Scaling rule in low-energy atom-atom scattering

An energy-scaling rule for atom-atom scattering is derived. It is shown to be approximately accurate to third order in the parameter $\frac{\Delta M}{M_{av}}$, where Δ M is the difference in the reduced masses between two different atomic pairs. The energy scaling depends upon the energy itself and on the interaction potential. The method is illustrated by the He-He scattering problem using the isotopes He₃ and He₄ in their three different combinations.