Ground-State Properties of Nonspherical Nuclei

Abstract

A knowledge of the wave functions and energy levels for nucleonic motion in a nonspherical force field is a prerequisite to any detailed comparison between the empirical data and the model of Bohr and Mottleson. An extremely simple model for this internal problem is provided by the motion of independent particles in an ellipsoidal oscillator potential. By minimizing the total energy of such a system, we are led to conclude (1) that deformations which possess an axis of symmetry are always preferred by low-lying states, and (2) that this model cannot reproduce the observed preponderance of positive quadrupole moments. This last conclusion is unaltered by the surface tension or Coulomb repulsion.The oscillator model is too simplified for the computation of spins or magnetic moments ($\ensuremath{\mu}$). We have therefore determined wave functions and energy levels for independent-particle motion in a spheroidal square well, including spin-orbit coupling. Because of the approximations made in this calculation, we cannot use our levels to compute the equilibrium shape, but only to assign the ground-state level and wave function. Thus the present scheme is intended to replace that of the shell model in the region of large deformations.With the exception of ${\mathrm{W}}^{183}$, we have been able to correctly assign the spin to all the odd-$A$ nuclei for which $150\ensuremath{\lesssim}A\ensuremath{\lesssim}190$. The ease and consistency with which such an assignment can be made leads us to conclude that our scheme is essentially correct for those nuclei which possess large deformations, and whose odd particle lies in the 50-82 shell or the lower part of the 82-126 shell. In addition, we have computed $\ensuremath{\mu}$ in this region. Here the agreement is more limited, but there are a number of important successes: we reproduce the large difference in $\ensuremath{\mu}$ for the europium isotopes, the ratio of the magnetic moments of the two hafnium isotopes, both $\ensuremath{\mu}$ and the decoupling coefficient for ${\mathrm{Tm}}^{169}$, and $\ensuremath{\mu}$ for ${\mathrm{Yb}}^{171,173}$, Re, ${\mathrm{Eu}}^{151}$, and ${\mathrm{Os}}^{189}$. The properties of the ground state and first excited state of ${\mathrm{Ag}}^{109}$ are also found to be in good agreement with the theory.