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

A single $(n,\ell )$ electron outside an $(N-1)$-electron atomic core is bound as long as $Z>Z_c=N-1$. A conjecture is examined, according to which the quantum defect of the outermost electron satisfies ${\lim }_{Z\to Z_c}{\delta }_{n,\ell }\left(Z\right)=N_{\ell }$, where $N_{\ell }$ is the number of occupied or partially occupied orbitals with angular momentum quantum number $\ell$ 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.