Adiabatic Perturbation Theory on Reactions Induced by Weakly Bound Particles at Higher Incident Energies

Abstract

We develop an adiabatic perturbation theory to clarify the relation of the adiabatic approxima­ tion of-Johnson and Soper to the method of continuum discretized coupled-channels (CDCC). The theory presents a systematic way of evaluating corrections to the adiabatic approximation. It is proven on the basis of the adiabatic theorem that the corrections vanish in the adiabatic limit of motions of nucleons in the projectile nucleus, or in the limit of high incident energies. The correc· tions do not include any geometrical quantity such as Berry's phase, because of no diabolic point in the .parameter space. 4 ),5) for reactions induced by weakly bound particles at higher incident energies. These approximations are performed with the following common procedure. First the total system is considered to be composed of slow and fast subsystems interacting through a coupling between them. The total Hamiltonian H is then separated into a sub-Hamiltonian Hs of the slow subsystem and the remainder Hf describing motions of the fast subsystem interacting with the slow one through the coupling. The coupling in Hf depends on intrinsic coordinates r of the slow subsys­ tem, but the coordinates work only as parameters in Hf, since Hf does not include any kinetic energy operator on r. So the fast subsystem is assumed to be in an eigenstate I())m(r» of H f, with a common quantum number ni at each r. The eigenstate has different energies Em(r) at each r in the cases of (a) and (b). In (c), the energy is an incident energy E po of the projectile independently of r, and then the quantum number m represents an incident momentum of the projectile. Next His approximated into Hs+ Em(r) by replacing Hf by Em(r), and the slow subsystem is considered to be in an eigenstate ¢(r) of Hs+ Em(r). Eventually the total wave function 1Jf is given approx­ imately by 1JfAD= ¢(r)1 ())m(r». The separation of H is often called the adiabatic separation approximation. In (c), 1JfAD satisfies the equation, (Hf - Ep,) 1JfAD=O, i.e., (Hf + eo-E) 1JfAD=O, since the total energy Eis the sum of Epo and the eigenvalue eo of Hs in its ground state. This indicates that the adiabatic Hamiltonian, Hf+eo, can be simply defined from H by replacing Hs by eo. So far the approximation (c) was defined with the replace-