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

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→Zcδn,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.