Phonons and the Specific Heat

Here m, and mj are the effective masses and is the band gap. Then the interaction is seen to contain a factor There arc similar terms from other 

In conclusion, the origin and nature of the long-range interactions in semiconductors seem to be fairly well understood, though accurate inclusion of its effects has only been accomplished for simple situations. It is interesting that the same method, Eq. (9-15), can be used to calculate the interaction between atoms in metals, where there is no gap we shall in fact do just that when we discuss metals. Where there is no gap, the evaluation of the asymptotic form is different, and the evaluation gives an oscillatory form cos(2/ ,. r)/(/c,./ ) rather than the exponential decay. 

Distribution of vibralioiial frequencies in GaA.s. The frequencies of the two main peaks arc identified with characteristic frequencies of the spectrum. The figure suggests the Einstein approximation of replacing the distribution by two sharp peaks. [After Dolling and Waugh, 1965, p. 19.] 

The distribulion of fi cquencics for one branch of the vibration spectrum, described in the Debye approximation by treating the crystal as an elastic continuum. The Debye frequency is 

Indeed, the Debye approximation is more appropriate in any problem where the modes of lowest frequency are important, as they are in thermal properties at low temperatures. In cases were all modes are important, such as in the evaluation of the total zero-point energy, the simpler Einstein model may be preferable. Notice that even within the Debye approximation the frequencies are concentrated near the highest frequency, called the Debye frequency. This is illustrated in Fig. 9-7. 

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Hence the heat transport, in this case, depends on the dimension and shape of the liquid container. As we can see in Fig. 2.13, the thermal conductivity (and the specific heat) of liquid 4He decreases when pressure increases and scales with the tube diameter. At temperatures below 0.4 K, the data of thermal conductivity (eq. 2.7) follow the temperature dependence of the Debye specific heat. At higher temperatures, the thermal conductivity increases more steeply because of the viscous flow of the phonons and because of the contribution of the rotons. 

A number of studies are based on YNi2B2C samples with differing stoichiometry or homogeneity. Lipp et al. (2000) concluded from measurements of the electrical resistivity and the specific heat that both, the electron density of states and the phonon spectrum, change with the boron content. Yang-Bitterlich and Kramer... 

Here U and Cy are the lattice internal energy and the specific heat respectively, D(cd) is the phonon density of states (DOS) and... 

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...

We will now discuss the density of states and the specific heat of 3 AgI as derived from observed and calculated phonon dispersions [3.15]. The hexagonal unit cell and the crystal structure of 3-AgI are shown in Fig.3.7a Fig.3.7b illustrates the corresponding Brillouin zone. The observed and calculated phonon dispersions are depicted in Fig.3.12. 3-AgI contains 4 ions in the unit cell which gives rise to 12 branches. A valence-shell model (Chap.4) has been used to calculate 03.(q) and the parameters of this model... 

The overall specific heat of a polymer is given by a combination of the various contributions to the specific heat of longitudinal and transversal phonons. At temperatures below 1K, the linear contribution due to the TLS must be added. 

In this temperature range, the number of phonons is small, and their scattering is due to lattice defects or to crystal boundaries. Of the two processes of scattering, the latter is of more importance since, at low temperatures, the dominant phonon wavelength is larger than the size of the lattice imperfections. As a consequence Aph is usually temperature independent. Hence, the temperature dependence of the thermal conductivity is that of the specific heat ... 

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. 

Fig. 5.11 Electronic part of the specific heat of Si P as a function of temperature for three different doping levels n/nc. The dashed lines represent the subtracted phonon contribution AT (0D=640K) and the solid lines are the expected specific heat y0T for degenerate electrons with effective mass m = 034m.

Fig. 47. (a) Temperature dependence of the specific heat C as aC/T-vs.-T2 plot for TmNi2B2C. The maximum at 7"c indicates the transition to superconductivity and the low-temperature upturn is related to magnetic ordering. The solid line is calculated taking into account contributions from phonons and crystal field levels (b) specific heat of TmNi2B2C at low temperatures with a maximum at Tn (after Movshovich et al. 1994). 

Schottky anomaly is determined from the difference between an RY compound and LaX or LuX compound. Then the crystal field parameters are deduced from the Schottky anomaly data. The accuracy of the method is limited by spin-phonon interactions and exchange effects in rare earth ions which affect the Schottky effect, ft is used to find crystal field parameters, W, x which fit the specific heat data as shown in Fig. 8.4. The figure refers to a plot of C/Rq vs. T for TmAF [19]. 

Once we have obtained the dispersion curves for the metal, wc may proceed to other properties just as we did with the covalent solids. In particular, we may quantize the vibrations as was done for the covalent solids and obtain the appropriate contribution to the specific heat. We shall not repeat that analysis now for the simple metals but shall wish to use the customary terminology by referring to the vibrations as phonons. 

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