Specific heat electronic term

However, most of the examples quoted in these earlier papers do not include the higher melting-point elements such as W, where a detailed treatment shows that the total entropy (at least of the solid phases) must include many other components such as the electronic specific heat, anharmonicity terms and the temperature dependence of 9d (Grimwall et al. 1987, Moroni et al. 1996). An estimate for the Debye temperature of the high-temperature 0 phase was included in the seminal... 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

Free electrons contribute to specific heat with a term which, at least at low temperatures, is ... 

The orbitals of the d states in clusters of the 3d, 4d, and 5d transition elements (or in the bulk metals) are fairly localized on the atoms as compared with the sp valence states of comparable energy. Consequently, the d states are not much perturbed by the cluster potential, and the d orbitals of one atom do not strongly overlap with the d orbitals of other atoms. Intraatomic d-d correlations tend to give a fixed integral number of d electrons in each atomic d-shell. However, the small interatomic d-d overlap terms and s-d hybridization induce intraatomic charge fluctuations in each d shell. In fact, a d orbital contribution to the conductivity of the metals and to the low temperature electronic specific heat is obtained only by starting with an extended description of the d electrons.7... 

It may of course be unnecessary to consider all these terms and the equation is much simplified in the absence of magnetism and multiple electronic states. In the case of Ti, it is possible to deduce values of the Debye temperature and the electronic specific heat for each structure the pressure term is also available and lambda transitions do not seem to be present. Kaufman and Bernstein (1970) therefore used Eq. (6.2), which yields the results shown in Fig. 6.1(c). 

The right-hand side can be separated into five parts. The first part is the enthalpy at 0 K, the second represents the zero point energy, the third is the Debye energy term, the fourth is an approximation for the Cp — C correction while the last part arises from the difference in electronic specific heats. 

The excess term should allow the total Gibbs energy to be fitted to match that of Eq. (6.3) while at the same time incorporating a return to the inclusion of f 6) and f i) in the lattice stabilities. With the increased potential for calculating metastable Debye temperatures and electronic specific heats from first principles (Haglund et al. 1993), a further step forward would be to also replace Eq. (6.5) by some function of Eq. (6.8). 

It should be pointed out here that the measured specific heat [25, 54,55] of AU55 showed no trace of a linear term which would normally indicate the presence of an electronic contribution to the specific heat, at least down to 60 mK. We will return to this point in Sect. 4.5. 

If metallic conductivity within the cluster should be present, with mobile, delocalized electrons, one might hope to see a linear electronic term in the specific heat [54, 56]. Considering the results above, such a linear term should not be expected for AU55. 

The coordination numbers based on this structure work extremely well for describing the microscopic physical properties of this material, including the Mossbauer I.S.s of the surface sites and of the specific heat of the clusters below about 65 K. No linear electronic term in the specific heat is seen down to 60 mK, due to the still significant T contribution from the center-of-mass motion still present at this temperature. The Schottky tail which develops below 300 mK in magnetic fields above 0.4 T has been quantitatively explained by nuclear quadrupole contributions. 

We now give a simple application of the present method to Plutonium which is a good test case. Pu lies between light actinides with itinerant 5/ electrons and heavy actinides with localized 5/ electrons. The competition between these two electronic regimes in Pu is responsible for a lot of unusual properties as large values of the linear term in the specific heat coefficient and of the electrical resistivity or a very complex phase diagram. 

The transport of heat in metallic materials depends on both electronic transport and lattice vibrations, phonon transport. A decrease in thermal conductivity at the transition temperature is identified with the reduced number of charge carriers as the superconducting electrons do not carry thermal energy. The specific heat and thermal conductivity data are important to determine the contribution of charge carriers to the superconductivity. The interpretation of the linear dependence of the specific heat data on temperature in terms of defects of the material suggests care in interpreting the thermal conductivity results to be described. 

Fig, 15. Effect of nickel content on the rate of hydrogenation of styrene by alloy catalysis. Curve A . Hydrogen uptake. Curve B Number of holes per atom in the Sd-band. Curve C Coefficient of the electronic specific heat term [D. A. Dowden and P. W, Reynolds, Disc. Faraday Soc. 8, 187 (1950)]. 

A recent specific-heat measurement is shown in Fig. 2.31 for -(ET)2l3 [198]. At Tc = 3.4 K (3.5 K with magnetization measurements at the same crystal) a clear anomaly in C/T vs T can be seen. The height of the jump at Tc is AC sa 103 mJ/molK. In a small field applied perpendicular to the ET planes the anomaly of C becomes smaller and much broader. In an overcritical magnetic field of Bx = 0-5 T (not shown here) the normal-state specific heat was measured. Besides the usual linear electronic and cubic Debye specific heat a hyperfine contribution at low temperatures and an appreciable T phononic term had to be taken into account. Therefore, below 5K C was fitted by... 

S2, the electronic term, has been calculated from the electronic specific heat of Pdto be — lOJK mol . Boureau et al. point out that the temperature dependence of the electronic specific heat means that this value overestimates S2 by a factor of 2 or 3. They calculate S2 to be —6.7 JK mol but use an average value of —8.3 JK mol ... 

Data for the temperature dependence of the enthalpy of UO2, obtained using drop calorimetry methods, showed a clear peak in the specific heat at 2610 K [87]. However, the structural origin of this feature was questioned by some groups, who proposed an alternative explanation in terms of electronic disorder (small polarons of... 

Fig. 5.28. Non-electronic low-lying contributions ydis to the 7-linear term of the specific heat (5.13). Au-Sn [5.67] and A Cu-Sn metallic glasses [5.116]...

For the actinides the crystal entropies follow approximately the decreasing average radius produced by f-electron participation in metallic bonding. They are also clearly shown to be non-magne-tic, as the f s are itinerant. However, the entropy correlation itself cannot predict these values, since there is no model in terms of a like metal that can be used to compare these totally unique early actinides. There are also of course perturbations due to the high electronic specific heats, caused by high densities of states at the Fermi level. 

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