Modeling oxygen isotope shifts and line-broadening effects of the 502-${\mathrm{cm}}^{\mathrm{-}1}$ Raman mode of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$

Raman scattering experiments have been carried out on sintered pellets of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3^{18}$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ and ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3^{16}$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$, obtained both by gas exchange and by growth with substituted oxides. The frequencies of the modes at 340, 435, and 500 ${\mathrm{cm}}^{\mathrm{-}1}$, which involve motion of the oxygen atoms and which shift significantly upon oxygen-isotope substitution, have been measured for several sets of samples. The 500-${\mathrm{cm}}^{\mathrm{-}1}$ line of the ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ samples is observed to be isotope shifted to a lesser degree than the 340- and 435-${\mathrm{cm}}^{\mathrm{-}1}$ phonons, and is significantly broadened. To model the observed behavior of the 500-${\mathrm{cm}}^{\mathrm{-}1}$ mode, a linear chain of O and Cu atoms is used to compute first-order k≃0 Raman cross sections from 400 to 600 ${\mathrm{cm}}^{\mathrm{-}1}$. Disorder is introduced by randomly substituting ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ atoms for ${}_{}{}^{16}{}_{}{}^{}\mathrm{O}$ atoms at positions in the chain corresponding to O(4) in the ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ lattice. Comparison of computed and experimental spectra suggest that the maximum linewidth in the 500-${\mathrm{cm}}^{\mathrm{-}1}$ peak occurs for a relative ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ concentration ${[}^{18}$O(4)]/${[}^{16}$O(4)] between 24% and 36%. In addition, the 502-${\mathrm{cm}}^{\mathrm{-}1}$ peak position is found to depend nonlinearly on the relative ${}_{}{}^{18}{}_{}{}^{}\mathrm{O}$ concentration. The implications of these results are discussed in terms of the currently accepted models for oxygen-isotope exchange in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$