We consider background gas collision shifts in ion-based clocks. We give both a classical and quantum description of a collision between an ion and a polarizable particle with a simple hard-sphere repulsion. Both descriptions give consistent results, which shows that a collision-shift bound is determined by the classical Langevin collision rate reduced by a readily calculated factor describing the decoupling of the clock laser from the ion due to the recoil motion. We also show that the result holds when using a more general Lennard-Jones potential to describe the interaction between the ion and its collision partner. This leads to a simple bound for the collision shift applicable to any single-ion clock without resorting to large-scale Monte Carlo simulations or determination of molecular potential energy curves describing the collision. It also provides a relatively straightforward means to measure the relevant collision rate.

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• Atomic & molecular collisions
• Atomic, optical & lattice clocks
• Scattering of atoms, molecules, clusters & ions
• Scattering theory