Theory of Off-Center Displacements in Solid Solutions of Ionic Crystals

The minimum-energy configurations for impurity displacements along the $〈111〉$, $〈110〉$, and $〈100〉$ crystal symmetry axes were determined for KCl:${\mathrm{Li}}^{+}$, KBr:${\mathrm{Li}}^{+}$, and CsF:${\mathrm{Na}}^{+}$, through calculations based on a nearest-neighbor Born-Mayer-type model. The off-center configurations were all found to have lower energies than the centrosymmetric configurations. For all three materials the $〈111〉$ displacement gave the lowest and the $〈100〉$ the highest minimum energy, in agreement with the KCl:${\mathrm{Li}}^{+}$ experimental results. The CsF:${\mathrm{Na}}^{+}$ potential wells were about twice as deep as those in KCl:${\mathrm{Li}}^{+}$, whereas the KBr:${\mathrm{Li}}^{+}$ wells were only half as deep, indicating rapid tunneling between adjacent minima, which might explain the absence of an electrocaloric effect in KBr:${\mathrm{Li}}^{+}$. The calculated static dipole moments of the $〈111〉$ configurations were 6.23, 5.90, and 6.72 D, respectively, for KCl:${\mathrm{Li}}^{+}$, KBr:${\mathrm{Li}}^{+}$, and CsF:${\mathrm{Na}}^{+}$. The electricfield-gradient tensor was calculated at the impurity nucleus position for the three configurations. For the $〈111〉$ configuration, the quadrupole coupling constants $e^2\mathrm{qQ}$ for ${\mathrm{Li}}^7$ in KCl and KBr were -0.1196 Mc/sec and -0.0834 Mc/sec, respectively, and for ${\mathrm{Na}}^{23}$ in CsF it was 3.965 Mc/sec. An analysis of the expected quadrupole splittings of nuclear magnetic resonance (NMR) lines for various relative orientations of applied electric and magnetic fields indicates that such studies can distinguish between different possible minimum-energy configurations for these materials.