Electrical Resistivity due to Dislocations

Regarding a dislocation as a row of weak scatterers, each having spherically symmetric potential, a solution of the linearized Boltzmann equation in the presence of a set of parallel dislocations is obtained. The method consists of expanding the distribution function in terms of spherical harmonics and, as developed here, it is exact for the monovalent metals. For polyvalent metals there are additional contributions to the resistivity tensor which arise from the diffraction peaks which occur when the electron wavelength is smaller than twice the interatomic distance - these terms are estimated by the variational method. By allowing for other mechanisms (impurities) of scattering to be present, it is found that the dislocation resistivity ${\rho }_D$ depends on the resistivity due to other mechanisms also. Numerical results for these deviations from Matthiessen's rule and for ${\rho }_D$ for randomly oriented dislocations in aluminum are obtained by taking, after Harrison, the scatterers constituting the dislocations to be vacancies.