Copper Debye temperature

Low-temperature (T < 1K) heat conduction of a pure metal, like copper of our experiment (Cu Debye temperature 0D 340K), is mostly electronic [27] and the phonon contribution should be negligible. With the latter hypothesis, in the 30-150 mK temperature range ... 

The experimental constant-pressure heat capacity of copper is given together with the Einstein and Debye constant volume heat capacities in Figure 8.12 (recall that the difference between the heat capacity at constant pressure and constant volume is small at low temperatures). The Einstein and Debye temperatures that give the best representation of the experimental heat capacity are e = 244 K and D = 315 K and schematic representations of the resulting density of vibrational modes in the Einstein and Debye approximations are given in the insert to Figure 8.12. The Debye model clearly represents the low-temperature behaviour better than the Einstein model. 

Even good heat conductors such as silver or copper come off badly when compared to diamond. The thermal conductivity of the latter passes through a maximum of 175 W cm" K" at 65 K (about 1 /30 of the Debye temperature). In the range of temperatures about 300 K as relevant for practical applications, the value is still 15-30Wcm" K". ... 

Solution. Equation (3.17) may be used to determine the electron heat capacity, while Eq. (3.10) may be used to calculate the lattice specific heat provided T/6 is less than 1/12. Table 3.3 gives a Debye temperature of 310 K for copper, for copper from Table 3.5 is 0.011 J/kgK. The molecular weight of copper is 63.55. 

Solution. Assume that the Debye temperature for copper is applicable to beryllium copper ... 

For copper, the heat capacity at constant volmne O at 20 K is 0.38 J/mohK and the Debye temperature is 340 K. Estimate the specific heat for the following ... 

Figure 10.14 Graph showing the limiting behavior at low temperatures of the heat capacity of (a), krypton, a nonconductor, and (b). copper, a conductor. The straight line in (a) follows the prediction of the Debye low-temperature heat capacity equation. In (b), the heat capacity of the conduction electrons displaces the Debye straight line so that it does not go to zero at 0 K.

Both the Einstein and Debye theories show a clear relationship between apparently unrelated properties heat capacity and elastic properties. The Einstein temperature for copper is 244 K and corresponds to a vibrational frequency of 32 THz. Assuming that the elastic properties are due to the sum of the forces acting between two atoms this frequency can be calculated from the Young s modulus of copper, E = 13 x 1010 N m-2. The force constant K is obtained by dividing E by the number of atoms in a plane per m2 and by the distance between two neighbouring planes of atoms. K thus obtained is 14.4 N m-1 and the Einstein frequency, obtained using the mass of a copper atom into account, 18 THz, is in reasonable agreement with that deduced from the calorimetric Einstein temperature. 

Figure 3.20 Heat capacity of copper (0 = 343 K [22]), magnesia (0 = 946 K [23]), and diamond (0 = 2230 K [23]) as a function of temperature, as predicted by Debye theory ...

The hardness curve and diffraction patterns of Fig. 9-3 illustrate these changes for an alpha brass, a solid solution of zinc in copper, containing 30 percent zinc by weight. The hardness remains practically constant, for an annealing period of one hour, until a temperature of 200°C is exceeded, and then decreases rapidly with increasing temperature, as shown in (a). The diffraction pattern in (b) exhibits the broad diffuse Debye lines produced by the cold-rolled, unannealed alloy. These lines become somewhat narrower for specimens annealed at 100° and 200°C, and the Ka. doublet becomes partially resolved at 250°C. At 250°, therefore, the re-... 

At sufficiently low temperatures, Debye s equation reduces to Eq, (3.10). This fits quite well with the measured specific heat curves of many substances, provided the temperature is below 0 >/12 for copper and aluminum this is about 30 K. On further reduction of the temperature, to the boiling point of helium, departures from the T law become evident and the specific heat is given by an equation of the form... 

Assume that the expansivity data are completely lacking for copper, but that the specific heat is known as a function of temperature either from the Debye function or from experimental observations. Estimate P, and the coefficient of linear thermal expansion for copper at 300 and 50 K and compare these estimated values with experimentally determined ones. Determine the percent difference and tabulate the results. 

Compare the contribution to Cv from electrons and phonons at 10 K for copper. Tlie Fermi temperature is 8.2 10 K. At higher temperatures the following expression is obtained for the heat ce acity within the Debye model... 

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