Quantum defect asymptotics at the critical charge: A study of the integrality conjecture

Abstract

A single (n, ℓ) electron outside an (N −1)-electron atomic core is bound as long as Z > Zc = N − 1. A conjecture is examined, according to which the quantum defect of the outermost electron satisfies limZ!Zc δn,ℓ(Z) = Nℓ, where Nℓ is the number of occupied or partially occupied orbitals with angular momentum quantum number ℓ within the (N − 1)-electron core. Specifically, the 3s quantum defect is inspected for the different occupancies of the n = 1 and n = 2 shells. The conjecture is found to hold in all the cases considered.