Theoretical investigation of the variety of first-order transitions between prototypic and ferroic phases

Although theorists have so far investigated first-order transitions much less than second-order transitions, first-order transitions are much richer in variety. Conceivable are a first-order transition between a prototypic phase and a ferroic phase derived from it, between two ferroics derived from the same prototypic, and between a prototypic or one of its ferroics and another prototypic or one of its ferroics. The present paper deals with first-order transitions, each between a prototypic and one of its ferroics. These first-order transitions are still rich in variety. For them the concept sense is introduced; they are divided into second-sense transitions, thirdsense transitions, and so on. First-sense transitions are also defined; they are seen to be identical with second-order transitions. The thermodynamics of sense is developed. Especially the (third) ideal case, for which all of $C_k$'s with $k>~5$ and none of $C_k$'s with $2<~k<~4$ are equal to zero ($C_k$ is the $2k$ th degree coefficient in the free-energy function), is investigated systematically, as well as the first and second ideal cases where all of $C_k$'s with $k>~3 or 4$ and none of $C_k$'s with $2<~k<~2 or 3$ are equal to zero. In the third ideal case the first, second, and third senses are possible. For experimentally discerning between the second-and third-sense transitions. several theoretically deduced relations are usable, an example of which is that $4<\mathrm{}\frac{{\kappa }_P}{{\kappa }_F}<6\mathrm{}$ or $6<\mathrm{}\frac{{\kappa }_P}{{\kappa }_F}<\infty$ according to the second- or third-sense transition (${\kappa }_P$, ${\kappa }_F$ are the electric susceptibilities of the prototypic and ferroic, respectively, at the transition temperature).