Specific heat, electronic contribution

Table III presents integral excess entropies of formation for some solid and liquid solutions obtained by means of equilibrium techniques. Except for the alloys marked by a letter b, the excess entropy can be taken as a measure of the effect of the change of the vibrational spectrum in the formation of the solution. The entropy change associated with the electrons, although a real effect as shown by Rayne s54 measurements of the electronic specific heat of a-brasses, is too small to be of importance in these numbers. Attention is directed to the very appreciable magnitude of the vibrational entropy contribution in many of these alloys, and to the fact that whether the alloy is solid or liquid is not of primary importance. It is difficult to relate even the sign of the excess entropy to the properties of the individual constituents.

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

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... 

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 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. 

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. 

The density of states at the Fermi-level N(pp) is responsible for many electronic properties, e.g. the electronic contribution to the low-temperature specific heat of a solid, and the Pauli paramagnetic moment of conduction electrons. The specific heat contribution may be written as ... 

Table 3. Density of states at the Fenni level for actinide metals from band calculations (model) from the electronic contribution y to the specific heat from magnetic susceptibility measurements. The increasing values indicate a decreasing 5 f bandwidth pinned at Ep for americium metal (not shown) there is a sudden decrease in N(np)...

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. 

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... 

Fig. 5.11 Electronic part of the specific heat of Si P as a function of temperature for three different doping levels n/nc. The dashed lines represent the subtracted phonon contribution AT (0D=640K) and the solid lines are the expected specific heat y0T for degenerate electrons with effective mass m = 034m.

Fig. 57. Specific heat contribution y(H) of the vortex core electrons in the mixed state (normalized by the Sommerfeld parameter /n) of the Yj[Lu jrNi2B2C samples from fig. 56 as function of the applied magnetic field (normalized by //c2(0)). The straight line y(H)

Fig. 61. Magnetic field dependence of the specific heat contribution y(H) of the vortex core electrons in the mixed state for Y(Nio.7sPto.25)2B2C. The dashed line is a fit according to eq. (8) with /) = 0.17, the solid line corresponds to the y(H)

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

The specific heat of a semiconductor has contributions from lattice vibrations, free carriers and point and extended defects. For good quality semi-insulating crystals only the lattice contribution is of major significance. Defect-free crystals of group III nitrides are difficult to obtain, and thus the specific heat measurements are affected by the contributions from the free carriers and the defects. While the specific heat of AIN is affected by the contribution of oxygen impurities, the data for GaN and InN are affected by free electrons, especially at very low temperatures. 

A number of properties of the electrons in a metal can be found from our model. In particular, we can find the specific heat of the electrons and can see in a qualitative way what their contribution to the equation of state should be. For the heat capacity, Eq. (5.9), Chap. V, gives... 

A thermodynamic quantity not very often measured for organic superconductors is the specific heat, C. Usually the crystal sizes are rather small and consequently a high sensitivity of the apparatus is needed. In most experiments, therefore, an assembly of many pieces of material is necessary to gain better resolution. In addition, the jump of C at Tc is expected to be rather small especially for compounds with higher transition temperatures because of the comparatively large lattice contribution to C owing to the low electron density and the low vibrational frequencies. 

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... 

Specific heat. In the Fermi liquid theory, the expression for the electronic contribution to the specific heat is linear in temperature CXT) = yT, where the Sommerfeld constant is... 

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