Phonons thermal expansion coefficient

The possibility of negative thermal-expansion coefficient (TEC) values along a direction of strong coupling in layered or chain structures (the so-called membrane effect") was suggested for the first time by Lifshitz [4] for strongly anisotropic compounds. In the phonon spectra of such compounds... 

The Raman active phonon of high quality single crystal silicoi occurs at 520.7cm l. Onder toisile stress the band shifts to lower energies under conpressive stress the band shifts to higher energies.(12) Because of the values of the thermal expansion coefficients of siliccm and its oxide, the silicon film re-crystallized over the oxide will experience tensile stress, m order to maximize the accuracy with thich the stress could be measured, the Raman Epectra were recorded digitally with O.lcm" between data points. (Instrumental repeatability was also 0.1cm l). 

Because of the term 5K, when we evaluate dco(q,a)/dV, as demanded by eqn (5.72) in our deduction of the thermal expansion coefficient, it will no longer vanish as it would if we were only to keep K. The phonon frequencies have acquired a volume dependence by virtue of the quasiharmonic approximation which amounts to a volume-dependent renormalization of the force constant matrix. 

We are now in a position to revisit the analysis of the thermal expansion presented earlier. In particular, we note that the thermal expansion coefficient may now be evaluated by attributing a particular form to the volume dependence of the phonon frequencies. In order to reveal the application of the developments advanced above, we undertake an analysis of the thermal expansion in the... 

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. 

The propagation of acoustic phonons in amorphous media depends on the mechanical and thermal moduli. We will denote the modulus of compression by K, the shear modulus by G, the longitudinal modulus by M = K + 4/3G, the thermal conductivity by c, the thermal expansion coefficient by a, and the ratio of specific heats by y = Cp/Cy. 

The linear thermal expansion coefficient a can be determined from the derive of the temperature dependence of the cell dimension as a = d ln L(T) /dT. Here, L(T) is the cell dimension at the corresponding temperature T. The volumetric TEC can be expressed via fundamental parameters, such as the Griineisen parameter Yv, the bulk modulus K, the molar voliune Vm, the heat capacity C, phonon frequencies o), and the internal energy U ... 

We note that through the Griineisen parameters anharmonic corrections of the potential energy are involved in the thermal expansion coefficient. The approximation that the vibrational free-energy function depends on the volume of the crystal through the change of the phonon frequencies described by the first-order approximation... 

I tested the GAP models on a range of simple materials, based on data obtained from Density Functional Theory. I built interatomic potentials for the diamond lattices of the group IV semiconductors and I performed rigorous tests to evaluate the accuracy of the potential energy surface. These tests showed that the GAP models reproduce the quantum mechanical results in the harmonic regime, i.e. phonon spectra, elastic properties very well. In the case of diamond, I calculated properties which are determined by the anharmonic nature of the PES, such as the temperature dependence of the optical phonon frequency at the F point and the temperature dependence of the thermal expansion coefficient. Our GAP potential reproduced the values given by Density Functional Theory and experiments. 

Up to now we have discussed the influence of the crystal field eflect on (due to phonon resonance scattering by the rare earth ions) and the heat capacity (Schottky effect). It is known that the crystal field can significantly influence the behaviour of the elastic constants (Luthi et al. 1973), the thermal expansion coefficient (Ott and Liithi 1976), the magnetostriction and thermoemf (Sierro et al. 1975), the electrical conductivity (Liithi et al. 1973, Friederich and Fert 1974, Andersen et al. 1974) and the electron part of the thermal conductivity (Smirnov and Tamarchenko 1977, Wong 1978, Matz et al. 1982, Muller et al. 1982). 

One physical effect which is related to phonon modes and to the specific heat is the thermal expansion of a soUd. The thermal expansion coefficient a is defined as the rate of change of the linear dimension L of the solid with temperature T, normaUzed by this linear dimension, at constant pressure P ... 

When changes in the internal energy E are exclusively due to phonon excitations, we can use E from Eq. (6.53) to obtain for the thermal expansion coefficient... 

