Analytical calculation of cold atom scattering

The interaction between atoms behaves as $-\alpha /r^n$ at large distances,and, owing to the large reduced mass $\mu $ of the collision pair, allows semiclassical treatment within the potential well. As a result, thelow-energy scattering is governed by two large parameters: the asymptoticparameter $\gamma =\sqrt{2\mu \alpha }/\hbar \gg a_0^{(n-2)/2}$ ($a_0$ is theBohr radius), and the semiclassical zero-energy phase $\Phi \gg 1$. In ourprevious work [Phys. Rev. A {\bf 48}, 546 (1993)] we obtained an analyticalexpression for the scattering length $a$, which showed that it has 75\%preference for positive values for $n=6$, characteristic of collisionsbetween ground-state neutral atoms. In this paper we calculate the effectiverange and show that it is a function of $a$, $r_e=F_n-G_n/a+H_n/a^2$, where$F_n$, $G_n$ and $H_n$ depend only on $\gamma $. Thus, we know the $s$ phaseshift at low momenta $k\ll \gamma ^{-2/(n-2)}$ from the expansion$k\cot \delta _0\simeq -1/a+\frac{1}{2}r_ek^2$. At $k\gg \gamma ^{-2/(n-2)}$the phase shift is obtained semiclassically as$\delta _0=\Phi +\frac{\pi}{4}-I_n \gamma ^{2/n} k^{(n-2)/n}$,where $I_n=\frac{n}{n-2}\Gamma \left( \frac{n-1}{n}\right)\Gamma \left( \frac{n+2}{2n}\right) /\sqrt{\pi}$. Therefore, $\gamma $and $\Phi $ determine the $s$ wave atomic scattering in a wide range ofmomenta, as well as the positions of upper bound states of the diatomicmolecule.