ANALYTICAL CALCULATIONS OF SCATTERING LENGTHS IN ATOMIC PHYSICS

We describe a method for evaluating analytical long-range contribution s to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles cons ists of two parts. The form of a short-range component, vanishing beyo nd some distance from the origin (a core radius), need not be given. I nstead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potent ial is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent p otential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with so lutions expressed in terms of the Bessel, Whittaker and Legendre funct ions, respectively. We utilize this fact and derive exact analytical f ormulae for the scattering lengths. The expressions depend on the shor t-range scattering lengths, the core radius and parameters characteriz ing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also disc ussed and formulae for scattering lengths linear in strengths of the p erturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approxima te, take very simple forms and contain only polynomial and trigonometr ic functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating converge nce of scattering lengths computed for e(-)-Xe and Cs-Cs systems.