Theory of the Degree of the Free-Energy Function for Ferroelectrics

The free-energy function in the thermodynamical theory of ferroelectrics is often put approximately equal to a finite power series in the components of the polarization vector. It is obvious that in general, in order to be able to give an account of ferroelectric phenomena, the free-energy function has to be of fourth degree at the lowest. In this paper, a free-energy function $\Phi$ of a certain degree is said to be able to describe a ferroelectric, when $\Phi$ permits the ferroelectric (i) to exist as a stable or metastable phase and (ii) to transform to a paraelectric phase. These two stipulations are the minimum condition necessary for any ferroelectric phase to be properly a "ferroelectric phase." The lowest degree necessary for the free-energy function to be able to describe a ferroelectric is referred to, briefly, as the "describability limit" of the ferroelectric. It is anticipated that the describability limits of all the ferroelectrics which belong to one and the same species have a lower limit (in the mathematical sense) which may not generally be 4. This lower limit is referred to as the describability limit of the species. The describability limit of a ferroelectric may depend on whether it is under zero stress or under constant strain, but the describability limit of a species does not. A determination is made of whether the describability limit of each of the 55 species is 4 or larger. As a result, it is found that 28 species—$\mathrm{F}\overline{1}\mathrm{}\left(1\right)\mathrm{}\mathrm{A}1$, etc.—are of describability limit 4, and that the describability limits of the other 27 species—$\frac{\mathrm{F}2}{m\left(2\right)\mathrm{}\mathrm{A}1}$, etc.—must be larger than 4; they are not all 6.