Specific heat from lattice vibrations

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. 

Fig. 15 Low-temperature molar specific heat of tetragonal filled triangles), orthorhombic (dots), and depolymerized C60 (open symbols), plotted as Cp/T3. Reprinted with permission from A Inaba, T Matsuo, A Fransson, and B Sundqvist, Lattice vibrations and thermodynamic stability of polymerized C60 deduced from heat capacities , J. Chem. Phys. vol. 110 (1999) 12226-32 [105]. Copyright 1999 American Institute of Physics...

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

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... 

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). 

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