For a free surface, a = a, and b = 0. There are some difficulties however to use this equation, as remarked on by Hwang et al. [29] since the thermal expansion coefficient within the temperature range T — Tg is hardly detectable. The value of B in Eq. (15.6) is given by the difference of the phonon negative dispersion and the size-dependent surface tension. Thus, a positive value of B indicates that the phonon negative dispersion exceeds the size-dependent surface tension and consequently causes the redshift of phonon frequency. On the contrary, if the size-dependent surface tension is stronger than the phonon negative dispersion, blue-shift occius. In case of balance of the two effects, i.e., B = 0, the size dependence... 

Fig. 42.1 The NTE of a H2O (Reprinted with permission from [37]), b graphite (Rejninted with permission from [12]), and c ZrW20g (Reprinted with permission from [10]), with a (open circles) and Grlineisen parameter y = 3aB/C (crosses). These NTEs share the same identity at different range of temperatures, which evidence the essentiality of two types of short-range interactions with specific-heat disparity, d ZrW20g phonon density of states measured at T = 300 K. Parameters a, B, and C correspond to thermal expansion coefficient, bulk modulus, and the specific heat at constant volume, respectively...

When before free electrons, which polarize their dielectric surrounding, generate traps by electron-phonon interaction, well-defined distortions of the icosahedra result, which depend on the number of pairs of phonons involved. Because of the thermal excitation of phonons required for this interaction, the formation of specific traps is maximum at defined temperatures, when electrons are available, e.g., by optical excitation (65,66,96). In thermal equilibrium the temperature range between about 500 and 600 K is of particular interest because the thermal energy concerned is sufficient to exchange electrons between all the levels in Fig. 14 within rather short relaxation times. With this consideration, Werheit et al. (71) were able to interpret the previously unexplained hysteresis of the thermal expansion coefficient (115) and the maximum of internal friction (116,117) in this temperature range consistently. 

As in aH solids, the atoms in a semiconductor at nonzero temperature are in ceaseless motion, oscillating about their equilibrium states. These oscillation modes are defined by phonons as discussed in Section 1.5. The amplitude of the vibrations increases with temperature, and the thermal properties of the semiconductor determine the response of the material to temperature changes. Thermal expansion, specific heat, and pyroelectricity are among the standard material properties that define the linear relationships between mechanical, electrical, and thermal variables. These thermal properties and thermal conductivity depend on the ambient temperature, and the ultimate temperature limit to study these effects is the melting temperature, which is 1975 KforZnO. It should also be noted that because ZnO is widely used in thin-film form deposited on foreign substrates, meaning templates other than ZnO, the properties of the ZnO films also intricately depend on the inherent properties of the substrates, such as lattice constants and thermal expansion coefficients. 

Because both excited vibrational phonon modes and Brownian motion contribute to the dynamic excluded volume of the embedded CNT, as the temperature is increased their contributions towards excluded volume increase significantly. The cross-linking of polymer with CNT was not allowed in these initial simulations. It is possible that the cross-linking of polymer matrix with embedded CNTs may further reduce the motions of polymer molecules or the CNT the predicted changes in the glass transition temperature and the thermal expansion coefficients in that scenario could be different. The increase in glass transition temperature and thermal expansion coefficients of carbon nanotube polymer composites has been also observed in experiments [55]. 

There are a variety of thermal properties of materials such as heat capacity, thermal conductivity, thermal expansion coefficient, and many more. For purposes of this text we will focus on the thermal conductivity as it is critical to many electronic devices. The reader is referred to the suggested readings for details of the full spectrum of thermal properties and for details not described here. The thermal conductivity, k, of a material results from transport of energy via electrons or via lattice vibrations (phonons). The total thermal conductivity can thus be written simply as ... 

At high temperature, TTF TCNQ is metallic, with a(T) oc T-2 3 since TTF TCNQ has a fairly high coefficient of thermal expansion, a more meaningful quantity to consider is the conductivity at constant volume phonon scattering processes are dominant. A CDW starts at about 160K on the TCNQ stacks at 54 K, CDW s on different TCNQ chains couple at 49 K a CDW starts on the TTF stacks, and by 38 K a full Peierls transition is seen. At TP the TTF molecules slip by only about 0.034 A along their long molecular axis. 

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