Theory of Isotope Effects in Ferroelectrics, Based on the Concept ofF-Modes of Lattice Vibration

Abstract

On the basis of the concept of ferroelectric modes ($F$-modes) of lattice vibration which was introduced in a recent paper by the author, a theoretical investigation is made into the effects of isotopic substitution on the spontaneous polarization at $T=0$, the Curie temperature ${T}_{c}$, the Curie constants in the paraelectric and ferroelectric regions, the coefficient of proportionality of ${{\ensuremath{\Omega}}_{+}}^{2}$ to $T\ensuremath{-}{T}_{c}$, and other constants characteristic of a crystal that can become ferroelectric. ($T$ stands for the absolute temperature. "$F$-mode" mentioned here is conceptually different from "ferroelectric mode" introduced and so named by Cochran, Landauer, and Thomas. ${\ensuremath{\Omega}}_{+}$ is the frequency of the ferroelectric mode in the latter sense.) The elements which undergo isotopic substitution are not restricted to hydrogen, or rather, may preferably be other than hydrogen. The crystal considered is assumed to have a single $F$-mode of lattice vibration. As a result, the expressions are obtained for computing the rates of change of the matter constants with respect to the mass of an ion when the masses and electric charges of all ions and the constants of the interionic short-range forces are given. Also, a few sum rules are obtained which hold among differential coefficients relating to individual ions. From these formulas, several qualitative conclusions are drawn. For example, it is found that if a certain ion does not move in the $F$-mode of lattice vibration, the matter constants are unchanged by the isotopic replacement of this ion.