Editing Drude model (section)

=== Thermal conductivity of metals ===
One great success of the Drude model is the explanation of the [[Wiedemann-Franz law]].  This was due to a fortuitous cancellation of errors in Drude's original calculation.  Drude predicted the value of the Lorenz number: 
<math display="block"> \frac {\kappa}{\sigma T} = \frac{3}{2}\left(\frac{k_{\rm B}}{e}\right)^2 = 1.11 \times 10^{-8} \, \mathrm{W{\cdot}\Omega/K^2}</math>
Experimental values are typically in the range of <math>2-3 \times 10^{-8} ~ \mathrm{W{\cdot}\Omega/K^2}</math> for metals at temperatures between 0 and 100 degrees Celsius.<ref group="Ashcroft & Mermin" name=":19">{{harvnb|Ashcroft|Mermin|1976|pp=18 table 1.6}}</ref>
{{math proof|title=Derivation and Drude's errors<ref group="Ashcroft & Mermin" name=":17">{{harvnb|Ashcroft|Mermin|1976|pp=17}}</ref>|proof=

Solids can conduct heat through the motion of electrons, atoms, and ions. Conductors have a large density of free electrons whereas insulators do not; ions may be present in either. Given the good electrical and thermal conductivity in metals and the poor electrical and thermal conductivity in insulators, a natural starting point to estimate the thermal conductivity is to calculate the contribution of the conduction electrons.

The thermal current density is the flux per unit time of thermal energy across a unit area perpendicular to the flow.  It is proportional to the temperature gradient.
<math display="block">\mathbf{j}_q = - \kappa \nabla T </math>
where <math>\kappa</math> is the thermal conductivity.
In a one-dimensional wire, the energy of electrons depends on the local temperature <math>\varepsilon[T(x)]</math>
If we imagine a temperature gradient in which the temperature decreases in the positive x-direction, the average electron velocity is zero (but not the average speed). The electrons arriving at location {{math|''x''}} from the higher-energy side will arrive with energies <math>\varepsilon[T(x-v\tau)]</math>, while those from the lower-energy side will arrive with energies <math>\varepsilon[T(x+v\tau)]</math>.  Here, <math>v</math> is the average speed of electrons and <math>\tau</math> is the average time since the last collision.

The net flux of thermal energy at location {{math|''x''}} is the difference between what passes from left to right and from right to left:
<math display="block">\mathbf{j}_q  = \frac{1}{2} n v \big( \varepsilon[T(x-v\tau)] - \varepsilon[T(x+v\tau)] \big)</math>
The factor of {{sfrac|2}} accounts for the fact that electrons are equally likely to be moving in either direction.  Only half contribute to the flux at {{math|''x''}}.

When the mean free path <math>\ell = v \tau</math> is small, the quantity
<math> \big( \varepsilon[T(x-v\tau)] - \varepsilon[T(x+v\tau)] \big) / 2 v \tau</math>
can be approximated by a derivative with respect to {{math|''x''}}.  This gives
<math display="block">\mathbf{j}_q  = n v^2 \tau \frac {d \varepsilon}{dT} \cdot \left(-\frac{dT}{dx} \right)</math>
Since the electron moves in the <math>x</math>, <math>y</math>, and <math>z</math> directions, the mean square velocity in the <math>x</math> direction is <math>\langle v_x^2 \rangle = \tfrac{1}{3} \langle v^2 \rangle</math>.  We also have <math>n \frac {d\varepsilon}{dT}=\frac{N}{V}\frac {d\varepsilon}{dT} = \frac{1}{V} \frac {dE}{dT} = c_v</math>, where <math>c_v</math> is the specific heat capacity of the material.

Putting all of this together, the thermal energy current density is
<math display="block">\mathbf{j}_q  = -\frac{1}{3} v^2 \tau c_v \nabla T</math>
This determines the thermal conductivity:
<math display="block">\kappa = \frac{1}{3} v^2 \tau c_v</math>
(This derivation ignores the temperature-dependence, and hence the position-dependence, of the speed {{math|''v''}}.  This will not introduce a significant error unless the temperature changes rapidly over a distance comparable to the mean free path.)

Dividing the thermal conductivity <math>\kappa</math> by the electrical conductivity <math>\sigma = \frac{n e^2 \tau} {m}</math> eliminates the scattering time <math>\tau</math> and gives
<math display="block">\frac{\kappa}{\sigma} = \frac{c_v m v^2}{3n e^2}</math>

At this point of the calculation, Drude made two assumptions now known to be errors.  First, he used the classical result for the specific heat capacity of the conduction electrons: <math> c_v= \tfrac{3}{2}n k_{\rm B}</math>.  This overestimates the electronic contribution to the specific heat capacity by a factor of roughly 100.  Second, Drude used the classical mean square velocity for electrons, <math>\tfrac{1}{2}mv^2=\tfrac{3}{2}k_{\rm B} T</math>.  This underestimates the energy of the electrons by a factor of roughly 100.  The cancellation of these two errors results in a good approximation to the conductivity of metals.  In addition to these two estimates, Drude also made a statistical error and overestimated the mean time between collisions by a factor of 2.  This confluence of errors gave a value for the Lorenz number that was remarkably close to experimental values.

The correct value of the Lorenz number as estimated from the Drude model is<ref group="Ashcroft & Mermin" name=":20">{{harvnb|Ashcroft|Mermin|1976|pp=25 prob 1}}</ref>
<math display="block">\frac {\kappa}{\sigma T} = \frac{3}{2}\left(\frac{k_{\rm B}}{e}\right)^2 = 1.11 \times 10^{-8} \, \text{W}\Omega/\text{K}^2.</math>
}}