Wiedemann-Franz Law

Laws Wiedemann-Frantz is a physical law that states that for metals the ratio of the thermal conductivity coefficient (or the thermal conductivity tensor) $K$ ${\ displaystyle K}$ $K$ electrical conductivity (or conductivity tensor) $\sigma$ ${\ displaystyle \ sigma}$ $\ sigma$ in proportion to temperature :

{\frac {K}{\sigma }}=LT.

{\ displaystyle {\ frac {K} {\ sigma}} = LT.}

{\ displaystyle {\ frac {K} {\ sigma}} = LT.}

In 1853, German scientists G. Wiedemann (1826–1899) and R. Franz (1827–1902), on the basis of experimental data, found that for different metals at the same temperature the ratio $K/\sigma$ ${\ displaystyle K / \ sigma}$ ${\ displaystyle K / \ sigma}$ practically unchanged. The proportionality of this relationship to the thermodynamic temperature was established by L. Lorenz in 1882 . In his honor, the coefficient $L$ ${\ displaystyle L}$ $L$ is called the Lorentz number , and the law itself is sometimes called the Wiedemann-Franz-Lorentz law.

The interconnection of electrical conductivity and thermal conductivity is explained by the fact that both of these properties of metals are mainly due to the motion of free electrons .

The coefficient of thermal conductivity increases in proportion to the average velocity of the particles, since energy transfer is accelerated. The electrical conductivity, on the contrary, decreases, because collisions at a high velocity of particles significantly impede the transfer of energy.

Drude , applying the classical kinetic theory of gases , obtained the value of the coefficient $L$ ${\ displaystyle L}$ $L$ :

L=3\left({\frac {k}{e}}\right)^{2}\approx 2{,}22\times 10^{-8}~{\text{W}}\Omega {\text{K}}^{-2},

{\ displaystyle L = 3 \ left ({\ frac {k} {e}} \ right) ^ {2} \ approx 2 {,} 22 \ times 10 ^ {- 8} ~ {\ text {W}} \ Omega {\ text {K}} ^ {- 2},}

{\ displaystyle L = 3 \ left ({\ frac {k} {e}} \ right) ^ {2} \ approx 2 {,} 22 \ times 10 ^ {- 8} ~ {\ text {W}} \ Omega {\ text {K}} ^ {- 2},}

Where $k$ ${\ displaystyle k}$ $k$ - Boltzmann constant , $e$ ${\ displaystyle e}$ $e$ - electron charge.

In his initial calculation, Drude was mistaken by 2 times, while receiving the correct order of magnitude. In fact, the classic statistics gives the result

L={\frac {3}{2}}\left({\frac {k}{e}}\right)^{2}\approx 1{,}11\times 10^{-8}~{\text{W}}\Omega {\text{K}}^{-2}.

{\ displaystyle L = {\ frac {3} {2}} \ left ({\ frac {k} {e}} \ right) ^ {2} \ approx 1 {,} 11 \ times 10 ^ {- 8} ~ {\ text {W}} \ Omega {\ text {K}} ^ {- 2}.}

{\ displaystyle L = {\ frac {3} {2}} \ left ({\ frac {k} {e}} \ right) ^ {2} \ approx 1 {,} 11 \ times 10 ^ {- 8} ~ {\ text {W}} \ Omega {\ text {K}} ^ {- 2}.}

Only with the help of quantum statistics Sommerfeld was obtained the value of the coefficient $L$ ${\ displaystyle L}$ $L$ well consistent with the experiment:

L={\frac {\pi ^{2}}{3}}\left({\frac {k}{e}}\right)^{2}\approx 2{,}47\times 10^{-8}~{\text{W}}\Omega {\text{K}}^{-2}.

{\ displaystyle L = {\ frac {\ pi ^ {2}} {3}} \ left ({\ frac {k} {e}} \ right) ^ {2} \ approx 2 {,} 47 \ times 10 ^ {- 8} ~ {\ text {W}} \ Omega {\ text {K}} ^ {- 2}.}

{\ displaystyle L = {\ frac {\ pi ^ {2}} {3}} \ left ({\ frac {k} {e}} \ right) ^ {2} \ approx 2 {,} 47 \ times 10 ^ {- 8} ~ {\ text {W}} \ Omega {\ text {K}} ^ {- 2}.}

Wiedemann-Franz law was a triumph for the theory of free electrons.

Content

Inaccuracies of classical theory

The classical theory, leading to a practically correct final result, gave it a wrong interpretation. In it the proportionality between $K$ ${\ displaystyle K}$ and $\sigma$ ${\ displaystyle \ sigma}$ was due to the fact that the average kinetic energy of the electron gas is ${\tfrac {3}{2}}kT$ ${\ displaystyle {\ tfrac {3} {2}} kT}$ , that is proportional to the absolute temperature. Actually, the law is explained by the fact that the absolute temperature is proportional not to the average energy, but to the heat capacity of the electron gas. The classical theory made a mistake, overestimating the heat capacity of the electron gas by 100 times, but this error was accidentally compensated by another error. The speed of electrons participating in heat exchange is determined by their kinetic energy on the Fermi surface: ${\sqrt {2E_{F}/m}}$ ${\ displaystyle {\ sqrt {2E_ {F} / m}}}$ , - whereas in the classical theory it was believed that this velocity is of the order of the classical average thermal velocity ${\sqrt {3kT/m}}$ ${\ displaystyle {\ sqrt {3kT / m}}}$ . Thus, the average square of the velocity of electrons participating in heat exchange was underestimated by a factor of 100 (as well as the heat capacity), and the final result was correct.

Area of Applicability

The validity of the Wiedemann-Franz law is not limited to Sommerfeld’s theory of free electrons. In the semiclassical theory of conductivity, it is shown that if we neglect the thermoelectric field, an expression similar to that obtained by Sommerfeld will be valid if we replace the thermal conductivity and conductivity by tensors of the corresponding quantities. However, it should be emphasized that in semiconductors there is no reason to expect such a simple connection.

The experiment shows that, in reality, the Wiedemann-Franz law is well executed at high (above room) and low (several kelvins ) temperatures. In the intermediate area he is unjust.

Its applicability is related to the applicability of the relaxation time approximation. With a strict conclusion of this law, it is implicitly assumed that all collisions are elastic, that is, the energy is conserved in a collision. If inelastic collisions take place, then there will necessarily be scattering processes that can reduce the heat flux without reducing the electric current (the heat flux is determined, in addition to the electron energy, also by chemical potential). If such processes give energy losses of the order of $k T$ ${\ displaystyle kT}$ as it happens at intermediate temperatures, then we should expect a violation of the Wiedemann-Franz law.

Literature

Kalashnikov S. G. Electricity. - 6th ed., Stereo. - M .: FIZMATLIT, 2004. - 624 seconds - ISBN 5-9221-0312-1 .
Sivukhin D.V. General Physics Course. In 5 t. T III. Electricity. - M .: FIZMATLIT; Publishing House of Moscow Institute of Physics and Technology, 2004. - 654 seconds - ISBN 5-9221-0227-3 .
Ashcroft N., Mermin N. Physics of a solid body.

Wiedemann-Franz Law

Content

Inaccuracies of classical theory

Area of Applicability

Literature

See also

More articles:

Wiedemann-Franz Law

Content

Inaccuracies of classical theory

Area of ​​Applicability

Literature

See also

More articles:

Area of Applicability