Structure of incommensurate ammonium tetrafluoroberyllate studied by structure refinements and the maximum entropy method.

Incommensurately modulated ammonium tetrafluoroberyllate (AFB) occurs in a narrow temperature interval between the paraelectric room-temperature phase with space group Pnma (Ti = 178 K) and the ferroelectric low-temperature phase with space group Pna2(1) (Tc = 173 K). The structure is determined from accurate single-crystal X-ray diffraction data collected with synchrotron radiation at 175 K. The superspace group of the structure is Pnma(alpha00)0ss with alpha = 0.4796 (4). Both structure refinements and the maximum entropy method lead to the same structure model, which involves only single harmonic modulations. The building units of the structure are BeF4(2-) and NH4+ complex ions with approximately tetrahedral point symmetry. They are relatively rigid and the modulations consist mainly of translations of the tetrahedra and their rotations around a fixed axis. The modulation is related to changes in the network of the hydrogen bonds. The low-temperature superstructure can be described as a commensurately modulated structure with the same superspace symmetry. The first harmonic modulations of the low-temperature and incommensurate phases are related by a scale factor with a value of approximately two. In addition, the low-temperature phase exhibits a second harmonic modulation that is responsible for shifts along c and the ferroelectricity in this phase. The experimental data of the incommensurate phase do not contain any evidence for the presence of a second harmonic in the modulation functions. This suggests that the development of the second harmonic, i.e. the development of the spontaneous polarization, is responsible for the lock-in transition.