Electrical and Thermal Conductivity. Thermoelectricity

To explain electrical conductivity, we must suppose the electrons in a metal to have a free path In fact, if we were to adhere in the problem of electrical conductivity to the idea of perfectly free electrons (that this is only a first approximation we have already emphasized above), the result would be an infilnitely great conductivity. To explain finite resistance, therefore, we must take into account the fact that the electrons, in the course of their motion through the metal, collide from time to time with the ions of the lattice, and are thus deflected from their path, or are retarded the average distance which an electron traverses between two collisions with the lattice ions is called, by analogy with the similar case in the kinetic theory of gases, the mean free path. 

It has been shown by Sommerfeld (1928) that we can calculate the general behaviour of electrical and thermal conductivities without necessarily making special hypotheses as to the free path. The Wiedemann-Eranz law follows from this theory and we can explain in the same way the Joule heat, the Peltier and Thomson thermoelectric effects, and other phenomena. 

e refinements of the theory, which have been worked out in particular by Houston, Bloch, Peierls, Nordheim, Fowler and Brillouin, have two main objects. In the first place, the picture of perfectly free electrons at a constant potential is certainly far too rough. There will be binding forces between the residual ions and the conduction electrons we must elaborate the theory sufficiently to make it possible to deduce the number of electrons taking part in the process of conduction, and the change in this number with temperature, from the properties of the atoms of the substance. In principle this involves a very complicated problem in quantum mechanics, since an electron is not in this case bound to a definite atom, but to the totality of the atomic residues, which form a regular crystal lattice. The potential of these residues is a space-periodic function (fig. 10), and the problem comes to this— to solve Schrodinger s wave equation for a periodic poten-tial field of this kind. That can be done by various approximate methods. One thing is clear if an electron 

These possible energy levels become filled up by electrons, two of which always fall into each state, on account of the spin. It turns out that such a strip contains exactly 2V electronic states, if V is the 

We come now to the second main problem (cf. p. 224)— to determine by wave mechanics the free paths of the electrons—a problem which cannot be solved by the classical theory it is a question of the scattering of the electronic waves which traverse the lattice of the metal, by the ions situated at the lattice points and of the transference of their energy to the ionic lattice. The calculation gives thoroughly satisfactory results, bringing out correctly, for instance, 

---