# Oom Lecture 06

### Thermal Conductivity

- 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:

, so

This estimate is a little low: Al is , Cu is 2, Fe is 10, and liquid Hg is 100.

### Magnetic Diffusivity

- 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.

### Fluid Mechanics

- Navier Stokes

We’ll assume is constant , and that we are moving sub-sonically.