Ειδικές λύσεις της εξίσωσης Bohr στα πλαίσια των συμμετριών κρίσιμου σημείου της πυρηνικής δομής

Abstract

The concept of critical point symmetry refers to the special solutions of the Bohr Hamiltonian, known as E(5) and X(5), that were proposed six years ago by F. Iachello, in order to describe, in the framework of Bohr and Mottelson’s collective model, the changes in nuclear structure that appear as shape changes in the ground state, as the nucleon number changes. The main assumption in both models is the presence of an infinite square well potential with respect to the β variable that measures the degree of deformation of the nucleus. In E(5) the potential is independent of the γ variable (that measures the degree of asymmetry of the nuclear shape) and the solution is exact, while in X(5) the potential in γ is that of a harmonic oscillator around zero and the solution is approximate for γ ≈ 0. Furthermore, by considering energy ratios and ratios of quadrupole transition rates the results are parameter free, for direct comparison to experiment. In exactly the same spirit, a series of new solutions was proposed and their results were compared not only with those of E(5) and X(5), but with experimental data, as well. Initially, a family of potentials, known as Davidson potentials, were used in the Bohr Hamiltonian, that is thus exactly solved, while through a variational procedure, the results of E(5) and X(5) are confirmed. The Z(5) model is an approximate solution of the Bohr equation, in which the assumptions of X(5) hold, the only difference being that γ ≈ π/6. It could be stated that it is the X(5)-analogue for triaxial nuclei. The Z(4) model is the γ-rigid version of Z(5), since the γ degree of freedom stabilizes at exactly γ = π/6 and the degrees of freedom are reduced from five to four. The resulting solution is exact, while a connection with the structure of the Euclidean algebra in four dimensions E(4) is established. Finally, X(3) is the exact, γ-rigid version of X(5) with γ exactly γ = 0. This condition is imposed in the early stages of the construction of the Laplacian in the Bohr Hamiltonian and as a result, the dimensionality of the space is reduced from five to three, since besides γ, one of the Euler angles becomes redundant, due to axial symmetry.