Specific heat, Debye

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. 

FIGURE 1 Molar specific heat at constant pressure Cp of AIN, GaN and InN. Full circles, open circles and crosses are AIN [1-3], GaN [10,11] and InN [20] experimental data, respectively. Solid, dashed and dash-dotted lines Debye specific heat for 0D = 962 K, 0D = 700 K and 0D = 660 K. These data are discussed below. 

The Debye temperature can also be obtained from the elastic constants. The measurement of the elastic constants of polycrystalline AIN was used by Slack et al [8] to derive the Debye temperature, giving 0D = 950 K. Therefore, Slack et al have criticised the value of the AIN Debye temperature 0D = 800 + 2 K, derived from the heat capacity measurements by Koshchenko et al [6], as too low. Also, Slack s value differs considerably from Meng s result [7]. Since the cubic dependence T3 approximates the Debye specific heat well in the temperature range below T = 0d/5O [9], it is likely that the upper temperature limit used by Meng is too high and led to error and the difference from the results of Slack et al [8],... 

We used these data for temperatures above 100 K to fit the Debye specific heat. The resulting Debye temperature was 0D = 700 K. The specific heat obtained for this value of Debye temperature was plotted in FIGURE 1. 

A recent specific-heat measurement is shown in Fig. 2.31 for -(ET)2l3 [198]. At Tc = 3.4 K (3.5 K with magnetization measurements at the same crystal) a clear anomaly in C/T vs T can be seen. The height of the jump at Tc is AC sa 103 mJ/molK. In a small field applied perpendicular to the ET planes the anomaly of C becomes smaller and much broader. In an overcritical magnetic field of Bx = 0-5 T (not shown here) the normal-state specific heat was measured. Besides the usual linear electronic and cubic Debye specific heat a hyperfine contribution at low temperatures and an appreciable T phononic term had to be taken into account. Therefore, below 5K C was fitted by... 

R. Stratton. A Surface Contribution to the Debye Specific Heat. Philos. Mag. 44 519 (1953). 

The collisional frequency can also be dependent on temperature. Electron-defect and electron-boundary scattering are typically independent of temperature. Debye temperature is a characteristic temperature arising in the computation of Debye specific heat. It is defined as... 

In deahng with the effect of temperature, the Debye specific heat for constant volume may not be accurate. In fact, most of the measurements were conducted under constant pressure. The specific heat for constant pressure should be preferable. However, for solid state, the ratio of (Cp — Cv)/Cp is 3 % or less [4]. Therefore, assuming the specific heat to follow the Debye approximation is reasonably acceptable. 

An excellent way of deducing the specific heat of a crystal is to model it as an empty box containing a gas of phonons. The phonons are bosons so Bose-Einstein statistics must be used. It should be noted, however, that the total number of phonons is not conserved (so that a Bose-Einstein condensation is not to be expected), quite unlike a gas of, for example, helium atoms. This approach yields the famous Debye specific heat curve shown in Fig. 5. It is plotted, not as a function of T directly, but as a function of T/ D where = is the Debye characteristic... 

Unfortunately for ZnO, the samples contain large densities of free carriers and defects, which make the Debye specific heat expression unreliable. The Debye contribution to the specific heat of pure ZnO is also shown in Figure 1.30 as the curve labeled with a calorimetric Debye temperature of 0d = 399.5 K, and the deviation of the data below 5 K is due to the Schottky term and that above 10 K is due to the Einstein term. The latter has an exponential dependence and is given by... 

Fig.2.12. A comparison of the Einstein and Debye specific heat for longitudinal acoustic and optic branches of the linear NaCl crystal. 0 is either the Einstein or the Debye temperature depending on which curve is being examined. Both curves are normalized to approach the classical value 2Nkg. For T 0, the specific heat Cr(T) drops exponentially reflecting the difficulty in thermally exciting optical modes at low temperatures. Cq(T) is linear at low temperatures this is due to the thermal excitation of low frequency acoustic modes...

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