Phonon specific heat

With a lattice (phonon) specific heat varying as 7 at low temperatures, and an electron specific heat varying as 7, it is evident that the latter contribution will eventually dominate if the temperature is reduced sufficiently. For a typical metal, it is necessary to go down at least to temperatures near 1 K in order to measure the electronic contribution to the specific heat. 

Finally, we return to the specific heat. The effects of the phonon coupling on the ripplon spectrum can be taken into account in the same fashion as in the conductivity case. Here we replace the discrete summation in Eq. (38) by integration over the broadened resonances, as prescribed by Eq. (57). The bump, as shown in Fig. 15, is also predicted to be nonuniversal depending on Tg/oio-The predicted bump for Tg/(Od = 2 seems to match well the available data for... 

Hence the heat transport, in this case, depends on the dimension and shape of the liquid container. As we can see in Fig. 2.13, the thermal conductivity (and the specific heat) of liquid 4He decreases when pressure increases and scales with the tube diameter. At temperatures below 0.4 K, the data of thermal conductivity (eq. 2.7) follow the temperature dependence of the Debye specific heat. At higher temperatures, the thermal conductivity increases more steeply because of the viscous flow of the phonons and because of the contribution of the rotons. 

The overall specific heat of a polymer is given by a combination of the various contributions to the specific heat of longitudinal and transversal phonons. At temperatures below 1K, the linear contribution due to the TLS must be added. 

In this temperature range, the number of phonons is small, and their scattering is due to lattice defects or to crystal boundaries. Of the two processes of scattering, the latter is of more importance since, at low temperatures, the dominant phonon wavelength is larger than the size of the lattice imperfections. As a consequence Aph is usually temperature independent. Hence, the temperature dependence of the thermal conductivity is that of the specific heat ... 

Also in the CFL phase (T < 7).) there is a contribution to the specific heat of Goldstone-like excitations, cf. phonons in the ordinary condensed matter. For T nip.a, where mp.a is the mass of the pseudo-GoIdstone excitation, we get... 

Note that the full curve drawn in the figure corresponds to the specific heat calculated with no free parameters. While the agreement of this model to the specific heat is rather impressive, it is in fact the 3 inter-cluster degrees of freedom, i.e. motion of the whole cluster, which govern the phonon contribution to the specific heat at temperatures below about 15-20 K [99]. 

Measurement of the specific heat at Tc should yield quite informative data. Information can be gained about the binding energy of the electrons in the Cooper pairs as mediated by electron-phonon coupling (in the BCS theory). 

The original specific heat experiments on BaPb xB Og by Methfessel et al (60) immediately raised the prospect that an unusual mechanism was operative in this newly found system. Their finding of no heat capacity anomaly at Tc could actually have a number of possible interpretations, including an impurity phase giving rise to superconductivity, a non-phonon mechanism, or some new form of conductivity. 

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. 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.

Deducing the electronic entropy change from the specific heat, it appears that about one-third is due to the electrons. The rest must be the result of soft phonon modes. 

Fig. 47. (a) Temperature dependence of the specific heat C as aC/T-vs.-T2 plot for TmNi2B2C. The maximum at 7"c indicates the transition to superconductivity and the low-temperature upturn is related to magnetic ordering. The solid line is calculated taking into account contributions from phonons and crystal field levels (b) specific heat of TmNi2B2C at low temperatures with a maximum at Tn (after Movshovich et al. 1994). 

A number of studies are based on YNi2B2C samples with differing stoichiometry or homogeneity. Lipp et al. (2000) concluded from measurements of the electrical resistivity and the specific heat that both, the electron density of states and the phonon spectrum, change with the boron content. Yang-Bitterlich and Kramer... 

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