Threshold effects in the photoassociation of cold atoms: R-6 model in the Milne formalism

The general trends of the energy dependence of the photoassociation probability in the threshold range are obtained by introducing the simplest asymptotic model to describe the ground molecular state. It consists in a R-6 potential limited at short range by a repulsive wall, with a tuneable position chosen to reproduce the scattering length of the system. The introduction of reduced units of length and energy allows the systematic investigation of all pairs of alkali atoms. The use of the Milne phase-amplitude formalism is suitable to analyse the breaking of the quasi-classical approximation. The universal trends of the threshold behaviour of collisional properties are analysed and exhibit clearly the importance of the scattering length value. Two particular solutions of the amplitude equation are studied, corresponding to integration from inward or outward, and analytic expressions are given. Both solutions are smoothly connected through the whole range of internuclear distance, extending the results of different authors to the description of states with larger binding energy