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)]{ {}2M{α} /[{}(n-2)]^2/(n-2)[{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}/Rⁿ, (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 ³Σᵤ state, a{}=95.5aB has been obtained, where aB 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.