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Using the Landau Ginzburg Devonshire theory and scalar approximation, wederived analytical expressions for the singular points (zeros, complex ranges)of the acoustic phonon mode (A mode) frequency in dependence on the wave vectork and examined the conditions of the soft A modes appearance in a ferroelectricdepending on the magnitude of the flexoelectric coefficient f and temperatureT. We predict that if the magnitude of the flexocoefficient f is equal to thetemperature-dependent critical value fcr(T) at the temperature T=T_IC, the Amode frequency tends to zero at k=kr_0 and the spontaneous polarization becomesspatially modulated in a temperature range T<T_IC.The comparison of calculatedphysical properties with measured ones are performed for some ferroelectricswith SMP phases.In particular, temperature dependence of the calculated directand inverse static dielectric susceptibility is in an agreement withexperimental data in Sn2P2(SexS1-x)6 that gives us additional background topredict flexo-coupling induced soft acoustic amplitudon-type mode in the SMPphase.The available experimental data on neutron scattering in organicincommensurate ferroelectric (CH3)3NCH2COO*CaCl2*2H2O are in asemi-quantitative agreement with our theoretical results. To quantify thetheory, it is necessary to measure the frequency dependence of the A mode in auniaxial ferroelectric with a spatially modulated phase in the temperatureinterval near its occurrence.

Flexocoupling induced soft acoustic mode and the spatially modulated phases in ferroelectrics