Nuclear deformations in the pairing-plus-quadrupole model

Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

“…The quantities H ph and H (1) ph are the components of two vectors in this space. We are looking for a basis, or rather (as we change the collective variable) a continuous, differentiable one-dimensional manifold of bases, such that the two vectors are everywhere parallel to each other, and in so far as possible, also parallel to the tangent to the manifold.…”

“…(3.4) and (3.5) since covariant and contravariant derivatives are now identical. In the case of exact decoupling we would find that V ,α is parallel to the path g α , 1 . In the present model this can not be true, since we do not have exact decoupling.…”

“…By requiring, finally, that the application of a transformation followed by its inverse should be equivalent to the identity transformation, we deduce the relation Returning now to the primary object of this subsection, insertion of the transformation (2.14a) and (2.14b) into the Hamiltonian (2.1) and neglect of terms of higher than second order in the momenta cause the transformed Hamiltonian to take the form H(q, p) = 1 2 p µ f (0)µ ,α B αβ (g(q)) f (0)ν ,β + V ,γ (g(q)) g (1)γµν p ν + V (g(q)) ≡ 1 2 p µB µν p ν +V (q). (2.19) This establishes the main point concerning the admission of extended transformations.…”

“…It should also be noted that in this calculation we normalizef , and thus c i with the second method discussed in Sec. IV B 1, where we chose ∆ρ ph f (0) ph = ∆ρ ph f (1) ph . This choice of normalization condition for the Suzuki model leads to the collective operator being the quadrupole operator, and the inverse mass taking the value 8Q, so that the numerical calculation ties right on to the analytic one.…”

“…In the course of the latter investigation, we developed approximation methods especially suited to the application to Hamiltonians with separable interactions. For such interactions, the expansion of the selfconsistent cranking operators in a small basis of operators suggested by the Hamiltonian itself would certainly yield improvement with respect to cranking calculations of the type pioneered by Kumar and Baranger [1][2][3][4][5] and, at the same time, permit the inclusion of curvature effects within the framework of extended point transformations. Results of these studies will be given in Sec.…”