Lattice vibrations heat capacity

Lattice Vibrations Heat Capacity and Related Properties... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

Einstein9 was the first to propose a theory for describing the heat capacity curve. He assumed that the atoms in the crystal were three-dimensional harmonic oscillators. That is, the motion of the atom at the lattice site could be resolved into harmonic oscillations, with the atom vibrating with a frequency in each of the three perpendicular directions. If this is so, then the energy in each direction is given by the harmonic oscillator term in Table 10.4... 

The electronic contribution is generally only a relatively small part of the total heat capacity in solids. In a few compounds like PrfOHE with excited electronic states just a few wavenumbers above the ground state, the Schottky anomaly occurs at such a low temperature that other contributions to the total heat capacity are still small, and hence, the Schottky anomaly shows up. Even in compounds like Eu(OH)i where the excited electronic states are only several hundred wavenumbers above the ground state, the Schottky maximum occurs at temperatures where the total heat capacity curve is dominated by the vibrational modes of the solid, and a peak is not apparent in the measured heat capacity. In compounds where the electronic and lattice heat capacity contributions can be separated, calorimetric measurements of the heat capacity can provide a useful check on the accuracy of spectroscopic measurements of electronic energy levels. 

ALL CHANGES IN PHASE involve a release or absorption of calories. One reason for this is that each solid has its own heat capacity. That is, there is a characteristic heat content for each material which depends upon the atoms composing the solid, the nature of the lattice vibrations within it, and its structure. The total heat content, or enthalpy, of each solid is defined by ... 

When the temperature is such that hv kT, neither of the limiting cases described earlier can be used. For many solids, the frequency of lattice vibration is on the order of 1013 Hz, so that the temperature at which the value of the heat capacity deviates substantially from 3R is above 300 to 400 K. For a series of vibrational energy levels that are multiples of some fundamental frequency, the energies are 0, hv, 2hv, 3hi/, etc. For these levels, the populations of the states (n0, nu n2, etc.) will be in the ratio 1 e hl T e Jh,/Ikr e etc. The total number of vibrational states possible for N atoms is 3N... 

This model, the Einstein model for heat capacity, predicts that the heat capacity is reduced on cooling and that the heat capacity becomes zero at 0 K. At high temperatures the constant-volume heat capacity approaches the classical value 3R. The Einstein model represented a substantial improvement compared with the classical models. The experimental heat capacity of copper at constant pressure is compared in Figure 8.3 to Cy m calculated using the Einstein model with 0g = 244 K. The insert to the figure shows the Einstein frequency of Cu. All 3L vibrational modes have the same frequency, v = 32 THz. However, whereas Cy m is observed experimentally to vary proportionally with T3 at low temperatures, the Einstein heat capacity decreases more rapidly it is proportional to exp(0E IT) at low temperatures. In order to reproduce the observed low temperature behaviour qualitatively, one more essential factor must be taken into account the lattice vibrations of each individual atom are not independent of each other - collective lattice vibrations must be considered. 

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. 

The Kieffer approach uses a harmonic description of the lattice dynamics in which the phonon frequencies are independent of temperature and pressure. A further improvement of the accuracy of the model is achieved by taking the effect of temperature and pressure on the vibrational frequencies explicitly into account. This gives better agreement with experimental heat capacity data that usually are collected at constant pressure [9],... 

In contrast to crystalline solids characterized by translational symmetry, the vibrational properties of liquid or amorphous materials are not easily described. There is no firm theoretical interpretation of the heat capacity of liquids and glasses since these non-crystalline states lack a periodic lattice. While this lack of long-range order distinguishes liquids from solids, short-range order, on the other hand, distinguishes a liquid from a gas. Overall, the vibrational density of state of a liquid or a glass is more diffuse, but is still expected to show the main characteristics of the vibrational density of states of a crystalline compound. 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

Because the heat capacities of crystalline solids at various T are related to the vibrational modes of the constituent atoms (cf section 3.1), they may be expected to show a functional relationship with the coordination states of the various atoms in the crystal lattice. It was this kind of reasoning that led Robinson and... 

Kieflfer S. W. (1979a). Thermodynamics and lattice vibrations of minerals, 1 Mineral heat capacities and their relationships to simple lattice vibrational models. Rev. Geophys. Space Phys., 17 1-19. 

Kieflfer S. W. (1985). Heat capacity and entropy Systematic relations to lattice vibrations. In Reviews in Mineralogy, vol. 14, P. H. Ribbe (series ed.), Mineralogical Society of America. 

Aminomethylpyridine (picolylamine) is an important ligand in respect to spin cross-over, [Fe(2-pic)3]Cl2 being the key compound." Fleat capacity measurements on [Fe(2-pic)3]Cl2 EtOH gave values of 6.14kJmol and 50.59 JK moC for the spin eross-over entropy the determined entropy was analyzed into a spin contribution of 13.38, an ethanol orientational effeet of 8.97, and a vibrational contribution of 28.24 JK mol. " This compound exhibits weak cooperativity in the solid state." The heat capacity of [Fe(2-pic)3]Cl2 MeOH is consistent with very weak cooperativity." [Fe(2-pic)3]Br2 EtOH shows a lattice expansion significantly different from that expected in comparison with earlier-established behavior of [Fe(2-pic)3]Cl2 EtOH." ... 

