Wiedemann-Franz ratio

The fact that the ratio of the thermal conductivity to the electrical conductivity of any metal is a constant times the absolute temperature was observed by Wiedemann and Franz and this relationship is known as the Wiedemann-Franz ratio. This relationship works because the collision time t for the electron carriers is the same in both models and cancels out when taking the ratio of the two conductivities. From Chapter 17, the classical electronic thermal conductivity was found in Equation 17.33 to be K = 4nl(fT/m n)T. The classical electrical conductivity from the Drude model is given by Equation 18.15 and the Wiede-mann-Eranz ratio becomes 

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. 

Also note that the Wiedemann-Franz law assumes that all of the heat is carried by the electrons and therefore it only applies to metals that are good thermal conductors (or where the heat conduction from the electrons is much greater than the conduction by phonons). 

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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... 

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 number of energy states in a minuscule interval dE is termed the electron level density or density of states n(E) and this is proportional to E (Figure 1.7). This highly simplitied theory worked quite well for metals having only s and p electrons (sodium, magnesium, aluminium, tin), and provided the first reasonable interpretation of their electronic specific heats it also led to a precise expression for the Wiedemann-Franz ratio. ... 

Cook, J.G., 1979, The Wiedemann-Franz Ratio of Liquid Metals, in Proc. 16th Int. Conf. on Thermal Conductivity, ed. D.C. Larsen (Plenum, New York) pp. 305-316. 

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. 

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. 

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. 

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. 

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. 

Wiedemann-Franz law The ratio of the thermal conductivity of any pure metal to its electrical conductivity is approximately constant at a ven temperature. The law is fairly well obeyed, except at low temperatures. The law is named after Gustav Wiedemarm and Rudolph Franz, who discovered it empirically in 1853. 

In solid materials that are not significantly electrically conductive, molecular vibrations known as phonons are the means of heat conduction. In metals that have free electrons available to conduct electric current, these same electrons provide another means of heat conduction. The electrical conductivity and electronic component of thermal conductivity are related by the Wiedemann-Franz-Lorenz ratio L, as shown in Eq. (1.10) ... 

The ratio of the thermal conductivity to the electrical conductivity times the absolute temperature is known as the Wiedemann and Franz ratio and involves only universal physical constants. Therefore, this ratio should be the same for any metal (provided that the heat is predominately carried by the electrons). This relationship depends on the fact... 

Wiedemann-Franz law—for metals, the ratio of thermal conductivity and the product of the electrical conductivity and temperature should be a constant... 

The fact that the thermal conductivity in a pure metal is dominated by the free electron contribution was Ulustrated in 1853 by Gustav Wiedemann (1826-1899) and Rudolf Franz (1827-1902), who showed that Xei and the electrical conductivity, (Tei, are proportionally related (Wiedemann and Franz, 1853). A few years later Danish physicist Ludvig Lorenz (1829-1891) realized that this ratio scaled hnearly with the... 

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