Question #42934

Two identical atomic ions, each with a positive charge $+Q$, interact via an effective interatomic potential $U_{int}(R)$. In the absence of other charges, this potential has a minimum at an equilibrium separation $R_0$. For all relevant separations $R$, it can be approximated as a harmonic well $U_{int}(R) = \frac{1}{2}k(R-R_0)^2$, where $k$ is the effective spring constant. Note that $U_{int}(R)$ already accounts for the Coulomb repulsion between the two ions. A point charge $-q$ ($q > 0$) is placed exactly at the midpoint between the two ions. By the principle of superposition, the new total potential energy includes the electrostatic interaction between $-q$ and the ions. As $q$ is increased, the equilibrium separation $R$ decreases. If $q$ exceeds a critical value $q_{max}$, the symmetric equilibrium becomes unstable and the system collapses. Find $q_{max}$. Assume all quantities are in standard SI units and $\epsilon_0$ is the vacuum permittivity.