S-wave scattering for deep potentials with attractive tails falling off faster than -1/r2

For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the s-wave phase shift δ₀. It exposes a universal dependence of δ₀(E) on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length a, or equivalently, the noninteger part Δ_th of the threshold quantum number n _th. The formula accurately reproduces δ₀(E) from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for δ₀(E) and demonstrate the validity of the formula for attractive potential tails proportional to 1/r6 or to 1/r4, and also for a mixed potential tail consisting of a 1/r4 term together with a non-negligible 1/r6 contribution.