Quantum theory of sticking

Abstract

We present an exact solution of a one-dimensional (1D) model: a particle of incident energy E colliding with a target which is a 1D harmonic ‘‘solid slab’’ with N atoms in its ground state; the Hilbert space of the target is restricted to the (N+1) states with zero or one phonon present. For the case of a short-range interaction V(z) between the particle and the surface atom supporting a bound state, an explicit nonperturbative solution of the collision problem is obtained. For finite and large N, there is no true sticking but only so-called Feshbach resonances. A finite sticking coefficient s(E) is obtained by introducing a small phonon decay rate η and letting N→∞. Our main interest is in the behavior of s(E) as E→0. For a short-range V(z), we find s(E)∼${\mathit{E}}^{1/2}$, regardless of the strength of the particle-phonon coupling. However, if V(z) has a Coulomb ${\mathit{z}}^{\mathrm{-}1}$ tail, we find s(E)→α, where 0s(E)→1 in both cases.] We conclude that the same threshold laws apply to 3D systems of neutral and charged particles, respectively. In an appendix we elucidate the nature of sticking by the behavior of a wave packet incident on a finite N target.