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 mu 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= root 2 mu alpha/h much greater than a(0)((n - 2)/2) (a(0) is the Bohr radius) and the semiclassical zero-energy phase Phi greater than or equal to 1. In our previous work [Phys. Rev. A 48, 546 (1993)] we obtained an analytical expression far 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 gamma. Thus, we know the s-phase shift at low momenta k much less than gamma(-2/(n - 2)) from the expansion k cot delta(0)similar or equal to-1/a + 1/2r(e)k(2). At k much greater than gamma(-2/(n - 2)) the phase shift is obtained semiclassically as delta 0 = Phi + pi/4-I(n)gamma(2/n)k((n - 2)/n), where I-n = [n/(n - 2)]Gamma((n - 1)/n)Gamma((n + 2)/2n)/root pi. Therefore, gamma and Phi 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. [S1050-2947(99)03603-3].