Change of Resistance in a Magnetic Field

Abstract

The general theory of the Hall effect and the change of resistance in a magnetic field expresses these quantities in terms of a number of integrals over the surface of the Fermi distribution. The values of these integrals depend upon the form of the electron energy and the relaxation time as functions of the wave vector. If the free electron situation is assumed, the Hall effect has the right order of magnitude, but there is no change of resistance. This can be seen from a qualitative consideration of the effect of the fields on the distribution function. A general form for the functions in other cases can be obtained as an expansion in spherical harmonics with the symmetry of the crystal lattice. The results can then be expressed in terms of the coefficients in this expansion. When only the first two harmonics are retained, the computed change of resistance and Hall effect are close to the observed values. However, contrary to the available observations, the ratio of the transverse to the longitudinal change of resistance shows a minimum value of about four. It seems improbable that this result could be changed in any material way by the inclusion of higher series members, so that if the experimental results are to be taken as reliable, doubt is thrown on the general method of treatment.