Analytical solutions of the Bohr Hamiltonian with the Morse potential

TL;DR

This paper derives analytical solutions for the Bohr Hamiltonian using the Morse potential, providing accurate energy spectra for various nuclei with closed-form expressions via the Asymptotic Iteration Method.

Contribution

It introduces exact analytical solutions for the Bohr Hamiltonian with Morse potential in both $eta$-unstable and rotational cases, enabling precise modeling of nuclear spectra.

Findings

Successfully fitted over 50 nuclei with the models

Accurately reproduced bandheads and energy spacings

Demonstrated effectiveness of AIM in solving Bohr equations

Abstract

Analytical solutions of the Bohr Hamiltonian are obtained in the $\gamma$-unstable case, as well as in an exactly separable rotational case with $\gamma\approx 0$, called the exactly separable Morse (ES-M) solution. Closed expressions for the energy eigenvalues are obtained through the Asymptotic Iteration Method (AIM), the effectiveness of which is demonstrated by solving the relevant Bohr equations for the Davidson and Kratzer potentials. All medium mass and heavy nuclei with known $\beta_1$ and $\gamma_1$ bandheads have been fitted by using the two-parameter $\gamma$-unstable solution for transitional nuclei and the three-parameter ES-M for rotational ones. It is shown that bandheads and energy spacings within the bands are well reproduced for more than 50 nuclei in each case.