Classical perturbation theory of the thermal accommodation coefficient in n dimensions

Abstract

The theory of the thermal accommodation coefficient is studied using a classical perturbation theory, in which the solid is represented by various n-dimensional models. A general formula is found for the effective thermal accommodation coefficient of a single gas atom in terms of a summation over the normal modes of vibration of the model; this formula applies to any classical harmonic model of the solid. From this formula are derived expansions in closed form for the effective thermal accommodation coefficient which are valid in the regimes of high and low incident gas atom speed; attempts are made to define the conditions under which these expansions are valid. The theory restricts all motion to the direction normal to the solid surface, and all gas atom-surface atom collisions are head-on. A modified exponential repulsive gas-solid interaction potential function which includes realistically a long-range gas-solid attraction is used; a consequence of the form of this potential function is that the perturbation theory is not restricted in general to small accommodation coefficients. In the low-speed regime the theory is not restricted in general to small gas atomic mass, but this restriction applies in the high-speed regime. The theory is applied in turn to the following models of the solid: the n-dimensional lattice with different central and non-central spring force constants; the one-dimensional lattice with a surface impurity; the n-dimensional lattice with various types of surface impurity; the n-dimensional continuum. Where appropriate, the physical significance of the results of these models is considered, and relations among the results are studied; in this way, for example, the role of the three-dimensional continuum theory of Landau is clarified. It is shown that great care must be taken in using such perturbation theory expansions, as they apply only to a very limited group of experimental systems. It is concluded that, in fact, no satisfactory comparison with experimental data is possible at present, and the need is stressed for a more general theory. That the much-used one-dimensional model is not useful in any regime for realistic studies of gas-surface interactions is conclusively established in various ways; similar remarks will apply also to one-dimensional quantum-mechanical models.