Electronic heat capacity enhancement

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. 

Figure 1 The coefficient 7 in the electronic heat capacity = jT plotted versus the average number of valence electrons per atom for some refractory compounds, j is obtained through Eq. (3) from the electron density of states N E) in ah initio electron structure calculations for 3d metal compounds (5) and for NbN andTaN (2) i.e., without the enhancement factor (1 + that is present only at low temperatures (approximately for T < 0d/3).

The quantity Xei-pi, enters two important physical relations. It gives the electron-phonon enhancement of the low-temperature electronic heat capacity as Q = jT = ybandT(l + Xei ph), where JiwiT is the heat capacity obtained from the electron band structure without regard to the electron-phonon interaction. The other important relation, of particular interest here, is the connection between Xei.ph and the transition temperature T. An accurate calculation of requires the solution of the Eliashberg equations, which explicitly take into account the shape of a F(co). In many practical applications one instead obtains F from a semiempirical relation containing a F(co) in the form of the average Xei.ph, Eq. (25). The most frequently used such relation is that due to McMillan (31). It reads... 

Figure 4 shows the solutions 0s(F) and 0c(F) to Eqs. (39) and (40) for TiC when the heat capacity and the entropy are taken from recommended experimental data in the JANAF tables (18) of thermodynamic functions. Because the experiments give the sum of vibrational and electronic contributions, an electronic part has been subtracted assuming thaty = 0.5 nJ/(mol K ) at all T and (for simplicity) without correction for the electron-phonon enhancement. In fact, the enhancement goes to zero with increasing temperature and can be ignored when T > 0d/ 3. [A table with data for TiC at a more dense set of temperatures than in the printed JANAF... 

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