  The phonon-assisted sticking rate of slow moving atoms impinging on an
elastic membrane at nonzero temperature is studied analytically using a model
with linear atom-phonon interactions, valid in the weak coupling regime. A
perturbative expansion of the adsorption rate in the atom-phonon coupling is
infrared divergent at zero temperature, and this infrared problem is
exacerbated by finite temperature. The use of a coherent state phonon basis in
the calculation, however, yields infrared-finite results even at finite
temperature. The sticking probability with the emission of any finite number of
phonons is explicitly seen to be exponentially small, and it vanishes as the
membrane size grows, a result that was previously found at zero temperature; in
contrast to the zero temperature case, this exponential suppression of the
sticking probability persists even with the emission of an infinite number of
soft phonons. Explicit closed-form expressions are obtained for the effects of
soft-phonon emission at finite temperature on the adsorption rate. For slowly
moving atoms, the model predicts that there is zero probability of sticking to
a large elastic membrane at nonzero temperature and weak coupling.
