CALCULATION OF THE SCATTERING LENGTH IN ATOMIC-COLLISIONS USING THE SEMICLASSICAL APPROXIMATION

A simple analytical formula, a = aBAR [1 - tan pi/n-2 tan (PHI - pi/2(n-2))], is obtained for the scattering length in atomic collisions. Here aBAR = cos[pi/(n - 2)] {square-root 2Malpha/[hBAR(n - 2)]}2/(n-2) GAMMA(n-3/n-2))/(GAMMA(n-1/n-2)) is the mean scattering length determined by the asymptotic behavior of the potential U(r) approximately -alpha/R(n) (n = 6 for atom-atom scattering or n = 4 for ion-atom scattering), M is the reduced mass of the atoms, and PHI is the semiclassical phase calculated at zero energy from the classical turning point to infinity. The value of aBAR, 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 3SIGMA(u) state, aBAR = 95.5alpha(B) has been obtained, where a(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.