Thermal conductivity lattice part

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 ... 

This being so, we could conclude, from the fact that Z at low temperatures approaches the classical numerical value, that here the effect of the lattice vibrations on the thermal conductivity really recedes into the background, as Peierls theory also requires. On the other hand, the fact that at higher temperatures Z has larger values would have to be interpreted to mean that owing to the effect of the lattice vibrations the thermal resistance is raised by statical disturbances of the lattice to a smaller extent than would occur if the lattice vibrations had no effect. In order to understand the experimental details, it would be necessary—here I agree wholeheartedly with Eucken—to analyse accurately the part played by the lattice vibrations in the total flow of heat. According to Peierls theory, however, this would seem to be far from easy, as the conduction due to the electrons and the conduction due to the lattice cannot be combined by simple addition. 

When a material can crystallize into a different polymorph, the chemical nature of the species remains identical, however, the physical properties of the material can be different. For example, properties such as density, heat capacity, melting point, thermal conductivity, and optical activity can vary from one polymorph to another. Table 2.3 lists common materials that exhibit polymorphism. Looking at Table 2.3 we can see that density varies significantly for the same materials when the crystal structure has changed. In addition, the change in the crystal structure often means a change in the external shape of the crystal, which is often an important parameter in industrial crystallization that has to be controlled. Many substances crystallize into structure in which the solvent is present as part of the crystal lattice. These crystals are known as solvates (or hydrates when the solvent is water). A substance can have multiple solvates with different crystal structures as well as a solvent free crystal form with a unique crystal structure. The solvates are often referred to as pseudopolymorphs. They are not true polymorphs because of the addition of the solvent molecule(s) to the crystal lattice. Conformational polymorphism refers to the situation where the molecular conformation of the molecules of a given substance are different in each polymorph. 

Total thermal conductivity is a sum of the lattice and electronic parts, K = Ki + Ke- The lattice part of the thermal conductivity describes the scattering of phonons on the vibrations of atoms, whereas the electronic part describes thermal conductivity appearing due to conduction electrons and is related to the electrical conductivity Wiedemann-Franz equation, = a T Lo, where T is the absolute temperature and Lq is the ideal Lorenz number, 2.45 X 10 Wf2K [64]. The electronic part of the thermal conductivity is typically low for low-gap semiconductors. For the tin-based cationic clathrates it was calculated to contribute less than 1% to the total thermal conductivity. The lattice part of the thermal conductivity can be estimated based on the Debye equation /Cl = 1 /3(CvAvj), where C is the volumetric heat capacity, X is the mean free path of phonons and is the velocity of sound [64]. The latter is related to the Debye characteristic temperature 6 as Vs = [67t (7V/F)] . Extracting the... 

Debye temperature from either heat capacity or structural data and assuming that the mean free path of phonons is the average distance between the guest atoms, the lattice part of the thermal conductivity was calculated for several tin-based cationic clathrates to be in the range of 0.7-0.9 W m K , which is in good agreement with the experimental data [31, 32, 56, 58]. 

In theory, heat in crystalline solids is transferred by three mechanisms (i) electrons (ii) lattice vibrations and (iii) radiation [44], Since zirconia is an electronic insulator (electrical conductivity occurring at high temperatures by oxygen ion diffusion), electrons play no part in the total thermal conductivity of the system. Hence, thermal conduction in zirconia-based ceramics is mainly by lattice vibrations (phonons) or by radiation (photons). 

Fig. 3 shows the kinetic energy distribution following the heat pulse plotted as functions of time and lattice position. The interaction potential was assumed to be A of Fig. 2a for a-iron. The observable features L, L2, etc., are indicated. These features were superimposed on a surface which described the diffusive part of energy transport. This surface could be fitted to the continuum theory of thermal diffusion with a thermal diffusivity of 4.8x10 m s (thermal conductivity of 11.3 Wm K ) for the lattice of a-iron. The latter value may be compared with the estimated lattice thermal conductivity of 28.2 Wm K for Fe(99.5%)Ni-(0.05%) at 75 K [19]. In view of the fact that the interatomic potential for iron is not accurately known, we consider this result satisfactory. 

Figure 9.4 Thermal conductivity and figure of merit ZT of the series CaMn, ,M ,03 5 (x = 0.00

In metals the major contribution to the thermal conductivity comes from the conduction electrons. But it is an insulator, diamond, that has by far the highest thermal conductivity at room temperature, showing that conduction through phonons is not always small. The total thermal conductivity Km is the sum of an electronic part Kei and a phonon (lattice) part Kia ... 

Figure 5 The lattice contribution K,a, to the thermal conductivity of TiCo.gs and the total thermal conductivity Ktot = Kiat + Kei. The lattice part Kiat is deduced from the measured total thermal conductivity K,o, by subtracting an electronic part Kd that is calculated from the measured electrical resistivity and assuming the Wiedemann-Franz law with L = Lq. After experiments by Morelli (22).

What the addition of electron acceptors and donors means in the band picture can be easily understood from Figs. 6.128 and 6.129. The electron acceptors and donors enter the lattice of the semiconductor and introduce electron-energy levels between the valence and conduction bands. Thus, with an n-type of semiconductor (Fig. 6.128), only a small part of the electrons in the conduction band arise by thermal excitation from the valence band the rest come from the ionization of electron donors. The hole concentration, however, depends only upon the number of valence electrons that are excited into the conduction band. The hole concentration can therefore be made small. 

The structural interpretation of dielectric relaxation is a difficult problem in statistical thermodynamics. It can for many materials be approached by considering dipoles of molecular size whose orientation or magnitude fluctuates spontaneously, in thermal motion. The dielectric constant of the material as a whole is arrived at by way of these fluctuations but the theory is very difficult because of the electrostatic interaction between dipoles. In some ionic crystals the analysis in terms of dipoles is less fruitful than an analysis in terms of thermal vibrations. This also is a theoretically difficult task forming part of lattice dynamics. In still other materials relaxation is due to electrical conduction over paths of limited length. Here dielectric relaxation borders on semiconductor physics. 

In an ideal ionic crystal all ions are rigidly held in the lattice sites where they perform only thermal vibratory motion. Transfer of an ion between sites under the effect of electrostatic fields (migration) or concentration gradients (diffusion) is not possible in such a crystal. Initially, therefore, the phenomenon of ionic conduction in solid ionic crystals was not understood. Yakov I. Frenkel showed in 1926 that ideal crystals could not exist at temperatures above the absolute zero. Part of the ions leave their sites under the effect of thermal vibrations and are accommodated in the interstitial space leaving vacancies at the sites. 

The delocalized electrons can also conduct heat by carrying kinetic energy (in the form of vibrations) from a hot part of the metal lattice to a colder part of the lattice. The presence of delocalized electrons in metals accounts for the thermal and electrical conductivity of metals. 

Of course we cannot consider here, not even in a modest way, the quantum mechanical treatment of the problem of semiconduction. The modern theory of solids makes use of the band theory permitting us to modify the classical concept of bound electrons in valence crystals. Impurities may play a very important part in semiconduction. If one adds foreign atoms (impurities) we introduce new electronic levels in the forbidden regions. This substance will have semiconduction properties if the electron is thermally lifted to the empty band. Lattice defects may induce similar properties. The conductance depends on the number of electrons filling the band. 

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