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- Given the approximations for the included above
* we assumed no electromagnetic field: this is always smaller by a factor v/c given the additional Lorentz term in the equation of motion
* we assumed spatially uniform field: this is true if the field does not oscillate considerably across a few mean free paths of electrons. This is typically not the case: the mean free path is of the order of Angstroms corresponding to wavelengths typical of X rays.
The following are Maxwell's equations without sources , in Gaussian units:
Then
or
which is an electromagnetic wave equation for a continuous homogeneous medium with dielectric constant in the Helmholtz form
where the refractive index is and the phase velocity is
therefore the complex dielectric constant is
which in the case can be approximated to:
In SI units the in the numerator is replaced by in the denominator and the dielectric constant is written as . (en)
- Given
And the equation of motion above
substituting
Given
defining the complex conductivity from:
We have: (en)
- From the simple one dimensional model
Expanding to 3 degrees of freedom
The mean velocity due to the Electric field
To have a total current null we have
And as usual in the Drude case
where the typical thermopowers at room temperature are 100 times smaller of the order of microvolts. (en)
- 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.
where is the thermal conductivity.
In a one-dimensional wire, the energy of electrons depends on the local temperature
If we imagine a temperature gradient in which the temperature decreases in the positive x-direction, the average electron velocity is zero . The electrons arriving at location from the higher-energy side will arrive with energies , while those from the lower-energy side will arrive with energies . Here, is the average speed of electrons and is the average time since the last collision.
The net flux of thermal energy at location is the difference between what passes from left to right and from right to left:
The factor of accounts for the fact that electrons are equally likely to be moving in either direction. Only half contribute to the flux at .
When the mean free path is small, the quantity
can be approximated by a derivative with respect to . This gives
Since the electron moves in the , , and directions, the mean square velocity in the direction is . We also have , where is the specific heat capacity of the material.
Putting all of this together, the thermal energy current density is
This determines the thermal conductivity:
Dividing the thermal conductivity by the electrical conductivity eliminates the scattering time and gives
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: . 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, . 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 (en)
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