Change of the Order of a Phase Transition by the Influence of an External Parameter

It is investigated how, in a phase transition, the order of the transition may be changed from second to first or vice versa by the influence of a parameter Y. It is supposed that the free energy can be written as the sum of four terms: G1(η) a function only of the order parameter η; G2(X) a function only of X, the conjugate of Y; G3(X, η) a coupling function, and G4 = − YX. Some different examples are examined choosing a simple form for G2(X), and of G3(X, η). Also the case of two external parameters coupled to the order parameter is investigated. The results are discussed in connection with different materials like BaTiO3, PbTiO3, and improper ferroelectrics.