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 ${\sl s}(E)$ is obtained by introducing a small phonon decay rate $\eta$ and letting $N\to\infty$. Our main interest is in the behavior of ${\sl s}(E)$ as $E\to 0$. For a short range $V(z)$, we find ${\sl s}(E)\sim E^{1/2}$, regardless of the strength of the particle-phonon coupling. However, if $V(z)$ has a Coulomb $z^{-1}$ tail, we find ${\sl s}(E)\to\alpha$, where $0 < \alpha < 1$. [A fully classical calculation gives ${\sl s}(E)\to 1$ in both cases.] We conclude that the same threshold laws apply to 3D systems of neutral and charged particles respectively.