‘Diffuse scattering" of the Fermi electrons in monovalent metals in relation to their electrical resistivities

The main step in the calculation of the electrical resistivities of monovalent metals, in which the conduction electrons are almost completely degenerate, is the calculation of the relaxation time $\tau $ of the electrons at the Fermi surface, which in these metals is a sphere, and is well inside the first Brillouin zone. Since the wave-length $\lambda $, and hence the group velocity v, of the Fermi electrons is known, the calculation of $\tau $ means also the calculation of the mean free path l = v$\tau $ of these electrons. Now the finite mean free path of these electrons arises from the scattering-particularly the large-angle scattering-of these electrons in their passage through the crystal, by the thermally agitated atoms. Hence a detailed knowledge of the scattering coefficient of the crystal for the Fermi electrons, incident and scattered along different directions in the crystal, will enable us to calculate $\tau $ or l. Now the scattering coefficient depends on two factors. 1. The atom form factor for scattering, which in monovalent metals may be taken to be isotropic, i.e. independent of the direction of incidence or of scattering separately, but dependent on the angle of scattering $\phi $ between them, and on $\lambda $. (Extensive measurements are available on the scattering of slow electrons by the rare gases, which give us information regarding the atom form factors for the scattering of the Fermi electrons in the corresponding alkali metals, and the variation of these factors with $\phi $.) 2. The structure factor of the crystal, which, besides being a function of $\lambda $, will vary, even in a cubic crystal, with the direction of incidence and of scattering, but will, however, be independent of the nature of the waves, i.e. independent of whether they are X-rays, or electron or neutron waves. (The 'diffuse scattering' of X-rays of long wave-lengths in crystals has been studied in great detail, both theoretically and experimentally, from which one can calculate the structure factors of the monovalent metals for their respective Fermi wave-lengths, for different directions of incidence and of scattering in the crystals.) Using these data for the atom form factor and for the structure factor of the crystal, the mean free path of the Fermi electrons is calculated in detail in the present paper for different directions of incidence, for one typical monovalent metal, namely sodium crystal. The free path l is given by 1/l = $\Psi \nu ^{2}$kT$\beta \sigma $, where $\nu $ is the number of atoms per unit volume, $\sigma $ is the cross-section of the atom for total scattering in all directions, $\beta $ is the compressibility, and $\Psi $ is a numerical factor which varies from a maximum of about 2$\cdot $2 for incidence along [110] to a minimum of about 0$\cdot $9 for incidence along [100], its average value being close to the minimum, and nearly unity. With $\Psi $ actually unity, the right-hand side of the above expression for 1/l can be seen to be just the Einstein-Smoluchowski expression for the attenuation coefficient of a liquid medium for long waves: which shows that in sodium, and presumably in the other monovalent metals also, the mean free path of the Fermi electrons may be taken roughly as the reciprocal of the attenuation coefficient of the crystal due to scattering, and the scattering may be regarded as due almost wholly to the local thermal fluctuations in density, and the Fermi wave-length as long enough for the Einstein-Smoluchowski formula for density-scattering to be applicable.