Analysis of a rotational state in terms of the shell model states

Abstract

The wave function of a rotational state is analyzed in terms of shell model wave functions. The rotational wave function is constructed in the three dimensional model for even nuclei according to the method of the generator coordinates. For the single particle potential the deformed harmonic oscillator well is assumed, and the spin orbit coupling is neglected. The wave function of the rotational state is expressed in terms of the shell model wave functions. The distribution function of the shell model states of given excitation-energy is found to be of Poisson type. For light nuclei in lower excited states the rotational wave function is mainly composed of the degenerate lowest shell model states, Elliott's wave functions, while for heavy nuclei or light nuclei in higher excited states the main contribution comes from shell model states with excitation energy of a few ħω 0, where ω 0 is the angular frequency of the harmonic oscillator. The matrix element of the quadrupole transition is also calculated. Its value, taking only the Elliott states, is found to give half the correct value, while the calculation taking into account both the Elliott states and the interference terms with the states of excitation energy 2ħ;ω is found to give the right answer.