  The majority of magnetoelectric crystals show complex temperature-magnetic
field or temperature-pressure phase diagrams with alternating antiferromagnetic
incommensurate, magnetoelectric, and commensurate phases. Such phase diagrams
occur as a result of successive magnetic instabilities with respect to
different order parameters, which usually transform according to different
irreducible representations (IR) of the space group of the crystal. Therefore,
in order to build a phenomenological theory of phase transitions in such
magnetoelectrics one has to employ several order parameters and assume the
proximity of various instabilities on the thermodynamic path. In this work we
analyze the magnetoelectrics MnWO4, CuO, NaFeSi2O6, NaFeGe2O6, Cu3Nb2O8,
alpha-CaCr2O4, and FeTe2O5Br using the praphase concept and the symmetry of the
exchange Hamiltonian. We find that in all the considered cases the appearing
magnetic structures are described by IR's entering into a single exchange
multiplet, whereas in the cases of MnWO4 and CuO by a single IR of the space
group of the praphase structure. Therefore, one can interpret the complex phase
diagrams of magnetoelectrics as induced by a single IR either of the praphase
or of the symmetry group of the exchange Hamiltonian. Detailed
temperature-magnetic field phase diagrams of MnWO4 and CuO for certain field
directions are obtained and the magnetic structures of the field-induced phases
are determined.
