Theoretical studies of the temperature dependence of zero-field splitting of Cr3+ centers in ruby.

Abstract

Detailed theoretical studies of the temperature dependence of the zero-field splitting [characterized by \ensuremath{\Delta}D(T)=D(T)-D(0) or dD/dT] of ruby have been made by taking into account both the static contribution due to the thermal expansion of the lattice and the vibrational contribution due to the electron-phonon interaction. The static part is calculated by using the local thermal expansion coefficients ${\mathrm{\ensuremath{\alpha}}}_{\mathit{i}}$ in the vicinity of the ${\mathrm{Cr}}^{3+}$ ions, which are obtained from the local compressibilities ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ and the well-known relation ${\mathrm{\ensuremath{\alpha}}}_{\mathit{i}}$=${\mathit{C}}_{\mathit{v}}$\ensuremath{\gamma}${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$/V. The vibrational part includes the contributions from the acoustic and optical phonons. As in specific-heat studies of crystals, the vibrational contribution of phonons of the acoustic branches is calculated using the long-wavelength approximation, and that of phonons of the optical branches is calculated using a single-frequency model. The calculated results show that, to explain satisfactorily the temperature dependence of the zero-field splitting (ZFS), all of the contributions from the thermal expansion and the electron-phonon interaction should be considered. The static contribution is due mainly to the variations of bond angles as the temperature changes, which results in the static part of the temperature dependence of ZFS being opposite in sign to the observed temperature dependence. Thus the contribution due to the thermal expansion cannot be regarded as the main one. Among the various contributions, the contribution of acoustic phonons is the greatest in magnitude, and it has the same sign as the observed value. The contribution of optical phonons is the smallest and yet cannot be ignored. \textcopyright{}1996 The American Physical Society.