Triaxial nuclei and analytical solutions of the conformable fractional Bohr Hamiltonian with some exponential-type potentials

Prolate-oblate shape transitions and O(6) symmetry in even–even nuclei: a theoretical overview

“…Three-body interactions have been found [250] to improve the agreement between the theoretical predictions of O (6) and the data for 124−128 Xe 70-74 . Recent studies of entanglement entropy in 122−134 Xe 68-80 [221] show increasing entanglement entropy (i.e, increasing O(6) character) from 128 Xe 74 to 122 Xe 68 . 130 Ce 72 has also been suggested as lying close to the O(6) symmetry, based on experimental lifetimes [257].…”

“…Several variations of the Z(5) solution exist in the literature, taking advantage of its approximate solvability. In these variations the infinite well potential in the β variable is replaced by a sextic potential [126], a Kratzer potential [127], a Morse potential [128], a Tietz-Hua potential [128], or a multi-parameter exponential type potential [128]. In [127,128] a conformable fractional Bohr Hamiltonian is used, which is a generalization of the Bohr Hamiltonian in which the usual derivatives are replaced by conformable fractional derivatives [129], which allow for fractional orders of derivatives, while preserving the familiar properties of usual derivatives.…”

“…In these variations the infinite well potential in the β variable is replaced by a sextic potential [126], a Kratzer potential [127], a Morse potential [128], a Tietz-Hua potential [128], or a multi-parameter exponential type potential [128]. In [127,128] a conformable fractional Bohr Hamiltonian is used, which is a generalization of the Bohr Hamiltonian in which the usual derivatives are replaced by conformable fractional derivatives [129], which allow for fractional orders of derivatives, while preserving the familiar properties of usual derivatives. Conformable fractional derivatives, a special case of fractional derivatives [130][131][132] introduced in the study of critical point symmetries by Hammad [133], contain an extra parameter, the order of the derivative, thus being able to approach closer to the critical point than usual derivatives.…”

“…The second region of experimental manifestations of O(6) has been proposed in 1985, including [87,252,253], with further features explored in 126 Xe 72 [254], 128 Xe 74 [255],…”

Proxy-SU(4) symmetry in A = 60–90 region

“…More importantly, the subject of ʼsymmetries in nuclei' is undergoing a second renaissance in the last 5-10 years with several new directions in this subject are being explored. Some of these are (i) point group symmetries in heavy nuclei such as 152 Sm, 156 Gd, 236 U [8][9][10] and in lighter nuclei such as 12 C, 13 C and 16 O [11][12][13], (ii) symmetries for shape coexistence [14], (iii) multiconfiguration or composite symmetries for cluster states [15], (iv) symmetry adopted no-core-shell model based on SU(3) and Sp(6, R) algebras [16][17][18][19], (v) proton-neutron Sp(12, R) model [20][21][22], (vi) multiple algebras in shell model and IBM spaces giving for example multiple pairing and SU(3) algebras [23,24], (vii) pairing algebras with higher order interactions, generalized seniority and seniority isomers [24][25][26][27], (viii) proxy-SU(3) scheme within shell model [28,29], (ix) symmetry restoration methods in mean-field theories [30], (x) new techniques for obtaining Wigner coefficients involving SU(3) ⊃ SO(3), SO(5) ⊃ SO(3) and spin (S)-isospin(T) SU ST (4) ⊃ SU S (2) ⊗ SU T (2) [31][32][33] and (xi) studies involving Bohr Hamiltonian with sextic and other types of potentials for shape phase transitions and shape coexistence pointing out the need for three and higher-body terms in IBM and perhaps also in shell model [34][35][36][37][38][39]. (xii) random matrix ensembles with Lie algebraic ...…”