Multiphonon processes in a quantum-statistical theory of desorption

A complete fourth-order quantum-statistical calculation of the isothermal desorption time $t_d$ in a gas-solid system is presented showing localized physisorption including all one-phonon and two-phonon processes in fourth order. The multiphonon interaction terms arising from higher-order derivatives of the surface potential turn out to contribute significantly to the desorption rate. Extensive numerical work confirms that the relaxation time approach to desorption phenomena is acceptable for weakly coupled systems for which $\frac{h}{\left|E_0\right|}\ll t_d$ where $E_0$ is the energy of the bound state. Second-order calculations are sufficient as long as $\left|E_0\right|\lesssim k_{B}T\lesssim \hslash {\omega }_D$, where $\hslash {\omega }_D$ is the Debye energy of the solid. Fourth-order contributions become important for $\left|E_0\right|<\mathrm{}\hslash {\omega }_D$ and $k_{B}T\gtrsim \hslash {\omega }_D$. Moreover, for $\hslash {\omega }_D\le \left|E_0\right|\le 2\hslash {\omega }_D$ fourth-order terms are essential because second-order contributions are zero in this region of bound-state energies.