Intracellular potential relief and size effects in the lattice of a LaF$_3$ superionic conductor

The potential relief in the lattice of a LaF$_3$ crystal is calculated by quantum-mechanical methods for clusters containing from 24 to 1200 ions. For the dielectric phase, formation energy $E_a$ for defects of the vacancy-interstitial fluorine ion type and potential barrier $E_d$ preventing the motion of fluorine ions are found to grow from minimal values $E_a = 0.12$ eV and $E_d = 0.22$ eV for a cluster of 24 ions to maximal values $E_a = 0.16$ eV and $E_d = 0.26$ eV for clusters of 576 and 1200 ions. The values of $E_a$ and $E_d$ obtained in quasi-mechanical calculations are in good agreement with those obtained from Raman and quasi-elastic light scattering data.