Low-Energy Expansion of the Scattering Amplitude for Long-Range Quadrupole Potentials

For static potentials which are proportional asymptotically to qP₂(cosθ)r³, the low-energy expansion of the scattering amplitude is found through terms of $O(k)$, using a modification of the method developed by Levy and Keller for central potentials. The resulting expansion to lowest order in k is found to be f(θ, φ)~-A+(q/3)P₂(cosθK)+O(k), where A is the scattering length and θK is the coordinate of the momentum transfer vector. Applications are attempted first to electron-atom elastic scattering where results are somewhat more complicated than for the potentials above, secondly to transitions between magnetic quantum states of atoms caused by slow electrons.