A TEM and RUM study of the inherent displacive flexibility of the fresnoite framework structure type

A careful electron diffraction study has been made of the incommensurately modulated room-temperature phases of the fresnoites Ba₂TiGe₂O₈ (BTG) and Ba₂TiSi₂O₈ (BTS) and used to determine their (3 + 1)- and (3 + 2)-dimensional superspace group symmetries. The primitive primary modulation wave vectors in both materials are found to occur close to the same position in the parent Brillouin zone, near ∼ 0.3<110>^*_p + 1/2c^*_p. A rigid unit mode (R U M) analysis of the inherent displacive structural flexibility of the ideal fresnoite framework structure type is then carried out in an attempt to understand the significance of this particular modulation wave vector. Six zero-frequency R U M modes and two close to zero frequency quasi-R U M (Q-R U M) modes are found to exist for any modulation wave vector. These R U M modes are all primarily associated with rotations of the constituent TO₄ (T = Si or Ge) tetrahedra and TiO₅ square pyramids around in-plane i.e. perpendicular to c rotation axes. A seventh R U M mode involving rotation of the constituent rigid polyhedra around c combined with shifts in the basal plane is found but only at a very specific modulation wave vector q ∼ 0.30<110>^*_p, in close agreement with the condensed R U M mode found in the electron diffraction study. It is the condensation of just such a R U M mode that appears to play a major role in the various incommensurately modulated structures observed in Ba₂TiGe₂O₈, Ba₂TiSi₂O₈ and Sr₂TiSi₂O₈, respectively.