Anharmonic thermal vibrational model

The heat capacity models described so far were all based on a harmonic oscillator approximation. This implies that the volume of the simple crystals considered does not vary with temperature and Cy m is derived as a function of temperature for a crystal having a fixed volume. Anharmonic lattice vibrations give rise to a finite isobaric thermal expansivity. These vibrations contribute both directly and indirectly to the total heat capacity directly since the anharmonic vibrations themselves contribute, and indirectly since the volume of a real crystal increases with increasing temperature, changing all frequencies. The constant volume heat capacity derived from experimental heat capacity data is different from that for a fixed volume. The difference in heat capacity at constant volume for a crystal that is allowed to relax at each temperature and the heat capacity at constant volume for a crystal where the volume is fixed to correspond to that at the Debye temperature represents a considerable part of Cp m - Cv m. This is shown for Mo and W [6] in Figure 8.15. 

It is generally agreed that thermally induced vibrations of atoms in solids play a major role in melting [2.144]. The simple vibrational model of Linde-mann predicts a lattice instability when the root-mean-square amplitude of the thermal vibrations reaches a certain fraction / of the next neighbor distances. However, the Lindemann constant/varies considerably for different substances because lattice anharmonicity and soft modes are not considered, thus limiting the predictive power of such a law. Furthermore, Born proposed the collapse of the crystal lattice to occur when one of the effective elastic shear moduli vanishes [2.138], Experimentally, it is found instead that the shear modulus as a function of dilatation is not reduced to zero at Tm and would vanish at temperatures far above Tm for a wide range of different substances [2.145]... 

The crystal structure of synthetic hibonite, CaAli20i9, was determined by Kato and Saalfeld (1968) and was refined by Utsunomiya et al. (1988). Since anharmonicity of the thermal vibration of the A12 atom was observed, refinement of the temperature factors including anharmonic terms up to the fourth order was done (Utsunomiya et al. 1988). The result supports a split atom model where statistically occupies one of the two equivalent A12 sites at room temperature. 

In the phonon assisted tunnelling model the H-bond stretch of benzoic acid is anharmonically linked to the entire thermal bath of internal and external vibrations. No individual vibration has any exceptional role in this almost stochastic process and in particular the... 

Such a potential energy function gives rise to the familiar parabolic curve (Figure 22) where the curvature of the function is related to the force constant. The success of this simple harmonic model in treating surface atom vibrations lies in the relatively small displacement of surface atoms during a period of vibration. For some crystal properties, such as thermal expansion at elevated temperature, anharmonic contributions to the potential must be included for an accurate description. 

The thermal effect on the Young s modulus was modeled based on the third law of thermodynamics as anharmonic effects of the lattice vibrations [4]. Wachtman et al. and Champion et al. suggested an empirical equation for the temperature effect on the Young s modulus [5],... 

However, above a few kelvin the universal properties of glasses deviate from the predictions of the AHVP model. The thermal conductivity shows a plateau around 10 K, which cannot be understood in terms of a constant density of tunneling states [9]. The sound velocity decreases linearly with temperature above a few kelvin [10]. Furthermore, there are an additional increase in the specific heat, and in the low-frequency Raman scattering [11] indicating the existence of still another kind of low-frequency modes. Recent neutron measurements [12] have shown these to be soft harmonic vibrations with a crossover to anharmonicity at the low-frequency end (at frequencies corresponding to several kelvin). 

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