1. Petrică Buganu, and Radu Budaca IFIN-HH, Bucharest – Magurele, Romania International Workshop “Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects” (SDANCA – 15), 8 – 10 October 2015, Sofia, Bulgaria

2. The Bohr-Mottelson Hamiltonian: The γ-rigid Hamiltonian for γ=30 o : The γ-rigid Hamiltonian for γ=0 o : E(5): F. Iachello, Phys. Rev. Lett. 85 (2000) 3580. spherical vibrator to γ-unstable rotor X(5): F. Iachello, Phys. Rev. Lett. 87 (2001) 052502. spherical vibrator to axial rotor Y(5): F. Iachello, Phys. Rev. Lett. 91 (2003) 132502. axial rotor to triaxial rotor Z(5): D. Bonatsos, D. Lenis, D. Petrellis, and P. A. Terziev, Phys. Lett. B 588 (2004) 172. prolate rotor to oblate rotor?! Z(4): D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 621 (2005) 102. A. S. Davydov, and A. A. Chaban, Nucl. Phys. 20 (1960) 499. X(3): D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 632 (2006) 238. A. Bohr, Mat. Fyz. Medd. K. Dan. Vidensk. Selsk. 26 (1952) No. 14. A. Bohr, and B. R. Mottelson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 27 (1953) No. 16.

3. The potentials in the β variable and the γ rigidity values for the most recent γ-rigid solutions. D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 621 (2005) 102. D. Bonatsos, D. Lenis, D. Petrellis, P. A. Terziev, and I. Yigitoglu, Phys. Lett. B 632 (2006) 238. R. Budaca, Eur. Phys. J. A 50 (2014) 87. R. Budaca, Phys. Lett. B 739 (2014) 56. P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106. P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306.

4. Sextic oscillator potential Exact separation of the variables: X(3)-Sextic and Z(4)-Sextic

5. The quasi-exactly solution for the sextic potential A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics, (Institute of Physics Publishing, Bristol, 1994)

6. Numerical results Z(4)-SexticX(3)-Sextic Z(4)-Sextic: P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306. X(3)-Sextic: P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106.

7. Z(4)-Sextic: P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306. X(3)-Sextic: P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106. Degenerate states! A possible dynamical symmetry?! Z(4)-Sextic X(3)-Sextic Parameter free solutions

10. Z(4)-Sextic: P. Buganu, and R. Budaca, Phys. Rev. C 91 (2015) 014306. X(3)-Sextic: P. Buganu, and R. Budaca, J. Phys. G: Nucl. Part. Phys. 42 (2015) 105106. Z(4)-Sextic X(3)-Sextic Experimental realisation of the predicted shape phase transitions

11. Conclusions  Two new γ-rigid solutions have been proposed, called Z(4)-Sextic and X(3)-Sextic. For both of them, a sextic potential is used which leads to a quasi-exactly solvable equation.  Up to some scale parameters, the energies and the E2 transition probabilities depend on a single free parameter. For special cases when the term β 2 or β 4 cancels, parameter free solutions are obtained.  Varying the free parameter, shape phase transitions from an approximately spherical shape to a well deformed one are described. In the critical point the potential is flat leading to numerical results which are closed to those of X(3) and Z(4) for which an infinite square well was used.  In the critical point of X(3)-Sextic the states are approximately degenerate, indicating the presence of a symmetry which can offer answers for the unknown symmetry of X(5). The β bands of some X(5) candidate nuclei are well described in the present picture.  The plot of the free parameter as a function of the neutron number for isotopes of Xe, Pt, Sm and Nd reveales the presence of the proposed shape phase transitions in these chains.

12. Content Introduction Brief presentation of the new γ– rigid solutions Numerical results Conclusions

13. Introduction: Bohr Collective Model The excitation spectra of the nuclei are interpreted as vibrations and rotations of the nuclear surface: R 0 – radius of spherical nucleus, α λμ – surface collective coordinates, Y λμ (θ,φ) – spherical harmonics. Types of multipole deformations: monopole dipole quadrupole octupole hexadecupole A. Bohr, Mat. Fyz. Medd. K. Dan. Vidensk. Selsk. 26 (1952) No. 14. A. Bohr, and B. R. Mottelson, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 27 (1953) No. 16.

14. Quadrupole deformation: Wigner function Bohr-Mottelson transformation: Euler angles β=0.4 and γ=nπ/3 (n=0,1,2,3,4,5.): prolate(n=0,2,4), oblate (n=1,3,5) and triaxial in rest. L. Fortunato, Eur. Phys. J. A 26 (2005) 1-30. The stretching of the nuclear axis. W. Greiner, J. A. Maruhn, Nuclear Models, Springer-Verlag Berlin Heidelberg (1996).

15. Page  15 Exactly separation of variables for γ=30 0 Sextic oscillator with centrifugal barrier for the variable β

16. Page  16 Condition to have a potential independent of state: L – even L – odd Final form of the potential