel (b) that even for a moderate increase in the barrier, one has a slight splitting of the peak for the ground state, in comparison with the situation for the critical point, even if one has no node for its corresponding wave function. The situation can be interpreted within the frame of the present model as a transition from shape fluctuations of the critical point of the shape phase transitions to the shape coexistence with mixing. This behavior is more evidently presented in panels (e,f) where, for a higher height of the barrier, there is a clearer splitting, while the probability density distribution for $0_1^{+}$ and $0_2^{+}$ looks like in a reflection. The phenomenon was also manifested for excited states. In panel (c), the $2_1^{+}$ state is positioned below the barrier of the effective potential given by $L=2$, which leads to a splitting of the probability density distribution (panel (d)), even if one again has no node for the first excited state of the ground band. An interesting remark is that here, the probability density distribution for $0_2^{+}$ did not follow the form corresponding to one node anymore, as is the case in the panels of Figure 1, with a more dominant peak around the deformed minimum. Moreover, for a very high barrier, the small peak around the approximately spherical minimum vanished, and the $\beta$-vibration feature of the $0_2^{+}$ state was canceled. This latter case was interpreted within the present model as being shape coexistence without mixing because the states $0_1^{+}$ and $0_2^{+}$ were almost completely separated by the barrier and their plots for the probability density distribution of deformation did not overlap at all (did not mix). In Figure 2, panel (g), the shape coexistence without mixing appears because the barrier is higher and thicker compared with the situation with mixing. Moreover, the density probability distribution for the $2_1^{+}$ state had a single peak centered above the deformed minimum, which overlapped with that for $0_2^{+}$, even if the ground state preferred the spherical one. This behavior led to a very small B(E2) transition between these states, even if they belonged to the same band. The same thing was valid for excited states of the same band that manifested a different ${\beta }_2$ quadrupole deformation. This scenario is appropriate for describing anomalously small B(E2) transitions observed in different experiments and predicted by other models [43,44]. Figure 2 shows some representative situations related to shape coexistence and mixing, which can be described by the model. There are also other possible cases, as the second deformed minimum was much higher in energy than the spherical one or even lower in energy. All these representations of the model will be exploited in future applications using the experimental data.

4. Discussion

A new solution is proposed for the Davydov–Chaban Hamiltonian by numerically solving the corresponding eigenvalue problem for a sextic oscillator potential in the $\beta$ variable using Bessel functions of the first kind as a basis. The model is able to describe the phase transition, shape coexistence and mixing between an approximately spherical shape and a rigid triaxial one (${\gamma }_0={30}^{\circ }$), respectively anomalously small B(E2) transition in the ground band. This aspect is evidenced by some applications performed within the present study for isotopes of platinum (Pt) and ruthenium (Ru), but also from arbitrary selected sets of the potential parameters suggesting special cases, which can be found by future applications using the experimental data. The model can be further improved by relaxing the $\gamma$-rigidity or, in other words, by taking into account the $\gamma$-vibration and, consequently, by having a stable triaxial deformation.

Other arguments that support these conclusions were given by many previous applications to describe shape phase transitions and their critical points [19,25,26,27,28,29,31,33,34,46], shape coexistence and mixing [45,46,49,50,52,53,54], and anomalously small B(E2) transition ratios [53,54]. For example, by applying the model to certain isotopic chains, it was shown that the experimental energy level structure and the quadrupole electromagnetic transitions between the corresponding states were very well reproduced by considering that the isotopes followed a phase transition from an approximately spherical shape to a deformed one that crossed a critical point for which the sextic potential had a flat shape. In turn, for the large monopole transition between the first excited $0^{+}$ and ground state, the large quarupole transitions between states belonging to different bands, which were even greater than the quadrupole transitions in-band, and some back-bending in energy observed in the experiment could be well described within our model only by taking into account the presence of the shape coexistence with the mixing phenomenon. The presence of different shapes and the mixing of the shapes dramatically changed the picture for the structure of the states by comparison with their absence. The mixing of the states of different shapes and from different bands increased the value for the matrix element of the transition operator, which was otherwise very small. The presence of the coexistence with mixing within our model was illustrated by the fact that the states of the ground band and of the $\beta$ bands presents two and three peaks in the corresponding plots of the probability density distribution of deformation rather than one and two, which should be for the no-node and one-node conditions of the wave function, respectively. Thus, the states were characterized by a superposition of the two shapes that emerged from the two minima of the sextic potential. Equally interesting was also the situation when a state had different quadrupole deformation than the lower one from the same band and these states were separated by a high barrier of the potential such that the electromagnetic transition became more difficult to take place. This scenario seemed to fit very well with some unusually small quadrupole transition ratios observed experimentally. There was a correlation here between the back-bending in energy and the very small electromagnetic transition, which was well reproduced within our model.