Lorentz number

Another success story for the Drude model was the explanation for the Wiedemann19 -Franz20 law of 1858, which stated the empirical observation that the ratio of the thermal conductivity k to the electrical conductivity absolute temperature—that is, that the so-called Lorentz number k/oT was independent of metal and temperature. The thermal conductivity k (> 0) is defined by assuming that the heat flow JH is due to the negative gradient of the absolute temperature T (Fourier s law) ... 

As explained earlier, thermal conductivity and difilisivity (subfigures d, e) are calculated using the WFL law, assuming the validity and temperature independence of the theoretical Lorentz number L = 2.45 x 10" W 2 K and the isobaric heat capacity obtained. Literature data can be found in [123 (T), 68 (A)]. 

Lorentz-number anisotropy in heavy-fermion systems... 

H magnetic field Eeff effective Lorentz number... 

The temperature dependence, W =f T ), is a straight line with a slope of a. The intercept on the VFe-axis is p. The parameter a must be a constant for a certain class of materials and independent of sample purity (Smirnov and Tamarchenko 1977). The constant P, on the other hand, changes from sample to sample and depends indirectly (through po) on its purity. Figure 3 shows schematically the temperature dependence of the different contributions to eq. (8). Since Wi increases and Wq decreases with T, has a minimum at a certain temperature. The Lorentz number of metals of any purity for T> 0, and in samples with impurities and defects for 0, is a constant and equals Lq. At T = 0 for pure metals L = 0 (see fig. 4 and eq. (10)). In the intermediate temperature range, L is determined by the temperature and the purity of the sample. At temperatures both lower as well as higher than 0 the ratio L/Lq in metals could be presented in the form (Oskotski and Smirnov 1972)... 

A more complicated behaviour of the Lorentz number is observed in semiconductors. L/Lq behaves differently for elastic and inelastic scattering of electrons and in the presence of complicated energy band and interband scattering. 

The Lorentz number of a semiconductor with a parabolic band, due to elastic scattering of electrons can be written as (Oskotski and Smirnov 1972, Smirnov and Tamarchenko 1977)... 

Fig. 6. The dependence of the Lorentz number on y for the case of high degeneration of current carriers at S = 32 (1), 8 (2). (a), (b), and (c) depict different positions of the Fermi level, Sp, relative to the bottom of the band of heavy carriers (sq), y = (sp — fioV o T...

Fig. 25. The temperature dependence for ErNi (a) of the total thermal conductivity (Mori el al. 1984), (b) of the effective Lorentz number calculated using eq. (33) and (c) of k calculated using eq. (34).

Anisotropy of the Lorentz number is strongly revealed at low temperatures. 

Figure 97 shows data on L f/Lo=/(T) in PrAlj. is the so-called Lorentz number of the crystal field effect. It is determined by the equation... 

In the high-temperature region one can expect peculiarities of c. due to variation of the Lorentz number near an electron zone boundary of a material (see section 2) and to other scattering mechanisms of electrons. Peculiarities can arise in k, , due to bipolar, photon and exciton contributions and possible heat transport by induced vacancy diffusion. Known experimental data on k of rare earth compounds confirm the regularities observed in other types of materials. However, many effects are original, have their own distinctive features. We will separate these results into two groups ... 

Best estimates, from the experimental literature, of the thermodynamic parameters, (specific heat at constant pressure), a (thermal diffusivity), k (thermal conductivity), and L (Lorentz number). Data from the references cited were used in making these estimates. All values are for the liquids taken at the melting temperature. ( ) Estimated from values for the solid at the melting temperature. 

See Shimoji 1977.) Here T is the absolute temperature, and Lq is called the Lorentz number. Equation (36) should be valid for a wide variety of conditions, but does require that the electron-ion scattering be elastic. 

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