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

For static potentials which are proportional asymptotically to $\frac{q{P}_2\left(cos\theta \right)}{r^3}$, the low-energy expansion of the scattering amplitude is found through terms of $O\left(k\right)\mathrm{}$, 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 $f(\theta , \phi )\sim -A+\left(\frac{q}{3}\right)P_2\left({cos\theta }_K\right)+O\left(k\right)\mathrm{}$, where $A$ is the scattering length and ${\theta }_K$ 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.