On the low-velocity limit of the Bohr stopping formula

The low-velocity limit of the classical Bohr stopping model is investigated by applying non-perturbative methods. For the repulsive Coulomb interaction between a heavy projectile and a harmonically bound electron, the stopping cross-section S+(v) is found to scale as v5/3 in the limit of v ↦0, where v is the projectile velocity. This scaling is obtained by establishing a corresponding scaling law for the energy transfer T+(v,b) in a single collision with an impact parameter b, namely, that T+/v1/3 is a function of a scaled variable b/v2/3. For the opposite case of the Coulomb attraction, direct numerical calculations reveal that the energy transfer T-(v,b) exhibits sharp resonances along the b axes when v becomes sufficiently small. The latter results in a characteristic non-regular behaviour of S-(v) near maximum. Suitable fitting formulae are proposed for the corresponding stopping numbers L±(v).