Comparison of classical and quantal calculations of helium three-body recombination

A general method to study classical scattering in $n$-dimension is developed. Through classical trajectory calculations, the three-body recombination is computed as a function of the collision energy for helium atoms, as an example. Quantum calculations are also performed for the $J^{\Pi}$ = $0^{+}$ symmetry of the three-body recombination rate in order to compare with the classical results, yielding good agreement for $E\gtrsim$ 1 K. The classical threshold law is derived and numerically confirmed for the Newtonian three-body recombination rate. Finally, a relationship is found between the quantum and classical three-body hard hypersphere elastic cross sections which is analogous to the well-known shadow scattering in two-body collisions.