Low Energy Behavior of Quantum Adsorption

We present an exact solution of a 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 non-perturbative solution of the collision problem is presented. 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) ∼ E 1/2 , regardless of the strength of the particle-phonon coupling. However, if V (z) has a Coulomb z −1 tail, we find s(E) → α, where 0 < α < 1. [A fully classical calculation gives s(E) → 1 in both cases.] We conclude that the same threshold laws apply to 3D systems of neutral and charged particles respectively.