Wiedemann-Franz law

Lorenz number for copper (turquoise)

The Wiedemann-Franz law , also Wiedemann-Franz law , (named after Gustav Heinrich Wiedemann and Rudolph Franz ) is an empirical law that describes the relationship between thermal conductivity  and electrical conductivity in a metal as almost proportional to the temperature T , regardless of the considered metal: ${\displaystyle \lambda }$  ${\displaystyle \sigma }$ 

${\displaystyle {\frac {\lambda }{\sigma }}=L\cdot T}$

The constant of proportionality in the area (or ) is called the Lorenz number . ${\displaystyle L={\frac {\lambda }{\sigma T}}}$ ${\displaystyle 2{,}1\ldots 2{,}9\cdot 10^{-8}\,\mathrm {W\,\Omega \,K^{-2}} }$ ${\displaystyle \mathrm {V^{2}\,K^{-2}} \,}$

In the graphic, the reciprocal value of the electrical conductivity, namely the specific resistance  , and the thermal conductivity of copper are plotted as red and green lines. The product of the two quantities, the blue line, depends on the temperature. After dividing by the temperature you get the Lorenz number, light blue line. At 300 K it is 2.31 · 10 ⁻⁸  V ²  K ⁻² and increases up to 900 K by less than 5 percent to 2.41 · 10 ⁻⁸  V ²  K ⁻² . ${\displaystyle \rho =1/\sigma }$

meaning

The Wiedemann-Franz law attests to the fact that the charge carriers in metals are also carriers of thermal energy . It also applies to very low and very high temperatures (compared to the Debye temperature ). Deviations occur at mean temperatures between approximately 10 K and 200 K due to ballistic heat conduction . In addition, the Wiedemann-Franzsche law does not take into account the contributions of lattice vibrations ( phonons ) to the conduction of heat, since they transport heat but not charge.

history

The two namesake found out in 1853 that the ratio is approximately the same for all metals at the same temperature. Ludvig Lorenz established the linearity of this relationship to temperature in 1872 . ${\displaystyle \lambda /\sigma }$

The first theoretical explanation of the law was made around 1900 by Paul Drude , who calculated the following value for the Lorenz number using the Drude model named after him :

${\displaystyle L={\frac {3}{2}}\left({\frac {k_{\mathrm {B} }}{e}}\right)^{2}=1{,}11\cdot 10^{-8}\,\mathrm {W\,\Omega \,K^{-2}} }$

With

• the Boltzmann constant and ${\displaystyle k_{\mathrm {B} }}$
• the elementary charge . ${\displaystyle e}$

This value deviates by a factor of 2 from the experimentally determined values ​​due to incorrect assumptions in the Drude model, but already represents the relationship qualitatively correctly.

With the Drude-Sommerfeld theory , improved by Arnold Sommerfeld around 1933 , the Lorenz number is finally confirmed quantitatively:

${\displaystyle L={\frac {\lambda }{\sigma T}}={\frac {\pi ^{2}}{3}}\left({\frac {k_{\mathrm {B} }}{e}}\right)^{2}=2{,}44\cdot 10^{-8}\,\mathrm {W\,\Omega \,K^{-2}} }$

literature

• G. Wiedemann, R. Franz: About the heat conductivity of metals . In: Annals of Physics . tape 165 , no. 8 , 1853, p. 497-531 , doi : 10.1002 / andp.18531650802 . ( PDF ).
• L. Lorenz: Determination of the degree of warmth in absolute measure . In: Annals of Physics . tape 223 , no. 11 , 1872, p. 429-452 , doi : 10.1002 / andp.18722231107 . ( PDF ).
• Neil W. Ashcroft , N. David Mermin : Solid State Physics . Saunders College Publishing, New York 1976, ISBN 0-03-083993-9 , pp. 20-23, 52 .
• RW Powell: Correlation of metallic thermal and electrical conductivities for both solid and liquid phases . In: International Journal of Heat and Mass Transfer . tape 8 , no. 7 , 1965, pp. 1033-1045 , doi : 10.1016 / 0017-9310 (65) 90086-4 .