High-Temperature Crystallographic Study of Mg-Doped TbAlO₃ and YAlO₃

The structures of Mg-doped TbAlO₃ and YAlO₃ have been examined by time-of-flight powder neutron diffraction at room temperature and at 573, 773, 1023 and 1223 K. For both compounds, at all temperatures, structural and anisotropic thermal parameters were refined in space group Pnma (No. 62) by full-profile Rietveld methods to weighted-profile R values of less than 0.042. The changes in difference nuclear densities, Δρ, due to changes in temperature are illustrated by difference-density maps around the atoms. The difference-density maps are constructed from F_obs, phased by F_calc, and the high-temperature data are corrected by an isotropic temperature factor obtained locally around the atom in question. Δρ is computed as Δρ(low temperature) − Δρ(high temperature) with the coordinates of the atoms translated to the origin (0, 0, 0). The observed difference-density distributions are described by mainly zeroth- and second-order spherical harmonics (quadrupolar functions), the natures of which vary between atomic sites. The difference-density maps provide a direct picture of the average in space and time of atomic thermal vibrations. These observations are less biased by the least-squares method than refined parameters. The nonspherical components to the atomic thermal vibrations are similar in Mg-doped TbAlO₃ and YAlO₃ but the absolute magnitude is significantly higher in TbAlO₃.