Consider again the unidimensional gas discussed in our derivation of thermal conductivity in Section 5.5. We want to derive a formula for the con-vective transport of heat. We will disregard heat conduction; its effect can simply be superposed on the results obtained here.
We assume that there is a net flux, nv, of molecules and that the temperature is not uniform along the gas column. Take three neighboring points: 1, 2, and 3. Each molecule that moves from 1 to 2 carries an energy cTi. Here, c is the mean heat capacity of the molecule (i.e., c is 1/N of the heat capacity of N molecules). For each molecule that arrives at 2 coming from 1, another leaves 2 toward 3 carrying cT2 units of energy. Thus, the increase in energy at 2 owing to the flow of gas must be c(T\ — T2)nv joules per second per unit area.
If 1 and 2 are separated by an infinitesimal distance, then Ty — T2 = —dT and the energy is transported at a rate dP* = —cnvdT W/m2, (5.115)
where P* is the power density. If instead of a gas column, we have a free-electron conductor, then heat is convected by electrons and since J = qnv, dP* = — ~cdT W/m2 (5.116)
The above expression can be rewritten as or dP = rldT,
where t is the Thomson coefficient and has the dimensions of V/K. Clearly, t = - q. (5.119)
For conductors in which the carrier distribution is Maxwellian, c = | k, and the Thomson coefficient is
Many semiconductors do have a Thomson coefficient of about — 100 yU,V/K. However, a more accurate prediction of the coefficient requires the inclusion of holes in the analysis.
Electrons in metals do not obey Maxwellian statistics. As discussed in Chapter 2, they follow the Fermi-Dirac statistics: only a few electrons at the high energy end of the distribution can absorb heat. Those that do absorb energy of the order of kT units, but they represent only a fraction of about kT/Wp of the total population. Hence, the mean heat capacity of the electrons is roughly d f kT \ 2k2T
The ratio of the Fermi-Dirac heat capacity to the Maxwellian is about kT/WP.
At room temperature, kT is some 25 meV, whereas a representative value for Wp is 2.5 eV. Therefore, the quantum heat capacity is approximately 100 times smaller than the classical one. For this reason, the Thomson coefficient of metals is small compared with that of semiconductors.
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