Semiclassical description of the small angle differential cross section for elastic atom—atom scattering

Abstract

The scattering amplitude is investigated for small angles θ* ⩽ 8.5, with θ* = θ/θ o a reduced scattering angle. The scaling angle is θ o = (4π/ k 2 Q 3) 1 2 , with k the wavenumber and Q 3 the total cross section due to the attractive branch of the intermolecular potential. Both pure inverse power potentials and realistic intermolecular potentials are investigated. The method used is a least-square curve fit of the real and imaginary parts of quantum-mechanically calculated scattering amplitudes f(θ) with suitable model functions. The shape of these model functions is partially based on the classical and semiclassical results for small angle scattering. For the shape of the attractive contribution to the differential cross section σ 3(θ) in the case of an inverse power potential we give a new model function that describe the quantum oscillations (diffraction) modulating the classical result σ e1(θ) HE for the differential cross section in the high energy approximation, i.e. the approximation of a straight line trajectory. The first order correction on this straight line approximation is also derived and incorporated in our model function. For s = 6 and 2.5 ⩽ θ* ⩽ 8.5 we find a description of σ 3(θ)/σ e1(θ) HE with a rms deviation of 2.3 × 10 −3. For the shape of the differential cross section at small angles we give a second, new semi-empirical model functions, σ 3(θ)/σ 3(0) = [1 −1 c 1 sin ( c 2θ* 2) + c 3θ* 2] −(3 + 1)/3, with s the power of the potential. The asymptotic behaviour of this function for θ* ⪡ 1 and θ* ⪢ is in good agreement with the corresponding semiclassical and classical results. For s = 6 the parameters are c 1 = 3.75, c 2 = 0.556, and c 3 = 2.94, resulting in a description of σ 3(θ)/σ 3(0) with a rms deviation of 0.9 × 10 −3 for θ* ⩽ 4.0. A simple and accurate model function for the phase angle φ a(θ) = arg( f(θ)) enables us to include the glory behaviour in the case of a realistic intermolecular potential. For the analysis of the differential cross section of a realistic potential a suitable model function for the glory contribution is added, resulting in a description of σ(θ)/σ(0) with a rms deviation of 3 × 10 −3 for θ* ⩽ 4.0. A universal set of parameters is presented that can be used for predicting the differential cross section for a realistic intermolecular potential. It is also shown that the velocity dependency of the attractive and the glory contribution to the imaginary part of the scattering amplitude for θ* = 0, i.e. the total cross section, can be effectively used for predicting the small angle behaviour of the differential cross section.