Calculation of the thermodiffusion current

We imagine that an electron at position $\vec r$ has been scattered last at a distance given by the mean free path $l$, and we make the simplifying assumption that in this event the electron has assumed a velocity of absolute magnitude given by the mean value $\overline v(T)$ which corresponds to the local temperature $T$. It follows that an electron at $\vec r$ with direction of flight $\hat v$ has a velocity given by $\hat v \cdot \overline v (T (\vec r- l \cdot \hat v))$ (Fig. 9). Averaging over the direction $\hat v$, one obtains for the mean electronic velocity

$$\vec u (\vec r) \equiv \langle \vec v\rangle_{\vec r}$$

$$= \int \frac{d \Omega}{4 \pi} \hat v \cdot \overline v \left( T(\vec r - l \hat v)\right) \, .$$ (19)

Expanding

$$T(\vec r - l \hat v) \approx T(\vec r) - l \hat v \cdot \vec\nabla T(\vec r)$$ (20)

and

$$\overline v \bigl( T(\vec r - l \hat v)\bigr) \approx v (T(\vec r)) - l \hat v \cdot \vec\nabla T(\vec r) \frac{d \overline v}{d T}$$ (21)

one arrives at

$$\vec u (\vec r) = - \frac{l}{3} \frac{\partial \overline v}{\partial T} \cdot \vec\nabla T (\vec r) \, .$$ (22)

According to this result the different speeds, corresponding to different kinetic energies, of electrons arrived from different directions lead to a mean velocity in a direction opposite to the temperature gradient. This is the phenomenon of thermodiffusion! With (22) one obtains the corresponding current density as

$$\vec j (\vec r)\vert_{\rm {Thermodiff.}} = n \cdot \vec u (\vec r).$$ (23)

Introducing an off-diagonal transport coefficient $l_{12}$ this result can be written as

$$\vec j (\vec r)\vert_{\rm {Thermodiff.}} = -l_{12} \vec \nabla T(\vec r)/T \, .$$ (24)

Comparison of (24) with (22) and (23) yields for $l_{12}$

$$l_{12} = \frac{l}{3} n T \frac{d \overline v}{dT}\, .$$ (25)

Figure 9: Calculation of the current of thermodiffusion.