Calculation of the scattering length in atomic collisions using the semiclassical approximation

A simple analytical formula, a=a¯[1-tan(π/n-2)tan{Φ-[π/2(n-2)]}, is obtained for the scattering length in atomic collisions. Here a¯=cos[π/(n-2)]{ √2Mα /[ħ(n-2)]${\mathrm{\}}}^{2/\left(\mathit{n}\mathrm{-}2\right)}$[Γ(n-3)/(n-2)]/[T(n-1)/(n-2)] is the mean scattering length determined by the asymptotic behavior of the potential U(r)∼-α/${\mathit{R}}^{\mathit{n}}$, (n=6 for atom-atom scattering or n=4 for ion-atom scattering), M is the reduced mass of the atoms, and Φ is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of a¯, the average scattering length, also determines the slope of the s-wave phase shifts beyond the near-threshold region. The formula is applicable to the collisions of atoms cooled down in traps, where the scattering length determines the character of the atom-atom interaction. Our calculation shows that repulsion between atoms (a>0) is more likely than attraction with a ‘‘probability’’ of 75%. For the Cs-Cs scattering in the ${}_{}{}^3{}_{}{}^{}\mathrm{\Sigma }_{\mathit{u}}^{}{}_{}{}^{}$ state, a¯=95.5${\mathit{a}}_{\mathit{B}}$ has been obtained, where ${\mathit{a}}_{\mathit{B}}$ is the Bohr radius. The comparison of the calculated cross-section energy dependence with the experimental data gives evidence for a positive value for the Cs-Cs scattering length, which makes cesium Bose gas stable.