Wiedemann-Franz

As described above, quantum restrictions limit tire contribution of tire free electrons in metals to the heat capacity to a vety small effect. These same electrons dominate the thermal conduction of metals acting as efficient energy transfer media in metallic materials. The contribution of free electrons to thermal transport is very closely related to their role in the transport of electric current tlrrough a metal, and this major effect is described through the Wiedemann-Franz ratio which, in the Lorenz modification, states that... 

Estimated from electrical resistivity according to Wiedemann-Franz-Lorenz law b Arithmetic average of properties of alloy elements based on mole fractions c Properties of Ni... 

Classical Free-Electron Theory, Classical free-electron theory assumes the valence electrons to be virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are that (1) an electron can pass from one atom to another, and (2) in the absence of an electric field, electrons move randomly in all directions and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field, electrons drift toward the positive direction of the field, producing an electric current in the metal. The two main successes of classical free-electron theory are that (1) it provides an explanation of the high electronic and thermal conductivities of metals in terms of the ease with which the free electrons could move, and (2) it provides an explanation of the Wiedemann-Franz law, which states that at a given temperature T, the ratio of the electrical (cr) to the thermal (k) conductivities should be the same for all metals, in near agreement with experiment ... 

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

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...

Usually, the electronic thermal conductance re can be calculated from the Wiedemann - Franz law, re TG/e2. However, as shown in Ref. [8, 9] for the ballistic limit f > d, this law gives a wrong result for Andreev wires if one uses an expression for G obtained for a wire surrounded by an insulator. Andreev processes strongly suppress the single electron transport for all quasiparticle trajectories except for those which have momenta almost parallel to the wire thus avoiding Andreev reflection at the walls. The resulting expression for the thermal conductance... 

Adequate predictions of thermal conductivity for pure metals can be made by means of the Wiedemann-Franz law, which states that the ratio of the thermal conductivity to the product of the electrical conductivity and the absolute temperature is a constant. High-purity aluminum and copper exhibit peaks in thermal conductivity between 20 and 50 K, but these peaks are rapidly suppressed with increased impurity levels and cold work of the metal. The aluminum alloys Inconel, Monel, and stainless steel show a steady decrease in thermal conductivity with a decrease in temperature. This behavior makes these structural materials useful in any cryogenic service that requires low thermal conductivity over an extended temperature range. 

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. 

Formerly these metallic properties were attributed to the presence of free electrons. The classical theory of this electron gas (Lorentz) leads, however, to absurdities for instance, a specific heat of 3/2 R had to be expected for this monatomic gas, contrary to the experience that Dulong and Petit s rule (atomic specific heat 6/2 R) holds for both conductors and non-conductors. The calculated ratio of heat conductivity to electrical conductivity (Wiedemann-Franz constant) also did not agree with observation. 

Use the modihed Wiedemann-Franz-Lorenz law to estimate the phonon contribution to the thermal conductivity of a semiconductor at 298 K whose total thermal conductivity is 2.2 W m if the electrical conductivity is 0.4 x 10 S m ... 

As Peierls obtains a law for the electrical resistance in the limiting case, he concludes that the ratio of the electrical and thermal resistances does not decrease proportionally to T, but to 7, or in other words at low temperatures the Wiedemann-Franz-Lorenz quantity pjTw should not be constant, but should decrease proportionally to 7. 

The Wiedemann-Franz-Lorenz Law and the Linear Law connecting Thermal Resistance, Electrical Resistance, and Temperature... 

Experimentally it was found that at constant temperature the increase in the thermal resistance caused by deformation or formation of mixed crystals static disturbances of the lattice) is to a remarkable degree of accuracy proportional to the increase of the electrical resistance divided by the absolute temperature. If we compare the above formula with the Wiedemann-Franz-Lorenz law for pure metals,... 

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