The differential and transport cross sections (TCSs) for binary collision are fundamental parameters to depict the traversing of charged particles in matter, which are often obtained via classical mechanics. The Bohr criterion was proposed to judge when the treatment via classical mechanics is reliable. However, there are very few quantitative comparisons of the cross sections between classical and quantum methods under the Debye potential. In this work, such comparisons—using the partial wave method and the Wentzel-Kramers-Brillouin (WKB) approximation—are made for wide ranges of collision velocity, ion charge states and Debye length. For the differential cross section, a quantitative range of velocity is obtained for the Bohr criterion that is invalid for ion–ion collisions, and the criterion is found to be almost invalid for electron–ion collisions. For the TCS, the quantitative range of velocities is given for the criterion and found to be valid for both ion–ion and electron–ion collisions. Such ranges are related to the de Broglie wavelength of the collision system and Debye length, and they are also relevant to the complex classical trajectories due to the attractive Debye potential in electron–ion collisions. The reason for the failure of classical mechanics at low and high velocities is explored by reinterpreting the Bohr criterion.
This paper examines the validity of the Bohr criterion, which is used to determine when classical mechanics can reliably describe binary collisions. The authors compare classical and quantum mechanical methods for calculating differential and transport cross sections under the Debye potential, covering a wide range of collision velocities, ion charge states, and Debye lengths. They find that the Bohr criterion is often invalid for electron-ion collisions and provide quantitative ranges where it is valid for both ion-ion and electron-ion collisions.