Three-ion axial normal modes
= Three-ion axial normal modes
Three identical repelling charges in a one-dimensional harmonic trap have a translation mode $(1,1,1)$, a breathing mode $(1,0,-1)$, and a symmetric middle-against-outer mode $(1,-2,1)$. Their <frequencies> are $\omega$, $\sqrt3\omega$, and $\sqrt{29/5}\omega$. The translation <frequency> is unchanged by internal <Coulomb forces>. Linearizing each pair potential produces a positive spring coefficient $2q^2/(4\pi\epsilon_0d^3)$.