Thermopower und thermodiffusion

We first note that the difference (10) of the
*electrochemical* potential as measured in the voltmeter can
be expressed as a path integral of its gradient along the three
portions of metal between the exits and (see Fig. 1):

Since, according to (7), the electrochemical potential is continuous at the junctions of the two metals, there is no contribution to (11) from the junctions. On the other hand, the thermoelectric potential is given by the exact expression

In case the Seebeck coefficients of both metals are independent of temperature eqn. (12) reduces to the eqn. (1). Comparing the path integrals (11) and (12) we find the equation

which is the local form of the law for the thermoelectric potential. It is now shown that the relation (13) results from the thermodiffusion of the conduction electrons. This holds both for metals and semiconductors.