Effect of Uniaxial Stress on the Raman Spectra of Cubic Crystals: Ca${\mathrm{F}}_2$, Ba${\mathrm{F}}_2$, and ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{20}$

The effect of uniaxial stress on the first-order Raman spectra of the cubic crystals, Ca${\mathrm{F}}_2$, Ba${\mathrm{F}}_2$, and ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{20}$ is studied at $\sim 15$ °K using a quantitative-stress cryostat. Both Ca${\mathrm{F}}_2$ and Ba${\mathrm{F}}_2$ belong to the space group ${O_h}^5\left(Fm 3m\right)$ and possess a single triply degenerate Raman-active zone-center optical phonon of $F_{2g}$ symmetry. In contrast, ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{20}$ has a very rich first-order Raman spectrum consisting of lines $A$, $E$, $F$, and LO-TO-split $F$ modes. The effect of uniaxial stress on the $F_{2g}$ line of Ca${\mathrm{F}}_2$ and on Raman lines typical of the different symmetries in ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{20}$ is studied up to 7 kbar with compressive force $\overrightarrow{\mathrm{F}}$ along [001], [111] or [110]. The Raman line of Ba${\mathrm{F}}_2$ is studied up to 2.4 kbar for $\overrightarrow{\mathrm{F}}\parallel \mathrm{}\left[001\right]\mathrm{}$. The stress-induced splittings and polarization characteristics in each case can be understood on the basis of the reduced symmetry of the crystal under applied stress. Using a perturbing Hamiltonian linear in strain, the secular equation is derived for phonons of each symmetry in terms of phenomenological "deformation-potential constants." Phonons of $A$, $E$, and $F$ symmetry in $T$ and of $F_{2g}$ symmetry in $O_h$ are characterized by one, three, four, and three deformation-potential constants, respectively. Within the framework of this theory, the splittings as a function of the crystallographic orientation of the applied force can be correlated in terms of the deformation-potential constants, whereas the intensities of the stress-induced components can be calculated in terms of the zero-stress polarizability-tensor components. Experimentally, in all the cases the positions of the stress-induced components are observed to vary linearly with stress; the splittings, shifts, and polarization characteristics are consistent with the predictions based on the reduced symmetry of the crystal as well as the deformation-potential approach. On the basis of the observed polarization characteristics of the stress-induced components, appropriate eigenvalues have been associated...