math-ph — Mathematical Physics

Sharp Ionization Bounds for Helium and Lithium

Contributed by Alec Shelley, Charles Yang, Daniel Tartakovsky

How many electrons can a nucleus bind? In the non-relativistic Coulomb model, Lieb proved in 1984 that a nucleus of charge $Z$ binds fewer than $2Z+1$ electrons. While this bound has been improved, the sharp upper bound has remained unknown for any nucleus other than hydrogen. We solve this problem for $Z=2$ and $Z=3$ by proving that helium binds two or fewer electrons and lithium binds four or fewer. Both bounds are sharp. Our main tool is an adaptation of Lieb's argument with a quadratic weight, which reduces the nonbinding of doubly negative ions to finitely many inequalities that we verify in a computer-assisted proof.