Toward a New Theory of Spherical Nuclei. I

A tentative theory of the quadrupole moment ${Q_2}^{+}$ of the first $2^{+}$ excited state of spherical nuclei is developed as a logical extension of the methods used in the usual theory based on pairing plus quadrupole-quadrupole interaction. In the latter, in which the quasiparticle occupation amplitudes in the ground state are taken to be of zero order and the off-diagonal amplitudes connecting the ground state with the $2^{+}$ state to be of first order (and described by the random-phase approximation), ${Q_2}^{+}$ is nominally of second order. We find that, in a consistent calculation, the quadrupole deformation is driven by the off-diagonal amplitudes, but that there is also the possibility of a self-sustained deformation for sufficiently large quadrupole coupling constants. Numerically, one finds ranges of the latter, all in accord with the excitation energy and the $E2\mathrm{}$ transition probability, for which ${Q_2}^{+}$ shows extremely rapid variations, and in particular, also assumes values sufficiently large to contradict the whole basis of the calculation. The need for a self-consistent intermediate coupling calculation is indicated.