Lattice vibrations specific heat contribution

The transport of heat in metallic materials depends on both electronic transport and lattice vibrations, phonon transport. A decrease in thermal conductivity at the transition temperature is identified with the reduced number of charge carriers as the superconducting electrons do not carry thermal energy. The specific heat and thermal conductivity data are important to determine the contribution of charge carriers to the superconductivity. The interpretation of the linear dependence of the specific heat data on temperature in terms of defects of the material suggests care in interpreting the thermal conductivity results to be described. 

The specific heat of a semiconductor has contributions from lattice vibrations, free carriers and point and extended defects. For good quality semi-insulating crystals only the lattice contribution is of major significance. Defect-free crystals of group III nitrides are difficult to obtain, and thus the specific heat measurements are affected by the contributions from the free carriers and the defects. While the specific heat of AIN is affected by the contribution of oxygen impurities, the data for GaN and InN are affected by free electrons, especially at very low temperatures. 

A thermodynamic quantity not very often measured for organic superconductors is the specific heat, C. Usually the crystal sizes are rather small and consequently a high sensitivity of the apparatus is needed. In most experiments, therefore, an assembly of many pieces of material is necessary to gain better resolution. In addition, the jump of C at Tc is expected to be rather small especially for compounds with higher transition temperatures because of the comparatively large lattice contribution to C owing to the low electron density and the low vibrational frequencies. 

The solution arrived at in our linear elastic model may be contrasted with those determined earlier in the lattice treatment of the same problem. In fig. 5.13 the dispersion relation along an arbitrary direction in g-space is shown for our elastic model of vibrations. Note that as a result of the presumed isotropy of the medium, no g-directions are singled out and the dispersion relation is the same in every direction in g-space. Though our elastic model of the vibrations of solids is of more far reaching significance, at present our main interest in it is as the basis for a deeper analysis of the specific heats of solids. From the standpoint of the contribution of the thermal vibrations to the specific heat, we now need to determine the density of states associated with this dispersion relation. 

The specific heat (C) is the amount of energy required, per unit mass or per mole, to raise the temperature of a substance by one degree. This is the derivative of its internal energy dU/dT, and since magnetic levels make a contribution to this their separations can in principle be measured from C(T) measurements. However, the magnetic contribution to the specific heat must be disentangled from that of lattice vibrational modes. 

The total free energy of a system is the sum of the free energies of its components so that the total specific heat is the sum of these contributions. From the correct choice of the range of temperature where each of these contributions is dominant emerges the possibility of extracting a single contribution from the total specific heat. The main contributions to Cp in a solid at low temperatures are due to lattice vibrations (or... 

As mentioned, many experimental results have shown that the specific heat for composites increases sHghtly with temperature before decomposition. In some previous models, the specific heat was described as a Hnear function. Theoretically, however, the specific heat capacity for materials wiU change as a function of temperature, as on the micro level, heat is the vibration of the atoms in the lattice. Einstein (1906) and Debye (1912) individually developed models for estimating the contribution of atom vibration to the specific heat capacity of a sohd. The dimensionless heat capacity is defined according to Eq. (4.32) and Eq. (4.33) and illustrated in Figure 4.12 [25] ... 

Blackman, M. Contributions to the theory of the specific heat of crystals II. On the vibrational spectrum of cubic lattices and its application to the specific heat of crystals. Proc. Roy. Soc. London 148 A, 384 (1935). 

The first term is the contribution arising from the lattice vibrations, while the second term has been shown to arise from the specific heat of the conduction electrons. The latter is negligible at high temperatures in comparison with the lattice contribution but becomes significant at very low temperatures (see Example 3.4). 

Any description of a real solid in solid-state physics is based on its ground state properties at a temperature T = 0. However, it has also to include aU kinds of different excitations, for example, thermal excitations to account for a proper description at any finite temperature. Among the different excitations, lattice vibrations or phonons play an important role because of their relative high internal energy and their corresponding large contribution to the specific heat and, secondly, because of the fact that at ambient temperatures in most systems a large fraction of phonons is already excited. 

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