The state of polarization, and hence the electrical properties, responds to changes in temperature in several ways. Within the Bom-Oppenheimer approximation, the motion of electrons and atoms can be decoupled, and the atomic motions in the crystalline solid treated as thermally activated vibrations. These atomic vibrations give rise to the thermal expansion of the lattice itself, which can be measured independendy. The electronic motions are assumed to be rapidly equilibrated in the state defined by the temperature and electric field. At lower temperatures, the quantization of vibrational states can be significant, as manifested in such properties as thermal expansion and heat capacity. In polymer crystals quantum mechanical effects can be important even at room temperature. For example, the magnitude of the negative axial thermal expansion coefficient in polyethylene is a direct result of the quantum mechanical nature of the heat capacity at room temperature." At still higher temperatures, near a phase transition, e.g., the assumption of stricdy vibrational dynamics of atoms is no... 

Similar methods have been used to integrate thermodynamic properties of harmonic lattice vibrations over the spectral density of lattice vibration frequencies.21,34 Very accurate error bounds are obtained for properties like the heat capacity,34 using just the moments of the lattice vibrational frequency spectrum.35 These moments are known35 in terms of the force constants and masses and lattice type, so that one need not actually solve the lattice equations of motion to obtain thermodynamic properties of the lattice. In this way, one can avoid the usual stochastic method36 in lattice dynamics, which solves a random sample of the (factored) secular determinants for the lattice vibration frequencies. Figure 3 gives a typical set of error bounds to the heat capacity of a lattice, derived from moments of the spectrum of lattice vibrations.34 Useful error bounds are obtained... 

Fig. 3. Error bounds for the heat capacity of the harmonic vibrations of a body-centered cubic lattice with first- and second-nearest neighbor force constants.

Isochoric heat capacity of crystals can be presented as a sum of contributions from lattice vibrations (6 degrees of freedom) and intramolecular vibrations (3n-6 degrees of freedom, where n is a number of atoms in a molecule)... 

We must also consider the conditions that are implied in the extrapolation from the lowest experimental temperature to 0 K. The Debye theory of the heat capacity of solids is concerned only with the linear vibrations of molecules about the crystal lattice sites. The integration from the lowest experimental temperature to 0 K then determines the decrease in the value of the entropy function resulting from the decrease in the distribution of the molecules among the quantum states associated solely with these vibrations. Therefore, if all of the molecules are not in the same quantum state at the lowest experimental temperature, excluding the lattice vibrations, the state of the system, figuratively obtained on extrapolating to 0 K, will not be one for which the value of the entropy function is zero. 

The heat capacity of the trivalent lanthanide trihalides consists of a lattice component, arising mainly from the vibrations of the ions in the crystal, and an excess component (Westrum Jr. and Grpnvold, 1962 Westrum Jr., 1970 Flotow and Tetenbaum, 1981 Westrum Jr., 1983) ... 

In analogy with the approach that has been described in the section on the low-temperature heat capacity, the high-temperature heat capacity of the LnXj compounds can be described as the sum of the lattice and excess contributions (eq. (1)). However, whereas at low temperature the lattice heat capacity mainly arises from harmonic vibrations, at high temperatures the effects of anharmonicity of the vibrations, of thermal dilation of the lattice and of thermally... 

Cp.iatvib is the contribution from lattice vibrations, CPtintravaj the contribution from intramolecular vibrations, and CPimag is the magnetic or electronic heat capacity arising from thermal excitation of electrons. 

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...

Lattice vibrations are fundamental for the understanding of several phenomena in solids, such as heat capacity, heat conduction, thermal expansion, and the Debye-Waller factor. To mathematically deal with lattice vibrations, the following procedure will be undertaken [7] the solid will be considered as a crystal lattice of atoms, behaving as a system of coupled harmonic oscillators. Thereafter, the normal oscillations of this system can be found, where the normal modes behave as uncoupled harmonic oscillators, and the number of normal vibration modes will be equal to the degrees of freedom of the crystal, that is, 3nM, where n is the number of atoms in the unit cell and M is the number of units cell in the crystal [8],... 

The polymorphs of a substance can possess considerable different chemical and physical properties. Their melting points and heat capacities will be different. Their x-ray diffraction patterns will depend on the arrangement of molecules in the crystal lattice. In addition, vibrational spectra of the different polymorphic forms of a material will be different. These differences may be minor, but many times there are extensive differences, which can be used to identify the form and to understand its crystalline structure. 

At low temperatures, almost all lattice vibrations cease to contribute, leaving the thermal excitations of the electrons dominant [5]. The electronic term contributing to the constant volume heat capacity Cv is proportional to temperature T and the vibrational term is proportional to T3. Consequently, C is expressed as [5],... 

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