Cold and ultracold dipole-dipole collisions

Elastic collisions of ideal oriented dipoles are calculated using a variety of techniques appropriate to different cold and ultracold regimes of temperature. The elastic scattering cross section for two electric dipoles with moment $\vec\mu$ in an electric field $\vec E$ is obtained in the semi-classical Eikonal approximation, giving an exact result, $(4\pi\mu^2 /v)[1-(\hat k_i\cdot\hat E)^2]$, where $\vec k_i = m\vec v$ is the incident relative momentum. This result is expected to apply to collisions at temperatures above a few $\mu K$, encompassing recent experiments in the trapping and cooling of polar molecular gases. The Eikonal calculation contrasts sharply with the Born approximation which predicts an energy-independent cross section scaling as $\mu^4$ and which should be applicable at lower temperatures. A separate analysis of the threshold ultracold region is also presented. Numerical close-coupling results connect these various approximation methods, and demonstrate that the crossover between semi-classical and perturbative regimes occurs at the characteristic dipole energy scale, $E_0 = \hbar^6/m^3\mu^4$.