Landau model for commensurate-commensurate phase transitions in uniaxial improper ferroelectric crystals

We propose the Landau model for lock-in phase transitions in uniaxially mod ulated improper ferroelectric incommensurate-commensurate systems of class I. It includes Umklapp terms of third and fourth order and secondary order parameters representing the local polarization. The corresponding phase dia gram has the structure of a harmless staircase, with the allowed wave numbe rs obeying the Farey tree algorithm. Among the stable commensurate phases o nly those with periods equal to an odd number of lattice constants have fin ite macroscopic polarizations; These results are in excellent agreement wit h experimental findings in some A(2)BX(4) compounds.