Lattice vibrations anharmonicity

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. 

Displacive ferroeiectrics where a discrete symmetry group is broken at Tc and the ferroelectric transition can be described as the result of an instability of the anharmonic crystal lattice against soft polar lattice vibration (e.g., BaTiOs). 

In addition to the individual and uncorrelated particle motions, we also have collective ones. In a strict sense, the hopping of an individual vacancy is already coupled to the correlated phonon motions. Harmonic lattice vibrations are the obvious example for a collective particle motion. Fixed phase relations exist between the vibrating particles. The harmonic case can be transformed to become a one-particle problem [A. Weiss, H. Witte (1983)]. The anharmonic collective motion is much more difficult to treat theoretically. Correlated many-particle displacements, such as those which occur during phase transformations, are further non-trivial examples of collective motions. 

In the hydrate lattice structure, the water molecules are largely restricted from translation or rotation, but they do vibrate anharmonically about a fixed position. This anharmonicity provides a mechanism for the scattering of phonons (which normally transmit energy) providing a lower thermal conductivity. Tse et al. (1983, 1984) and Tse and Klein (1987) used molecular dynamics to show that frequencies of the guest molecule translational and rotational energies are similar to those of the low-frequency lattice (acoustic) modes. Tse and White (1988) indicate that a resonant coupling explains the low thermal conductivity. 

Brennan C, Nelson KA. Direct time-resolved measurement of anharmonic lattice vibrations in ferroelectric crystals. J Chem Phys 1997 107 9691-9694. 

In the weakly anharmonic molecular crystal the natural modes of vibration are collective, with each internal vibrational state of the molecules forming a band of elementary excitations called vibrons, in order to distinguish them from low-frequency lattice vibrations known as phonons. Unlike isolated impurities in matrices, vibrons may be studied by Raman spectroscopy, which has lead to the establishment of a large body of data. We will briefly attempt to summarize some of the salient experimental and theoretical results as an introduction to some new developments in this field, which have mainly been incited by picosecond coherent techniques. 

A scheme as described here is indispensable for a quantum dynamical treatment of strongly delocalized systems, such as solid hydrogen (van Kranendonk, 1983) or the plastic phases of other molecular crystals. We have shown, however (Jansen et al., 1984), that it is also very suitable to treat the anharmonic librations in ordered phases. Moreover, the RPA method yields the exact result in the limit of a harmonic crystal Hamiltonian, which makes it appropriate to describe the weakly anharmonic translational vibrations, too. We have extended the theory (Briels et al., 1984) in order to include these translational motions, as well as the coupled rotational-translational lattice vibrations. In this section, we outline the general theory and present the relevant formulas for the coupled... 

The temperature dependence of the linear thermal expansion coefficients a(T) of the investigated titanium silicides are illustrated in fig. 6. The complex hexagonal Ti5Si3 compound exhibits a (T) values lower than those of the disilicide TiSi2 with the closer packed C54 structure. Another reason is that the anharmonicity of the lattice vibrations -phonons- and the asymmetry of the lattice potential curves of the Ti-Si and Si-Si bonds of the C54 structure are more pronounced compared to that of the D8S lattice. 

Taking anharmonicity of the lattice vibrations into account leads to interaction of the phonons with one another. When this interaction turns out to be sufficiently strong, the formation of bound states of quasiparticles becomes possible in addition to the above-mentioned multiparticle states. Such states are... 

From equation 1.15 IF is proportional to T at high temperatures. Experience shows that experimentally this is usually not the case because of anhar-monicity of the lattice vibrations. The effects of anharmonicity have been discussed at length by Boyle and Hall [7] and will be met again in connection with the intensities of individual lines in hyperfine interactions. 

Given this description of the forces, there is no difficulty in carrying out the calculation of the full vibration spectrum for an ionic solid in the manner described in Chapter 9. At least for the compounds we have considered here the structures are sufficiently simple that the complications which arose in calculating the spectra for the mixed tetrahedral solids are not present. The elastic constants describe the low-frequency lattice vibrations and there is no reason to expect the description at high frequencies to be either worse or better. (Some discussion of the vibration spectra of ionic crystals is given, for example, by Wallis, 1965 a number of properties related to anharmonicity are described by Cowley, 1971.)... 

The mean free path A may be determined by many different scattering mechanisms but the dominant one at temperatures not too close to o °K is phonon-phonon scattering, the coupling taking place through the anharmonicity of the lattice vibrations. There are two possible types of phonon-phonon scattering processes normal processes in which total phonon wave vector is conserved, and umklapp processes in which the total wave vector after collision differs from that before collision by a vector of the reciprocal lattice. Since normal processes do not affect the total phonon momentum or energy, they do not contribute to thermal resistance and only umklapp processes need be considered. For an umklapp process to occur between two phonons of wave vectors q and q we must have a relation of the form... 

The free energy due to harmonic lattice vibrations (or equivalently the Debye temperature) is approximately the same for bcc, fee, and hep structures but with a significant tendency for the bcc value to be a few percent lower. The more open bcc structure has a transverse phonon mode with a particularly low frequency which causes a more rapid decrease in the free energy with temperature. On cooling, sodium and lithium transform partially from bcc to hep at very low temperatures (0.1-0.2 Tm). Calcium, strontium, beryllium, and thallium transform to a bcc phase at high temperatures (0.66-0.98 Tm) when there is a considerable anharmonic contribution to the free energy. 

Both expansion (a) and conductivity K) are linked to a cubic anharmonic term in the potential of a lattice vibration (Klemens 1958). An estimation can be obtained by the Leibfried-Schloemann formula (Leibfiied and Schloemann 1954)... 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

This also pertains to the phase transformation in a solid that is the result of a gradual displacement of atoms from their original position in a soft mode transition. By way of illustration, we adopt a one-dimensional model and suppose that the gradual shift in atomic location may be correlated with anharmonic terms in the lattice vibration of the participating atoms. This is quantified via the relation... 

At higher temperatures the anharmonicity of the lattice vibrations has to be considered also (Section II.4.S). As far as substances with larger thermal expansion are concerned, the anharmonic toms 0a and 0 in the progression (98) for the potential energy lead to an increase of the heat capacity, linear with temperature, that exceeds the value of Dulong and Petit. The terms 0 and 0 (and the next higher powers of 0a and cause a further increase of the heat capacity, proportio-... 

The influence of anharmonicity of lattice vibrations on heat capacities is a particular problem which has, in the case of linear high polymers, not been treated experimentally or theoretically. For polymers one must expect that anharmonicity even of higher order is of importance even far below the melting point. A detailed analysis of this problem parallel with investigations of lattice expansion and temperature dependence of the moduli seems to be urgently needed. 

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],... 

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