Thermodynamic Functions and Debye–Waller Factor for Adsorbed Particles
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A discussion is given of the effect of the particle masses on the thermal properties of an adsorbate-substrate system, with detailed calculations for model systems in which the adsorbate consists of a light or heavy monolayer. These models are more nearly realistic than those used in previous treatments in that there is no disagreement with the condition of rotational invariance. We have examined the dependence of the vibrational contributions to the energy, entropy, free energy, and specific heat upon the temperature, the surface orientation, and the particle masses. The mean-square amplitudes of vibration, which determine the Debye-Waller factor, were also studied. The calculated results for the free energy and the meansquare amplitudes are reproduced exactly at high temperatures and to a good approximation at low temperatures by a very simple model in which, roughly speaking, it is assumed that the vibrational frequencies associated with the adsorbate are proportional to the square root of the adsorbate mass. Arguments are given to show that this result should hold for any adsorbate-substrate system for which the quasiharmonic approximation is valid, regardless of structure or composition. Since the mass dependence can be almost exactly taken into account (in the quasiharmonic approximation), measurements of the temperature dependence of the free energy and Debye-Waller factor should provide direct information on adsorbatesubstrate interactions. In particular, the mean-square amplitudes at high temperatures are independent of the masses and are determined solely by the force constants.