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. He re aBAR = cos[pi/(n - 2)] {square-root 2Malpha/[hBAR(n - 2)]}2/(n-2) G AMMA(n-3/n-2))/(GAMMA(n-1/n-2)) is the mean scattering length determin ed by the asymptotic behavior of the potential U(r) approximately -alp ha/R(n) (n = 6 for atom-atom scattering or n = 4 for ion-atom scatteri ng), M is the reduced mass of the atoms, and PHI is the semiclassical phase calculated at zero energy from the classical turning point to in finity. The value of aBAR, the average scattering length, also determi nes the slope of the s-wave phase shifts beyond the near-threshold reg ion. 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 at oms (a > 0) is more likely than attraction with a ''probability'' of 7 5%. 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 leng th, which makes cesium Bose gas stable.