Collective excitations in even-even nuclei with a stepped inﬁnite square well potential

. A solution is worked out for a γ -unstable Bohr Hamiltonian with an inﬁnite square well potential having an adjustable step. The evolution of the model’s spectral characteristics are investigated with varying height and width of the additional step, revealing shape coexisting an mixing features. The description of the dynamical shape phase transition in the collective bands of the Te even-even nuclei is o ﬀ ered as a numerical application.


Introduction
The introduction of the critical point solution E (5) [1] gave the start for a variety of studies regarding algebraic and phenomenological description of critical phenomena in the collective excitations of atomic nuclei.Basically, the E(5) model is constructed on the solution of the fivedimensional γ-unstable Bohr Hamiltonian (BH) [2] with an infinite square well (ISW) potential only in the variable associated to axial deformation.The flat potential serves as good phenomenological picture for the critical point of the transition between deformed γ-soft rotor and the spherical harmonic vibrator pictures of the nuclear surface fluctuation.It's solution can also be used as a suitable basis state for diagonalization of double well collective potentials [3].Here we present a new extension of the E(5) by introducing a finite step in the ISW [4].

Theoretical framework
For the γ unstable case, the general five-dimensional Bohr Hamiltonian can be reduced, after integration on the S O(5) coordinates, to the following equation for the axial variable β: where v(β) = 2BV(β)/ℏ 2 and ϵ = 2BE/ℏ 2 are reduced potential and energies scaled by the mass parameter B, while τ is the so called seniority [5].For the states considered in this study, the L = 2τ correspondence to angular momentum holds.We solve the above equation for a ISW potential with a step: with v 0 > 0 and β s ∈ [0, 1] being the height and the starting position of the step.The general solution is a combination of Bessel functions of the first and second kind, with distinct behavior in specific regions of β values depending on the energy relative to the height v 0 .The wave function must vanish at external walls and match at the position of the step.These boundary and continuity conditions provide determining equations for the total energy and the wave functions.The phenomenological interpretation of the obtained solution is easily extracted from its correlation with the shape of the effective potential As can be seen in Fig. 1, the inclusion of the centrifugal effect induces a double-pocket shape even in the ground state.Such double well potentials are well known to be associated with shape coexisting and mixing phenomena [3,4,[6][7][8][9].

Model application
The energy is determined from a transcendental equation which is solved numerically.The E(5) critical point model [1] serves as an upper and lower limit for the energy spectrum when v 0 → 0 or ∞ and β s → 0 or 1.This property is used as a test for the numerical results.The E(5) low-lying states are most perturbed for wider steps and when are positioned near its height.The presence of the additional step usually compress the ground state band spectrum to resemble the vibrational level sequence, while simultaneously lowering more significantly the β excited states.
The even-even Te isotopes present both vibrationallike ground band and low-lying excited states and are therefore a perfect ground for testing the model.The model fits on few Te nuclei predicted a step at half ISW width with an increasing height with mass.Moreover, the model reproduces with great accuracy the very low β excited states and their rotational excitation.The low 0 + excited states is considered as a possible signature of shape coexistence [11].An example in this sense is provided in Fig. 1 for the 116 Te which has the lowest observed 0 + 2 state.As can be seen from the deformation probability distribution, there is a transition in this nucleus happening from a low deformed ground state to more deformed rotational states.The ground state also has a more extended profile, suggesting a sizable mixing between low and high deformation pockets of the effective potential.

Summary
The Bohr Hamiltonian for a γ-unstable ISW potential with an adjustable step is solved analytically.The resulted formalism constitutes an extension of the critical point model E (5) which is recovered at extreme limits of the step's width and height.The centrifugal contribution transforms the stepped potential into an effective double-pocket potential, making it suitable for treating shape coexistence and mixing phenomena.This aspect is state-dependent and can be used to describe the dynamical evolution of the rotational states from low to high deformation configurations.Moreover, due to its special construction, the model can attain very low excited β states.These features are demonstrated for the low-lying states of 116 Te nucleus, which was found to exhibit a shape coexisting ground state.

Figure 1 .
Figure 1.Theoretical effective potential v τ e f f with its isolated step (top), as well as the probability distribution (bottom) ρ τ0 (β) = |F(β)| 2 corresponding to the L = 2τ = 0, 2, 4, 6, 8 ground band states of the 116 Te nucleus.The solid and dashed lines denote the theoretical ground band and β excited band states, while circles and squares represent experimental data[10] for the ground and respectively β excited states within the same reference.