Sextic Oscillator Potential Research Articles

Bohr Hamiltonian with sextic potential for γ -rigid prolate nuclei with deformation-dependent mass term

The main aim of the present paper is to solve the eigenvalues problem with the Bohr collective Hamiltonian for $\ensuremath{\gamma}$-rigid nuclei within a model we have elaborated by combining two model approaches: the quantum mechanical formalism, namely, deformation-dependent mass formalism (DDM), and the anharmonic sextic oscillator potential for the variable $\ensuremath{\beta}$ and $\ensuremath{\gamma}=0$. The model developed in this way is conventionally called the sextic and DDM approach. Analytical expressions for energy spectra are conjointly derived by means of quasiexact solvability and a quantum perturbation method. Due to the scaling property of the problem, the energy and $B(E2)$ transition ratios depend on two free parameters apart from an integer number which limits the number of allowed states. Numerical results are given for 35 nuclei---$^{98--108}\mathrm{Ru}, ^{100--102}\mathrm{Mo}, ^{116--130}\mathrm{Xe}, ^{180--196}\mathrm{Pt}, ^{172}\mathrm{Os}, ^{146--150}\mathrm{Nd}, ^{132--134}\mathrm{Ce}, ^{154}\mathrm{Gd}, ^{156}\mathrm{Dy}$, and $^{150--152}\mathrm{Sm}$---revealing a good agreement with experiment. Moreover, as proved for the first time by Bonatsos et al. [D. Bonatsos, P. Georgoudis, D. Lenis, N. Minkov, and C. Quesne, Phys. Lett. B 683, 264 (2010)], the dependence of the mass on the deformation with the sextic potential moderates the increase of the moment of inertia with the deformation, removing an important drawback that has been revealed in the constant mass case [Buganu and Budaca, J. Phys. G: Nucl. Part. Phys. 42, 105106 (2015)]. Additionally, the correlation between the DDM and the minimal length formalism persists for the sextic potential. Finally, the DDM effects on the shape phase transition for the most numerous isotopic chains, namely, Ru, Xe, Nd, and Pt, have been duly investigated.

Extended study on the application of the sextic potential in the frame of X(3)-sextic

The main aim of the present paper is to extensively study the γ-rigid Bohr Hamiltonian with anharmonic sextic oscillator potential for the variable β and γ = 0. For the corresponding spectral problem, a finite number of eigenvalues are explicitly found, by algebraic means, the so-called quasi-exact solvability (QES). The evolution of the spectral and electromagnetic properties by considering higher exact solvability orders is investigated, especially the approximate degeneracy of the ground and first two β bands at the critical point of the shape phase transition from a harmonic to an anharmonic prolate β-soft, as well as the shape evolution within an isotopic chain. The numerical results are given for 39 nuclei, namely, 98–108Ru, 100–102Mo, 116–130Xe, 180–196Pt, 172Os, 146–150Nd, 132–134Ce, 152–154Gd, 154–156Dy, 150–152Sm, 190Hg and 222Ra. Across this study, it seems that the higher QES order improves our results by decreasing the root mean square, mostly for deformed nuclei. The nuclei 100,104Ru, 118,120,126,128Xe, 148Nd and 172Os fall exactly at the critical point.

Quasi-exact solutions for the Bohr Hamiltonian with sextic oscillator potential

A discussion on the quasi-exact solution of the Bohr Hamiltonian with sextic oscillator potential is made by attracting the attention on some recent results of its application to the phase transition from spherical vibrator to a γ-unstable system. More precisely, it is underlined the importance of the solvability order on the structure of the states, especially in the critical point, respectively, in the deformed region of the phase transition.

Extended study on a quasi-exact solution of the Bohr Hamiltonian

A unified presentation of the quasi-exact solutions of the Bohr Hamiltonian with a sextic oscillator potential is offered by evidencing the similarities and the differences between them. An extended study of the Bohr Hamiltonian with sextic potential for γ-rigid triaxial nuclei is conducted, focusing on the evolution of the energy spectra and B(E2) transition probabilities when the order of the quasi-exactly solvable system is increased. The effect is analyzed throughout the shape phase transition from a γ-rigid triaxial vibrator to an anharmonic one and especially in its critical point. The new results are tested by several applications for experimental data, improving the previous calculations and providing new possible triaxial nuclei.

