Thermal Ionization of Trapped Electrons

The rate of thermal ionization of an electron trapped on an impurity atom is treated on a quantum mechanical basis. From the standpoint of an adiabatic approximation, the multiphonon transitions are attributed to the dependence of the atomic vibrations on the electronic states. Approximate formulas based on an Einstein model are derived for the total ionization rate by using a generating function which is intimately related to density matrices. It is shown that the rate can be expressed as $256\left(\frac{m}{M}\right)\omega {\left(\frac{\rho }{2}\right)}^{\frac{{\epsilon }_0}{\hslash \omega -1\mathrm{}}}\exp (-\frac{{\epsilon }_0}{\mathrm{kT}})$ for low temperatures, where $m$ is the mass of electron, $M$ that of the atom, $\omega$ the frequency of atomic vibration, ${\epsilon }_0$ the energy of ionization, and $\rho$ is the fractional difference of the frequencies of atomic vibrations in the trapped and the ionized states, which can be of the order 0.1. For high temperatures we can expect a similar formula to that given by the activated states theory. Generally, we have reasons to expect much greater rates than those given by Goodman, Lawson, and Schiff.