The theory of the change in resistance in a magnetic field

Abstract

The resistance of a metal is in general increased by a magnetic field. For sufficiently small magnetic fields this dependence may, of course, be expressed by the equation ∆R/R = BH 2 . The calculation of the coefficient B has been the subject of many previous investigations. Sommerfeld has shown that if the electrons are regarded as moving in a constant potential, then the theoretical value of B is 10,000 times smaller than the observed value, and the temperature dependence is in the wrong direction. Peierls has suggested that the correct magnitude of B may be obtained if cognizance is taken of the fact that the electrons are moving in a periodic potential, as is usual in the modern theory of metals. Moreover, he has shown that the correct temperature dependence will then be obtained. Blochinzev and Nordheim have recently investigated in detail the charge of resistance of divalent metals from this standpoint. The periodicity of the lattice was, however, introduced in quite an idealized manner. In place of an actual metal they considered a simple cubic lattice. The surface of the Fermi distribution was assumed to be composed of sections each of which was a quadratic function of the quantum numbers ξ, η , ζ. The coefficient B was obtained in terms of the Fourier coefficient of the potential energy of an electron in the lattice, V 100 . Agreement with the experimental values of B was obtained by taking |V 100 | as small as several hundredths of an electron volt, while the correct value must be of the order of 1 volt. Since in their model B varies inversely as the square of |V 100 |, the comparison with experiment is far from satisfactory.