Sextic potential for -rigid prolate nuclei

The equation of the Bohr–Mottelson Hamiltonian with a sextic oscillator potential is solved for -rigid prolate nuclei. The associated shape phase space is reduced to three variables which are exactly separated. The angular equation has the spherical harmonic functions as solutions, while the equation is converted to the quasi-exactly solvable case of the sextic oscillator potential with a centrifugal barrier. The energies and the corresponding wave functions are given in closed form and depend, up to a scaling factor, on a single parameter. The and states are exactly determined, having an important role in the assignment of some ambiguous states for the experimental bands. Due to the special properties of the sextic potential, the model can simulate, by varying the free parameter, a shape phase transition from a harmonic to an anharmonic prolate -soft rotor crossing through a critical point. Numerical applications are performed for 39 nuclei: Ru, Mo, Xe, Ce, Nd, Sm, Gd, Dy, 172Os, Pt, 190Hg and 222Ra. The best candidates for the critical point are found to be 104Ru and Xe, followed closely by 128Xe, 172Os, 196Pt and 148Nd.

Analytical solution for the Davydov-Chaban Hamiltonian with a sextic potential forγ=30∘

An analytical solution for the Davydov-Chaban Hamiltonian with a sextic oscillator potential for the variable $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ fixed to ${30}^{\ensuremath{\circ}}$ is proposed. The model is conventionally called Z(4)-sextic. For the potential shapes considered, the solution is exact for the ground and $\ensuremath{\beta}$ bands, while for the $\ensuremath{\gamma}$ band an approximation is adopted. Due to the scaling property of the problem, the energy and $B(E2)$ transition ratios depend on a single parameter apart from an integer number which limits the number of allowed states. For certain constraints imposed on the free parameter, which lead to simpler special potentials, the energy and $B(E2)$ transition ratios are parameter independent. The energy spectra of the ground and first $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ bands as well as the corresponding $B(E2)$ transitions, determined with Z(4)-sextic, are studied as function of the free parameter and presented in detail for the special cases. Numerical applications are done for the $^{128,130,132}\mathrm{Xe}$ and $^{192,194,196}\mathrm{Pt}$ isotopes, revealing a qualitative agreement with experiment and a phase transition in Xe isotopes.

Application of the sextic oscillator potential together with Mathieu and spheroidal functions for triaxial and X(5) type nuclei

The Bohr-Mottelson Hamiltonian is amended with a potential which depends on both β and γ deformation variables and which allows us to separate the β variable from the other variables. The equation for the β variable is quasi-exactly solved for a sextic oscillator with centrifugal barrier potential. Concerning the γ equation, its solutions are the angular spheroidal and Mathieu functions for X(5) type and triaxial nuclei, respectively. The models developed in this way are conventionally called the Sextic and Spheroidal Approach (SSA) and the Sextic and Mathieu Approach (SMA). SSA and SMA was successfully applied for several nuclei, details being presented below.

Application of the sextic oscillator with a centrifugal barrier and the spheroidal equation for some X(5) candidate nuclei

The eigenvalue equation associated with the Bohr–Mottelson Hamiltonian is considered in the intrinsic reference frame and amended by replacing the harmonic oscillator potential in the β variable with a sextic oscillator potential with centrifugal barrier plus a periodic potential for the γ variable. After the separation of variables, the β equation is quasi-exactly solved, while the solutions for the γ equation are just the angular spheroidal functions. An anharmonic transition operator is used to determine the reduced E2 transition probabilities. The formalism is conventionally called the sextic and spheroidal approach (SSA) and applied for several X(5) candidate nuclei: 176, 178, 180, 188, 190Os, 150Nd, 170W, 156Dy and 166, 168Hf. The SSA predictions are in good agreement with the experimental data of the mentioned nuclei. The comparison of the SSA results with those yielded by other models, such as X(5) (Iachello 2001 Phys. Rev. Lett.87052502), infinite square well (Raduta et al 2009 Nucl. Phys.A 81946) and Davidson like potential (Raduta et al) for the β, otherwise keeping the spheroidal functions for the γ , and the coherent state model (Raduta et al 1981 Phys. Lett. B 99 444, Raduta et al 1982 Nucl. Phys.A 381 253, Raduta et al 1987 Phys. Rev. C 36 2111, Raduta et al 1983 Z. Phys. A 313 69, Raduta et al 1997 Phys. Rev. C 55 1747, Raduta et al 2002 Phys. Rev. C 65 064322, Raduta and Sabac 1983 Ann. Phys., NY 148 1, Raduta 2004 Recent Research Developments in Nuclear Physics vol 1) respectively, suggests that SSA represents a good approach to describe nuclei achieving the critical point of the U(5)→SU(3) shape phase transition.

Searching critical-point nuclei in Te- and Xe-isotopic chains using sextic oscillator potential

We have identified the nuclei in the Te- and Xe-isotopic chains lying close to the critical point, through which the shape phase transition occurs, by using the sextic oscillator potential formalism. It has been found that 110Te, 124Te, and 124Xe isotopes are the most promising candidates for the critical-point nuclei slightly above the Z = 50 proton shell closure.