- The thermal flux in metals is carried by mobile free electrons, and this makes them much better heat conductors than the insulators we have discussed previously.
Thus the thermal conductivity is . We can estimate some thermal conductivities. We’ll write as , with the factor of 3 reducing the diffusivity for 3 dimensions (it takes longer to randomly get somewhere than in 1D). Thus, the important factors are:
In comparison to insulators (4) is the same (), (3) is lower by , (2) is a lot bigger than in conductors (). This makes (2) higher by about a factor of 300. Finally, (1), the mean-free-path of an electron in a solid, must be much larger than the mfp of a phonon (as we know by the fact that metals are excellent thermal conductors). If is the spacing between ions, Kittel pg. 302-304 tells us that
where is the rms thermal displacement of an ion (their jiggling around). If all the ions were on a perfect lattice, a plane-wave of electrons would never scatter. An ad hoc rational for the above equation is as follows: each step in changes by . Thus, the number of steps for to change by order unity is . We can estimate this factor as:
Now , and , and , so we end up with
This means that the mean free path of an electron propagating through a lattice is (several hundred lattice spacings). Thus (1) is higher by a factor of 200:
And as it turns out, Cu and Al have , and Fe has . On the other hand, liquid Hg has because liquids have their long-range order broken, scattering electrons before thermal wiggling does.
- Notice also that though , (3) includes a factor of , so thermal conductivity should be temperature-independent, which it is until at low energies there aren’t any phonons to scatter electrons, and then instead scatter off of the same impurities that scatter phonons in insulators.
Electrical Conductivity (Resistivity)
- The current through a wire of cross-section A, length L, and voltage V is
where is the electrical conductivity, . Now is the current density, is the electric field, and .
- Electrons encounter resistance from ions: . The drift velocity is much smaller than the average forward velocity. . Thus:
This estimate is a little low: Al is , Cu is 2, Fe is 10, and liquid Hg is 100.
- Magnetic fields decay because currents decay. This decay is governed by Ohmic diffusion. Ampere’s law says:
and Ohm says:
and this gives us that
Using Faraday’s law and the fact that there are no magnetic monopoles:
and is the magnetic diffusivity.
- Thus, the Ohmic decay time is of order:
where L is the length scale over which B changes.
We’ll assume is constant , and that we are moving sub-sonically.