New view of the two commensurate phases in the typical incommensurate transition sequence (commensurate phase→incommensurate phase→commensurate phase)

Transition sequence Pr→IF→CF in dielectric substances is dealt with, where Pr, IF, and CF denote prototypic phase, incommensurate ferroic (or distorted) phase, and commensurate ferroic phase, respectively. The transition (or order) parameters are assumed to be two mutually conjugate complex quantities depending only on $x$, not on $y$ or $z$. They are denoted by $Q\left(x\right)\mathrm{}$ and $Q^{*}\mathrm{}\left(x\right)\mathrm{}$. In general, $Q\left(x\right)\mathrm{}$ can be expressed as a sum of harmonics of the form $Q_n{e}^{\mathrm{inhx}}$. Commonly, the lowest harmonic adopted is the first harmonic $Q_1{e}^{\mathrm{ihx}}$. In this paper, however, the zeroth harmonic $Q_0$ is taken into account. For simplicity, high harmonics are ignored so that $Q\left(x\right)=Q_0+Q_1{e}^{\mathrm{ihx}}$. Three views of CF are possible, i.e., CF may have {$Q_{\omega s}\ne 0$, $h_s=0\mathrm{}$}, {$Q_{0s}\ne 0$, $Q_{1s}=0\mathrm{}$, $h_s$ indefinite}, or {$Q_{0s}\ne 0$, $Q_{1s}=0\mathrm{}$, $h_s\ne 0$}, where $Q_{\omega }\equiv Q_0+Q_1$, and the subscript $s$ means "spontaneous." In the third view, $h_s$ is regarded as definite and assumed not to be nearly zero. The first view is conventional, in essence, while the third is new. Three views of Pr are possible, i.e., Pr may have {$Q_{\omega s}=0\mathrm{}$, $h_s=0\mathrm{}$}, {$Q_{0s}=0\mathrm{}$, $Q_{1s}=0\mathrm{}$, $h_s$ indefinite}, or {$Q_{0s}=0\mathrm{}$, $Q_{1s}=0\mathrm{}$, $h_s\ne 0$}. Only one view of IF is possible, i.e., IF has only {$Q_{0s}=0\mathrm{}$, $Q_{1s}\ne 0$, $h_s\ne 0$}. The transition Pr→IF and the imaginary direct transition Pr→CF are assumed to be second order. The temperatures of the respective transitions are denoted by $T_{\mathrm{tr}}$ and $T_{\mathrm{tr}}^{\prime }$. It is found that if the third views of Pr and CF are adopted, the transitions Pr→IF and Pr→CF are strongly second order, i.e., even $h_s$ is continuous at $T_{\mathrm{tr}}$ and $T_{\mathrm{tr}}^{\prime }$. Let $q_1\equiv \left|Q_1\right|$. Let $\Phi$ be the spatially averaged free-energy function. The third views of Pr and CF are thought to be reasonable from the investigation of the $T$ dependence of ${\left(\frac{{\partial }^2\Phi }{\partial q_1^2}\right)}_s$ in Pr near $T_{\mathrm{tr}}$ and in CF near $T_{\mathrm{tr}}^{\prime }$. A full account of this investigation is given.