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 μ of the collision pair, allows a semiclassical treatment within the potential well. As a result, the low-energy scattering is governed by two large parameters: the asymptotic parameter $\gamma =\sqrt{2\mu \alpha }/ħ\gg a_0^{\left(n-2\right)/2}$ ($a_0$ is the Bohr radius) and the semiclassical zero-energy phase $\Phi \gg 1.$ In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression for the scattering length $a,$ which showed that it has 75% preference for positive values for $n=6,$ characteristic of collisions between ground-state neutral atoms. In this paper we calculate the effective range 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 γ. Thus, we know the $s$-phase shift at low momenta $k\ll {\gamma }^{-2/\left(n-2\right)}$ from the expansion $k\cot {\delta }_0\simeq -1/a+\frac{1}{2}{r}_e{k}^2.$ At $k\gg {\gamma }^{-2/\left(n-2\right)}$ the phase shift is obtained semiclassically as ${\delta }_0=\Phi +\pi /4\mathrm{}-I_n{\gamma }^{2/n}{k}^{\left(n-2\right)/n},$ where $I_n=[n/(n-2\left)\right]\Gamma \left(\right(n-1)/n)\Gamma \left(\mathrm{}\right(n+2\mathrm{})/2n)/\sqrt{\pi }.$ Therefore, γ and Φ determine the $s$-wave atomic scattering in a wide range of momenta, as well as the positions of upper bound states of the diatomic molecule.