Question #43641

A projectile of mass $m$ and charge $+2e$ is fired head-on from infinity with initial kinetic energy $K_0$ towards a free, initially stationary target ion of mass $m$ and charge $+e$. To trigger an internal excitation of the target ion, the projectile must reach a distance of closest approach $d$. The electrostatic potential energy of two elementary charges $+e$ separated by $d$ is given as $U_0 = 20 \text{ eV}$. The target ion has internal energy levels described by $E_n = n E_0$ (where $n = 1, 2, 3, \dots$ and $E_0 = 15 \text{ eV}$). Due to quantum selection rules, the ion can only undergo transitions where the change in the quantum number $n$ (i.e., $\Delta n$) is an even integer. Assuming the target ion is initially in its ground state ($n=1$), what is the minimum initial kinetic energy $K_0$ required for the projectile to excite the target ion to its lowest allowed excited state?