Quantum defect model of a reactive collision at finite temperature
  We consider a general problem of inelastic collision of particles interacting with power-law potentials. Using quantum defect theory we derive an analytical formula for the energy-dependent complex scattering length, valid for arbitrary collision energy, and use it to analyze the elastic and reactive collision rates. Our theory is applicable for both universal and non-universal collisions. The former corresponds to the unit reaction probability at short range, while in the latter case the reaction probability is smaller than one. In the high-energy limit we present a method that allows to incorporate quantum corrections to the classical reaction rate due to the shape resonances and the quantum tunneling. 
Krzysztof Jachymski, Micha{\l} Krych, Paul S. Julienne and Zbigniew   Idziaszek
physics.atom-ph cond-mat.quant-gas quant-ph