Thermal phonon conductivity

Thermal conductivity. Phonon conductivity is most efficient in simple crystal structures formed by small atoms. 

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

The uTadiation-induced thermal conductivity degradation of graphites and CFCs will cause serious problems in fusion system PFCs. As with ceramics, the thermal conductivity of graphite is dominated by phonon transport and is therefore greatly... 

Fig. 10. The temperature dependence of thermal conductivity for pyrolytic graphite in three different conditions [66]. The reduction of thermal conductivity with increasing temperature is attributed to increasing Umklapp scattering of phonons.

This idea that the heat was transfered by a random walk was used early on by Einstein [21] to calculate the thermal conductance of crystals, but, of course, he obtained numbers much lower than those measured in the experiment. As we now know, crystals at low enough T support well-defined quasiparticles—the phonons—which happen to carry heat at these temperatures. Ironically, Einstein never tried his model on the amorphous solids, where it would be applicable in the / fp/X I regime. 

The thermal conductivity plateau has traditionally been considered by most workers as a separate issue from the TLS. In addition to the rapidly growing magnitude of phonon scattering at the plateau, an excess of density of states is observed in the form of the so-called bump in the heat capacity temperature dependence divided by T. The plateau is interesting from several perspectives. For one thing, it is nonuniversal if scaled by the elastic constants (say, co/)... 

Section V, other quantum effects are indeed present in the theory and we will discuss how these contribute both to the deviation of the conductivity from the law and to the way the heat capacity differs from the strict linear dependence, both contributions being in the direction observed in experiment. Finally, when there is significant time dependence of cy, the kinematics of the thermal conductivity experiments are more complex and in need of attention. When the time-dependent effects are included, both phonons and two-level systems should ideally be treated by coupled kinetic equations. Such kinetic analysis, in the context of the time-dependent heat capacity, has been conducted before by other workers [102]. 

According to the quantum transition state theory [108], and ignoring damping, at a temperature T h(S) /Inks — a/ i )To/2n, the wall motion will typically be classically activated. This temperature lies within the plateau in thermal conductivity [19]. This estimate will be lowered if damping, which becomes considerable also at these temperatures, is included in the treatment. Indeed, as shown later in this section, interaction with phonons results in the usual phenomena of frequency shift and level broadening in an internal resonance. Also, activated motion necessarily implies that the system is multilevel. While a complete characterization of all the states does not seem realistic at present, we can extract at least the spectrum of their important subset, namely, those that correspond to the vibrational excitations of the mosaic, whose spectraFspatial density will turn out to be sufficiently high to account for the existence of the boson peak. 

Coupling the motion of the mosaic cell (TLS and boson peak) to phonons is necesssary to explain thermal conductivity therefore the interaction effects discussed later follow from our identification of the origin of amorphous state excitations. The emission of a phonon followed by its absorption by another cell will give an effective interaction, in the same way that photon exchange leads to... 

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. 

These carriers of heat do not move balistically from the hotter part of the material to the colder one. They are scattered by other electrons, phonons, defects of the lattice and impurities. The result is a diffusive process which, in the simplest form, can be described as a gas diffusing through the material. Hence, the thermal conductivity k can be written as ... 

The main scattering processes limiting the thermal conductivity are phonon-phonon (which is absent in the harmonic approximation), phonon defect, electron-phonon, electron impurity or point defects and more rare electron-electron. For both heat carriers, the thermal resistivity contributions due to the various scattering processes are additive. For... 

In the following, we shall describe separately the temperature dependence of the contributions to the thermal conductivity for the two heat carriers . In the case of phonons, the Debye temperature 0D will be taken as a reference in analysing the temperature dependence of the thermal conductivity. 

In this temperature range, the phonon-phonon scattering dominates and Aph decreases with increasing temperature because the number of phonon increases [1,6-9,31,86] (see Figs 3.15 and 3.16). Hence, in this temperature range, the thermal conductivity decreases with increasing temperature. The thermal conductivity of several materials at T> 2K is shown in Fig. 3.16. 

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 ... 

The result of the considerations made in (a) and (b) is that the phonon thermal conductivity goes through a maximum as illustrated in Figs 3.15 and 3.16. It is to be noted that,... 

At high temperatures (T > 20K), the electron-phonon scattering is dominant and k decreases with T. Hence, we find a maximum of thermal conductivity (see Figs 3.16, 3.19 and 3.20) which is around 10 K for pure metals and 40 K for alloys. For example, in the case of A11050 A1 alloy, where the thermal conduction is mainly due to electrons ... 

It can be observed that these thermal conductances G(7) are typical of phonon conduction between two solids at very low temperature, as already reported [45], The value of the heat capacity was calculated from equation C = r G, where the thermal time constant r is obtained from the fit to the exponential relaxation of the wafer temperature. 

Low-temperature thermometers are obtained by cutting a metallized wafer of NTD Ge into chips of small size (typically few mm3) and bonding the electrical contacts onto the metallized sides of the chip. These chips are seldom used as calibrated thermometers for temperatures below 30 mK, but find precious application as sensors for low-temperature bolometers [42,56], When the chips are used as thermometers, i.e. in quasi-steady applications, their heat capacity does not represent a problem. In dynamic applications and at very low temperatures T < 30 mK, the chip heat capacity, together with the carrier-to-phonon thermal conductance gc d, (Section 15.2.1.3), determines the rise time of the pulses of the bolometer. 

The HEM is a thermal model which represents a doped semiconductor thermistor (e.g. Ge NTD) as made up of two subsystems carriers (electrons or holes) and phonons. Each subsystem has its own heat capacity and is thermally linked to the other one through a thermal conductance which takes into account for the electron-phonon decoupling (see Fig. 15.2). 

The phonon system is linked to the heat sink by a thermal conductance G s. 

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