Vibrational analysis of benitoite (BaTiSi3O9) and theSi3O9ring

Abstract

The normal modes of vibration and their frequencies are calculated for benitoite, a mineral whose crystal structure (space group ${\mathit{D}}_{3\mathit{h}}^{2}$) consists of three-membered silicate rings (${\mathrm{Si}}_{3}$${\mathrm{O}}_{9}$) linked by ${\mathrm{Ba}}^{2+}$ and ${\mathrm{Ti}}^{4+}$ ions. Factor-group analysis dictates that certain normal modes involve the motion of only the ring atoms. On the assumption that mode mixings and splittings due to inter-ring interactions are small, the normal frequencies of the isolated ring of ${\mathit{C}}_{3\mathit{h}}$ symmetry are determined by fitting to suitable averages of selected frequencies in the Raman spectra. A valence force potential consisting of only central interactions between nearest neighbors and bond-bending interactions centered at the silicon atoms is used. This potential is then extended to the full crystal structure by including interactions involving the ${\mathrm{Ba}}^{2+}$ and ${\mathrm{Ti}}^{4+}$ ions. The frequencies obtained are in excellent agreement with the infrared and Raman spectra, requiring only minor adjustment of the force constants obtained for the isolated ring. The identification of normal modes characteristic of three-membered silicate rings may prove to be a valuable guide in the interpretation of the infrared and Raman spectra of amorphous silicates, potentially leading to new information on the ring statistics of these materials.