The Binding Energy for a Self-Trapped Electron in NaCl

A calculation has been made of the binding energy of a self-trapped electron in NaCl. The potential well, caused by the ionic displacements, was assumed to be a sawed-off Coulomb field, the sides being given by $(\frac{1}{{\kappa }_0}-\frac{1}{\kappa })\frac{1}{r}$, where the $\kappa \text{'}\mathrm{s}$ are the dielectric constants. The depth of the well was calculated by the method of Mott and Littleton. In the present problem, however, we require a self-consistent solution; that is, the initial wave function which is employed in calculating the depth of the trapping potential hole must be identical to the one obtained from this potential hole on solving the Schroedinger equation. The electron is smeared over a sphere whose radius equals approximately three interatomic distance. The outer displacements are calculated by a semicontinuum theory, whereas the dipoles of the three ions nearest the trapping center are calculated from the condition of equilibrium. The optical dissociation energy is found to be 0.68 ev, whereas the thermal energy is 0.13 ev. The low value of the second quantity presumably explains why self-trapping has not been observed to date. This energy should be about twice as large in LiF. These results suggest that self-trapping should be sought at a very low temperature.