Microscopic Theory of Nuclear Collective Motion

A generalization of the Hill-Wheeler generator coordinate method is applied to collective deformations. The intrinsic wave function is constrained (as in constrained Hartree-Fock) to be characterized not only by a given deformation, but also by a deformation velocity. This is effected by a simple ansatz which involves operation on the singly constrained wave function by an exponentiated single-particle deformation operator containing an arbitrary function $\beta \left(\alpha \right)$, where $\alpha$ is the collective variable. The expectation value of the energy is minimized with respect to both $\beta \left(\alpha \right)$ and the Hill-Wheeler projection function $f\left(\alpha \right)$. This leads to an integral equation for $f$ which, upon invoking the collective nature of the intrinsic states, may be approximated by a second-order differential equation in the deformation coordinate $\alpha =〈Q〉$. In order to reduce this equation to the Schrödinger form, certain assumptions are introduced with regard to the approximate form of $f$. This procedure leads to two different differential equations for $f$ and to two mass parameters. One is valid in the classical region and one in the classically inaccessible tunneling region. This is to be contrasted to the cranking model where sufficient energy must always be available to drive the system. The expressions for the mass parameter are given in terms of expectation values of few-body operators. The case of uniform translation of the nucleus as a whole is studied in detail. The generalized Hill-Wheeler method as described above produces the correct mass (= total nuclear mass). This rigorous reproduction of a known result allows the study of approximations which become necessary for the general case of deformations. Comments are made about the potential energy of deformation surface, which is expected to lie lower than the expectation value of the Hamiltonian.