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Acta Cryst. (2014). A70, C54
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The description of a Landau phase-transition considers group-subgroup relations with respect to the initial phase at thermodynamic equilibrium. Accordingly, the Gibbs free energy is modified in terms of an order parameter Q [1]. Data collection on a particular phase at equilibrium is challenging in case of associated kinetic effects. Thus describing a phase-transition using the Landau theory leads quite often to a marked discrepancy between the observed and calculated data close to Tc. Evaluating an accurate order parameter consequently must be a hurdle. In some cases the order parameter is proportional to the unit-cell volume of the phase. Similar effect could be seen while evaluating the lattice volume of a phase using Debye-Einstein-Anharmonicity model [2] where the internal energy of a crystalline lattice volume is considered to be sum of the Einstein harmonic, Debye quasi-harmonic and anharmonic vibrational potentials. The DEA-model gives possibility to compare the experimental thermal expansion coefficient (TEC) with that of the theoretical one. For example, in dehydrated carbonate-nosean [3] the kinetic effect close to the P23-P23 phase-transition could be seen in terms of the decrease of the TEC over several data points around 800 K (Figure). In this case, a pure Landau transition would have sharply dropped the TEC leaving no intermediate values. These deviations in the experimental data could be theoretically modeled by overlapping the TEC's of the low-temperature (TECLT) and the high-temperature (TECHT) phase using a sigmoidal term fs(T) {0 < fs(T) < 1}: TECtotal(T) = fs(T)·TEClow(T) + (1- fs(T))·TEChigh(T). The slope of this sigmoidal function corresponds to the deviation of the model used for the volume calculation close to the phase-transition and therefore carries information about the reaction kinetics. The same could be calculated for the dehydration process of the hydrated carbonate-nosean |Na8(CO3)(H2O)4|[AlSiO4]6 around 400 K. Figure: Experimental temperature-dependent volume change of hydrated carbonate-nosean and respective DEA-model calculation results. The individual DEA-models as well as TEC changes including kinetic overlaps are given inserted.

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Acta Cryst. (2014). A70, C161
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The lattice thermal expansion of crystalline solids cannot be adequately modeled by Grüneisen approximation using either Einstein single harmonic frequency or Debye frequency spectrum because a true phonon spectrum does not follow either of the two kinds. The Debye model misfits to many observations due to the fact that real solids comprises of axial anisotropy, lattice waves with dispersion at the Brillouin zone boundaries, low/high frequency optical vibrations in excess of the Debye spectrum. The actual frequency distribution is a complicated function of frequency instead of a simple parabolic Debye spectrum. The frequency distribution can be simplified using power series [1] leading to singularities before and after the Debye cutoff frequency. Using multiple Debye or Einstein oscillators, or their mixtures, is also common practice to better describe the lattice expansion, however, these models extremely suffer from intrinsic anharmonicity in particular at high temperatures. It was demonstrated that even the noble monoatomic solids required inclusion of anharmonic terms in the harmonic model to better explain the observed values [2]. Worse even, when anharmonicity becomes dominant due to formation of vacancies and defects, anomalies of hard/soft modes or change of stereochemical activities of lone electron pairs (LEPs) as function of temperature. Herein we approach an extended Grüneisen approximation that includes harmonic, quasiharmonic and intrinsic anharmonic potentials to describe the internal energy of the crystal as function of temperature. The model has been applied to several complex oxides with LEPs (Bi2Ga4O9 [3]) along with axial negative thermal expansion (PbFeBO4) and rigid-unit-modes (KAsW2O9) reported here. The metric parameters were obtained from quality data collected from temperature-dependent neutron and X-ray powder diffractions.
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