Electron heat capacity

Electron heat capacity of a free electron metal, Q, is defined as 

is the electron number density. The above expression indicates electron capacity Cg to be a linear function of temperature. 

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Thus, even at temperatures well above absolute zero, the electrons are essentially all in the lowest possible energy states. As a result, the electronic heat capacity at constant volume, which equals d tot/dr, is small at ordinary temperatures and approaches zero at low temperatures. 

In Table 12.1, the contributions to the heat capacity Csp of the addendum are shown specific heat data references are reported in ref. [20], A factor 1/3 was attributed to the heat capacity contribution of the elements linking the crystal to the frame [15], Note that the electron heat capacity of the NTD Ge 31 sensor was derived from the electron... 

Table 8.2. Debye temperature ( p in K) and electronic heat capacity coefficient (see Section 8.4) (yin mJ K-1 mol-1) of the elements.

Using this simple argument, the electronic heat capacity, CE, of a free electron gas is... 

The electronic heat capacity thus varies linearly with temperature and is often represented as... 

The electronic heat capacity for the free electron model is a linear function of temperature only for T Tp = p / kp. Nevertheless, the Fermi temperature Tp is of the order of 105 K and eq. (8.46) holds for most practical purposes. The population of the electronic states at different temperatures as well as the variation of the electronic heat capacity with temperature for a free electron gas is shown in Figure 8.20. Complete excitation is only expected at very high temperatures, T>Tp. Here the limiting value for a gas of structureless mass points 3/2/ is approached. 

Figure 8.22 Variation of the electronic heat capacity coefficient with composition for the alloys Rh-Pdand Pd-Ag [17]. Solid and dotted lines represent the electronic DoS for the 5s and 4d bands, respectively.

The electronic heat capacity naturally has a pronounced effect on the energetics of insulator-metal transitions and the entropy of a first-order transition between an insulating phase with y = 0 and a metallic phase with y= ymet at Ttrs is in the first approximation Ains met5m = 7met7trs. 

Recall from Figure 1.15 that metals have free electrons in what is called the valence band and have empty orbitals forming what is called the conduction band. In Chapter 6, we will see how this electronic structure contributes to the electrical conductivity of a metallic material. It turns out that these same electronic configurations can be responsible for thermal as well as electrical conduction. When electrons act as the thermal energy carriers, they contribute an electronic heat capacity, C e, that is proportional to both the number of valence electrons per unit volume, n, and the absolute temperature, T ... 

It can be shown that the conduction electron net spin susceptibility is proportional to the temperature coefficient of the electronic heat capacity [cf. Eq. (4.42)] and, for free electrons in a single band, having the Fermi energy much lower than any band gap, is given by... 

We can now see why the experimental electronic heat capacity did not obey the classical result of fcB per electron. Following Pauli s exclusion principle, the electrons can be excited into only the unoccupied states above the Fermi energy. Therefore, only those electrons within approximately kBT of F will have enough thermal energy to be excited. Since these constitute about a fraction kBT/EF of the total number of electrons we expect the classical heat capacity of fkBN to be reduced to the approximate value... 

K, so that the electronic heat capacity at room temperature is dramatically reduced compared to the classical prediction for a free-electron gas. 

Fig. 7.7 A comparison of the theoretical and experimental 4d and 5d electronic heat capacities. The theoretical values were obtained directly from eqn (7.28) and Fig. 7.6, neglecting any changes in the density of states due to bandwidth variation within the 4d and 5d series.

Cp.iatvib is the contribution from lattice vibrations, CPtintravaj the contribution from intramolecular vibrations, and CPimag is the magnetic or electronic heat capacity arising from thermal excitation of electrons. 

ELECTRONIC HEAT CAPACITY DRUDE VERSUS FERMI—DIRAC... 

Finally, significant advances in the techniques of both thermal and thermochemical measurements have come to fruition in the last decade, notably aneroid rotating-bomb calorimetry and automatic adiabatic shield control, so that enhanced calorimetric precision is possible, and the tedium is greatly reduced by high speed digital computation. Non-calorimetric experimental approaches as well as theoretical ones, e.g., calculation of electronic heat capacity contributions to di- and trivalent lanthanides by Dennison and Gschneidner (33), are also adding to definitive thermodynamic functions. 

Fig. 3. Turnover frequency vs. the coefficient of the electronic heat capacity (T) Eq.2).

FIGURE 8.2 Thermal conductivity of aluminum and copper as a function of temperature [3]. Note that at low temperature, the thermal conductivity increases linearly with temperature. In this regime, defect scattering dominates and the mean free path is independent of temperature. The thermal conductivity in this regime depends on the purity of the sample. The linear behavior arises from the linear relation between the electronic heat capacity and temperature. As the temperature is increased, phonon scattering starts to dominate and the mean free path reduces with increasing temperature, lb a large extent, the thermal conductivity of a metal is independent of the purity of the sample. 

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