Relation between degrees of collectivity and repetition patterns of quadrupole transition probabilities in the candidates of shape coexistence

In this article, we study the possible relationship between degrees of freedom of collectivity in protons and neutrons' last orbitals and different patterns of quadrupole transition possibilities in the shape coexistence candidates. Because of the dependence of shape coexistence on the arrangement of nucleons in the nuclear structure, we classified the candidates of shape coexistence based on neutrons and protons' last orbitals using the shell-model configuration. In the orbitals corresponding to the neutron numbers in the N=40, 60, and 90 regions, there is a limitation in the degree of freedom for neutrons, and proton-induced shape coexistence occurs. Also, for the orbitals corresponding to the atomic numbers of the Z=40, 52, and 82 regions, the degree of freedom is limited for protons, and neutron-induced coexistence occurs. In this article, we study both the nuclei that belong to the neutron and proton-induced shape coexistence categories and the nuclei that are candidates for shape coexistence but do not belong to the mentioned two categories. The study of different patterns indicates that the $\frac{|B\left(E2;2_2^{+}\to 0_1^{+}\right)-B\left(E2;2_1^{+}\to 0_1^{+}\right)|}{B\left(E2;2_1^{+}\to 0_1^{+}\right)}$, $\frac{|B\left(E2;2_2^{+}\to 0_1^{+}\right)-B\left(E2;4_1^{+}\to 2_1^{+}\right)|}{B\left(E2;4_1^{+}\to 2_1^{+}\right)}$, $\frac{|B\left(E2;2_2^{+}\to 0_1^{+}\right)-B\left(E2;6_1^{+}\to 4_1^{+}\right)|}{B\left(E2;6_1^{+}\to 4_1^{+}\right)}$, $\frac{|B\left(E2;2_3^{+}\to 0_1^{+}\right)-B\left(E2;2_1^{+}\to 0_1^{+}\right)|}{B\left(E2;2_1^{+}\to 0_1^{+}\right)}$, $\frac{|B\left(E2;2_3^{+}\to 0_1^{+}\right)-B\left(E2;4_1^{+}\to 2_1^{+}\right)|}{B\left(E2;4_1^{+}\to 2_1^{+}\right)}$, and $\frac{|B\left(E2;2_3^{+}\to 0_1^{+}\right)-B\left(E2;6_1^{+}\to 4_1^{+}\right)|}{B\left(E2;6_1^{+}\to 4_1^{+}\right)}$ follow a similar repetition range)in most cases(for ratios related to the neutrons and protons' last orbitals classification. Also, in addition to the mentioned transition possibilities, in transitions related to the neutrons' last orbitals classification, the $\frac{|B\left(E2;2_3^{+}\to 2_1^{+}\right)-B\left(E2;4_1^{+}\to 2_1^{+}\right)|}{B\left(E2;4_1^{+}\to 2_1^{+}\right)}$ and $\frac{|B\left(E2;2_3^{+}\to 2_1^{+}\right)-B\left(E2;6_1^{+}\to 4_1^{+}\right)|}{B\left(E2;6_1^{+}\to 4_1^{+}\right)}$ have a repetition pattern. In the orbitals where neutrons and protons have fewer degrees of freedom (proton and neutron-induced shape coexistence occurs), the results show that the data related to repetition patterns are less correlated than in other orbitals. The study includes information on specific Nilsson orbitals, which are crucial in investigations related to shape coexistence. Correlating observations to these orbitals allows for a more accurate representation of the underlying microscopic picture, enhancing the quality of analysis and strengthening the connection between observations and the SC phenomenon.