Lorenz number

Figure 7. The Wiedemann-Franz ratio for solutions of lithium in ammonia at —33° C. The Lorenz number is 2.45 X 10 8 watt it/deg.2...

The Lorenz number as derived from thermal conductivity and electrical resistivity has small values just above Tc indicating different scattering mechanisms being important in the heat and charge transport for YNi2B2C, LuNi2B2C, and HoNi2B2C (Sera et al., 1996 Boaknin et al., 2000 Schneider, 2005). The shape of a typical minimum in the temperature dependence of the Lorenz number at about 40 K seems to be connected with the residual resistivity of the crystals (Boaknin et al., 2000). 

The thermal conductivities of U-ZrHi6o and U-ZrHi 90 by electronic conduction (/Le), plotted as (A, ) in Figs. 5 (a) and (b), were estimated from the relations of Zc=Tco7 , according to the Wiedemann-Franz rule. <7 is the electrical conductivity (a Mp), where p is the electrical resistivity, Le is the Lorenz number for the electronic conduction, assumed as fJ(p(n2/ i)(kH/e)2 A 2.45x 10 s [WO/K2], where kB and e are the Boltzmann constant and elementary electric charge. 

Using the relaxation time approximation of Boltzmann s equation, the expression for the properties (electrical conductivity, Seebeck coefficient, and Lorenz number) of the holes and electrons are given by... 

Estimate for thermal conductivity based on the Wiedemann-Franz relation with the Lorenz number, L Estimated thermal diftusivity based on the thermal conductivity or the electrical resistivity... 

There is a close relationship between electrical and thermal conductivity. From the simple jBree-electron model for metals, the ratio of the thermal conductivity and the electrical conductivity (reciprocal of resistivity) for metals is directly proportional to the temperature. This is called the Wiedemann-Franz-Lorenz (WFL) relation and the constant of proportionality yields the theoretical (Sommerfeld) Lorenz number, L = 7 l3- kjef = 2.45 x lO" W ft K [67], which was predicted to be independent of temperature (for temperatures significantly larger than the Debye temperature) and of the material. Assuming a known uid/or constant value of Z, the WFL relation can be used to obtain the thermal conductivity from pulse-heating data. 

Fig. 27. Resistivity p(T) (a), thermoelectric power S(T) and Lorenz number L T) (b), calculated with LNCA techniques for a sixfold degenerate Anderson lattice model in the Kondo regime (Cox and Grewe 1988). Impurity results, scaled with concentration are shown for comparison. The resistivity results exhibit the logarithmic increase with decreasing temperature and the coherence-derived decrease below T = to the residual value due to impurities, which is quadratic in the Fermi liquid regime T < T. S(T ) is positive definite for the simple model situation chosen.

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... 

For polycrystalline samples values of the total thermal conductivity X at 293 K, of the electronic component X(el) derived by the Friedemann-Franz law under the assumption that the Lorenz number L = 2.45x 10" V/K as for a degenerate electron gas, and of the lattice component X(lat) obtained by subtraction are, Zhuze et al. [15] ... 

In polycrystalline NdSe samples obtained from the melt, the heat capacity Cp has a maximum at 10.6 K associated with the magnetic ordering at the Neel temperature Tn values for Cp were not given. Fig. 48 shows the thermal resistance W = X between 2 and 100 K for these NdSe samples. The inset in the figure gives the positive deviation near T, as the magnetic contribution. Within the whole temperature range studied, the Lorenz number L in the... 

The constant of proportionality L = k/aT is called the Lorenz number. Putting in the appropriate values for the Boltzmann constant and the electronic charge, we get for the Lorenz number... 

However, when the corrected electronic conductivity K = (t /3) nl T/m)T (Equation 17.36) is used to compute the Lorenz number. 

Note that the temperature dependence in the Lorenz number has nothing to do with the temperature dependence of resistivity (Matthiessen s rule) that affects the collision time r because this effect is eliminated when taking the ratio of the conductivities. Instead this T originates with the first power dependence of the electronic heat capacity with temperature. 

Figure 4 The Lorenz number L T) as a function of T for Fe, after Ref. 21 (thick line), and with a tentative extrapolation to L = Lq at low T (thin line), where impurity scattering dominates.

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