Properties of the one-particle-displacement probability distribution in systems undergoing antiferrodistortive structural phase transitions

Abstract

Using the Hartree approximation, a high-temperature expansion, and the molecular-dynamics technique, we study some properties of the one-particle probability distribution ${F}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})$ of the displacement ${U}_{x}$ of particle one in a model system. The system is two dimensional and subjected to constraints in such a way that it exhibits antiferrodistortive structural phase transitions. It covers the displacive and order-disorder regime, including the Ising and displacive limit. We present evidence that ${F}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})$ or its symmetrized analog ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})=\frac{1}{2}[{F}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})+{F}_{1}(\ensuremath{-}{U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})]$, being a very useful property to elucidate the regime to which a particular antiferrodistortive transition belongs. In the displacive regime, the ratio ${a}_{s}=\frac{{\left(\frac{d}{d{U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}}}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})\right)}_{max}}{{\left(\frac{d}{d{U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}}}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})\right)}_{min}},$ for ${U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}}$ either negative or positive, is shown to diverge at some temperature ${T}^{*}$, because ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{F}}_{1}({U}_{x}^{\stackrel{\ensuremath{\rightarrow}}{1}})$ exhibits for $T<{T}^{*}$ a double-peak structure disappearing at $T={T}^{*}$. In the order-disorder regime, the ratio $\frac{{T}^{*}}{{T}_{c}}$ is infinite and decreases in the displacive regime by approaching the displacive limit to some value $\frac{{T}^{*}}{{T}_{c}}<1$. As M\"uller and Berlinger have shown, the key quantity ${a}_{s}$ can be measured, close to ${T}_{c}$ by means of the electron-paramagnetic-resonance technique.