Study of Bohr Mottelson Hamiltonian with minimal length effect for Woods-Saxon potential and its thermodynamic properties.

The Bohr Mottelson Hamiltonian with the variable of β collective shape for the Woods-Saxon potential in the rigid deformed nucleus for γ = 0 and the X(3) model was investigated in the presence of the minimal length formalism. The Bohr Mottelson Hamiltonian was solved approximately by proposing a new wave function. The q-deformed hyperbolic potential concept such that the rigid deformed nucleus of the Bohr Mottelson equation in the minimal length formalism for Woods-Saxon potential was used, so that the equation was reduced to the form of Schrodinger-like equation with cotangent hyperbolic potential. The hypergeometric method was used to obtain the energy spectra equation and the unnormalized wave function of the system. The results showed that the energy spectra were affected by the quantum number, the minimal length parameter, and the atomic mass. The larger mass of the atom affected the energy spectra to decrease, the increase of the values of the minimal length affected the increase of the energy spectra of all atoms. The energy spectra were used to determine the thermodynamic properties including the partition function, mean energy, specific heat, free energy, and entropy of the quantum system with the help of the imaginary error function.