Bound channels and quantum defect theory

An expansion used in electron collision theory is Psi = Sigma i Theta i + Sigma j Phi j C j , where the Theta i are products of target functions times radial functions for the colliding electron and are referred to as free channels, while the Phi j have the form of bound-state functions for the whole system and are referred to as bound channels. Previous formulations of quantum defect theory have used expansions in which the bound channels are omitted. It is shown that inclusion of bound channels does not alter the formal structure of the theory but may lead to rapid variations with energy of quantities which would otherwise be slowly varying. For the case of one free channel and a number of bound channels the dependence of the quantum defect mu on the energy E is given by tan( pi mu (E))= alpha - Sigma j beta j /(E- Delta j ) where alpha , beta j and Delta j are slowly varying functions of E. It is shown that experimental quantum defects for the 1 S and 1 D series in Ca I can be fitted to expressions of this type, with one bound channel for 1 S and three bound channels for 1 D.