Lattice heat capacity

The measurement of heat capacity of solids has been of great theoretical as well as practical interest because the departure of the observed heat capacity from the predictions based on classical concepts was one of the early hints that something was quite wrong with the classical models and that new models based on quantum concepts were necessary to understand what was going on. 

Since co2 =K/m, the mean potential and kinetic energy terms are equal and the total energy of the linear oscillator is twice its mean kinetic energy. Since there are three oscillators per atom, for a monoatomic crystal U m =3RT and Cy m =3R = 2494 J K-1 mol-1. This first useful model for the heat capacity of crystals (solids), proposed by Dulong and Petit in 1819, states that the molar heat capacity has a universal value for all chemical elements independent of the atomic mass and crystal structure and furthermore independent of temperature. Dulong-Petit s law works well at high temperatures, but fails at lower temperatures where the heat capacity decreases and approaches zero at 0 K. More thorough models are thus needed for the lattice heat capacity of crystals. 

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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. 

Figure 8.13 Lattice heat capacity of three different polymorphs of carbon C q [5], graphite and diamond.

When one has a model or related data to estimate the normal or lattice heat capacity in the transition region, the enthalpy of transition for a continuous transition can be obtained as an integration of the excess heat capacity, defined as... 

Figure 13.2 Heat capacities of (a), Hg near the melting temperature of 234.314 K showing the abrupt nature of the change in heat capacity for this first-order phase transition at this temperature [from R. H. Busey and W. F. Giauque, J. Am. Chem. Soc., 75, 61-64 (1953)] and (b), MnO showing the continuous magnetic transition (note inset). (Data obtained from Professor Brian Woodfield and co-workers at Brigham Young University.) The dashed line is an estimate of the lattice heat capacity of MnO.

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... 

As shown in Table I, lanthanum and lutetium oxides have Sq ground states and consequently their heat capacities should be attributed to lattice vibration. Data on these substances may be used to represent the lattice contribution to a first approximation for neighboring isostructural (and nearly so) sesquioxides. Cubic gadolinium oxide provides a midseries lattice heat capacity approximation at relatively high temperatures... 

Although the lattice heat capacity in a metal is much larger than its electronic contribution, the Fermi velocity of electrons (typically 106 m/s) is much larger than the speed of sound (about 103 m/s). Due to the higher energy carrier speed, the electronic contribution to the thermal conductivity turns out to be more dominant than the lattice contribution. For a semiconductor, however, the velocity is not the Fermi velocity but equal to the thermal velocity of the electrons or holes in the conduction or valence bands, respectively. This can be approximated as v /3kBT/m, where m is the effective electron mass in the conduction band or hole mass in the valence band. This is on the order of 105 m/s at room temperature. In addition, the number density of conduction band electrons in a semiconductor is much less than... 

We would like to find the contribution of the surface heat capacity to the total lattice heat capacity at a given temperature for a particle of a given size. Because the surface heat capacity is proportional to the surface area and the bulk term is proportional to the volume, the surface/volume ratio will clearly play an important role in determining the magnitude of the contribution of the surface heat capacity to the total heat capacity. The ratio of the bulk- and surface-heat-capacity tenns indicates both the temperature range and the thickness of the specimen for which the surface-heat-capacity contribution will become detectable. The ratio fora cube with sides of length L is approximately given by... 

Ce and Cl are the electronic and lattice heat capacities, g is the electron-phonon coupling constant, and LP(t) is the intensity profile of the exciting laser pulse dependent on the time t. 

For TmV04 and TmAs04 the ground state in each case is a non-Kramers doublet that can be split by the true Jahn-Teller effect. For the vanadate the cooperative transition occurs at To = 2.156 K in this region the lattice heat capacity is negligible, and the co-operative anomaly dominates, as shown by the measurements of Cooke et al. (1972) in fig. 11. The distinctive triangular shape is a typical result for a molecular field model in which the splitting J of the doublet varies with temperature as... 

The lattice heat capacity, Cl, at low temperatures can be expanded in odd powers of T, where the first term is often referred to as the Debye term. Thus... 

Lanthanum exists in modified hexagonal (a) and face-centred cubic ( 8) forms, each of which contributes independently to the electronic and lattice heat capacities. Both phases display superconductivity, the transition temperatures being Ta = 4.9 K and = 6.0 K, respectively (Finnemore et al., 1965 Finnemore and Johnson, 1966). 

Dugdale provides an introduction to entropy which considers both thermodynamics and statistics. Raja GopaP introduces elementary concepts of heat capacities such as lattice heat capacity, and electronic heat capacity. 

ForRXa (R = La-Lu) molecules, regularities in the behavior of various parameters in the equation describing the lattice heat capacity in quasiharmonic approximation are revealed from the available values of low-temperature heat capacity. These parameters have a linear dependence on molar volume for compoimds with identical crystalline structure. As a result, it becomes possible to define these parameters and to subsequently calculate heat capacities in a wide interval of temperatures for imexplored compounds. 

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