Low-energy limit for tunnelling subject to an Eckart potential barrier

Abstract

For two-body s-wave collisions subject to tunnelling though an Eckart potential barrier, cross-sections and rate coefficients in the low-energy regime can be evaluated in analytic form. These provide criteria for approach to the Wigner limit and a generic plot applicable to ultracold collisions (<1 mK). The Eckart barrier shape is found to fit fairly well accurate potential curves available for F(2P) + F(2P) and He(1S) + He*(1S). Also discussed are tunnelling-dominated ultracold chemical reactions, exemplified by F + H2. The very slow approach of the reactants allows averaging over rovibrational motions, so the effective adiabatic (‘dressed’) potential surface differs substantially from the Born–Oppenheimer (‘bare’) surface obtained from electronic structure calculations. The dressed barrier is more than two-fold lower and also thinner. For it the Eckart model gives an ultracold rate coefficient for forming HF in its v′ = 2, j′ = 0 rovibrational state (6.5 × 10−14 cm3/s) that is about 200-fold higher than for the bare barrier and of the same magnitude found from a full-scale 3-D quantum scattering calculation.