Heat capacity electronic contribution

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

In typical metals, both electrons and phonons contribute to the heat capacity at constant volume. The temperaPire-dependent expression... 

Figure A2.2.6. Electronic contribution to the heat capacity Cy of copper at low temperatures between 1 and 4 K. (From Corak et al [2]).

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elemental solid, which reflected the vibrational energy of a tliree-dimensional solid, should be equal to 3f JK moH The anomaly that the free electron dreory of metals described a metal as having a tliree-dimensional sUmcture of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add anodier (3/2)7 to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas die quanmtii theory of free electrons shows that diese quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

As described above, quantum restrictions limit tire contribution of tire free electrons in metals to the heat capacity to a vety small effect. These same electrons dominate the thermal conduction of metals acting as efficient energy transfer media in metallic materials. The contribution of free electrons to thermal transport is very closely related to their role in the transport of electric current tlrrough a metal, and this major effect is described through the Wiedemann-Franz ratio which, in the Lorenz modification, states that... 

The linear term in Cp m for metals results from the contribution to the heat capacity of the free electrons. It can become important at very low temperatures where the T3 relationship becomes very small. For example, the electronic contribution to the heat capacity of Cu metal is 1.2% at 30 K, but becomes 80% of the total at 2 K.e... 

Using equation (10.77), the electronic contribution to the heat capacity can be obtained by appropriate differentiation of the partition function ... 

The electronic contribution is generally only a relatively small part of the total heat capacity in solids. In a few compounds like PrfOHE with excited electronic states just a few wavenumbers above the ground state, the Schottky anomaly occurs at such a low temperature that other contributions to the total heat capacity are still small, and hence, the Schottky anomaly shows up. Even in compounds like Eu(OH)i where the excited electronic states are only several hundred wavenumbers above the ground state, the Schottky maximum occurs at temperatures where the total heat capacity curve is dominated by the vibrational modes of the solid, and a peak is not apparent in the measured heat capacity. In compounds where the electronic and lattice heat capacity contributions can be separated, calorimetric measurements of the heat capacity can provide a useful check on the accuracy of spectroscopic measurements of electronic energy levels. 

The heat capacity for liquid Pu02 has been estimated (21) as 96 J mol-l K- assuming no electronic contribution. If an electronic contribution is found by experiment to be present, the liquid heat capacity would be increased. 

The free-electron gas was first applied to a metal by A. Sommerfeld (1928) and this application is also known as the Sommerfeld model. Although the model does not give results that are in quantitative agreement with experiments, it does predict the qualitative behavior of the electronic contribution to the heat capacity, electrical and thermal conductivity, and thermionic emission. The reason for the success of this model is that the quantum effects due to the antisymmetric character of the electronic wave function are very large and dominate the effects of the Coulombic interactions. 

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... 

Figure 15.8 shows the thermal scheme of one detector there are six lumped elements with three thermal nodes at Tu T2, r3, i.e. the temperatures of the electrons of Ge sensor, Te02 absorber and PTFE crystal supports respectively. C), C2 and C3 are the heat capacity of absorber, PTFE and NTD Ge sensor respectively. The resistors Rx and R2 take into account the contact resistances at the surfaces of PTFE supports and R3 represents the series contribution of contact and the electron-phonon decoupling resistances in the Ge thermistor (see Section 15.2.1.3). 

Heat capacity contributions of electronic origin Electronic and magnetic heat capacity... 

The heat capacity of a molecule is equal to the sum of all the contributions from translation, rotation, vibration, and electronic degrees of freedom (table 4.19). 

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 ... 

Returning now to thermal conductivity, Eq. (4.40) tells us that any functional dependence of heat capacity on temperature should be implicit in the thermal conductivity, since thermal conductivity is proportional to heat capacity. For example, at low temperatures, we would expect thermal conductivity to follow Eq. (4.43). This is indeed the case, as illustrated in Figure 4.25. In copper, a pure metal, electrons are the primary heat carriers, and we would expect the electronic contribution to heat capacity to dominate the thermal conductivity. This is the case, with the thermal conductivity varying proportionally with temperature, as given by Eq. (4.42). For a semiconductor such as germanium, there are less free electrons to conduct heat, and lattice conduction dominates—hence the dependence on thermal conductivity as suggested by Eq. (4.41). 

A major achievement of the free-electron model was to show why the contributions of the free electrons to the heat capacity and magnetic susceptibility of a metal are so small. According to Boltzmann statistics, the contribution to the former should be nkB per unit volume. According to Fermi-Dirac statistics, on the other hand, only a fraction of order kBT/ F of the electrons acquire any extra energy at temperature T, and these have extra energy of order kBT. Thus the specific heat is of order nfcBT/ F, and an evaluation of the constant gives... 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

The heat capacity at constant volume Cv from the translational and rotational degrees of freedom are determined via Eqs. 8.124 and 8.128, the vibrational contributions to Cv are calculated by Eq. 8.129, and the electronic contribution to Cv is from Eq. 8.123. For an ideal gas, Cp = Cu + R, so Cp=41.418 J/mole/K. The experimental value is Cp=38.693 J/mole/K. Agreement with experiment gets better at higher temperature. At 1000 K, Cp from our calculation is 59.775 J/mole/K, compared to a value of 58.954 from the NIST-JANAF Tables. The difference between theory and experiment is due entirely to our use of the vibrational frequencies obtained from the ab initio results, rather than using the experimental frequencies. 

The thermodynamic properties of a substance in the state of ideal gas are calculated as the sums of contributions from translation and rotation of a molecule as a whole, vibrations and internal rotation in the molecule, and electronic excitation. For example, for entropy and heat capacity the following equations hold ... 

Fig. 10. Electronic contribution to the heat capacity divided by temperature vs. log Q T for a series of La doped alloys of CeR Sb]. The data has been corrected for a phonon contribution, using heat capacity data from LaRr Sb] and a nuclear quadrupo-lar contribution from 121 Sb and 123 Sb (Takeda and Ishikawa, 2001).

The excess contribution is due to the distribution of the valence electrons over the energy levels, and includes the splitting of the ground term by the crystalline electric field (Stark effect) and is called the Schottky heat capacity or Schottky anomaly. It can be calculated from... 

The variation in the heat capacity and entropy of the solid lanthanide trihalides can be described by a lattice contribution that linearly varies with atomic number within each crystallographic class of compounds, and an excess contribution that depends on the electronic configuration (crystal field) of the lanthanide ions. A distinct difference is observed between... 

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