Energy bands in the presence of an external force field—II Anomalous velocities

The theory of field-modified energy bands is extended to include the effect of weak scattering forces on the energy band structure. The modified current operator is found to contain terms giving anomalous currents of a type previously treated by and (Phys. Rev. 95, 1154 (1954) in connection with electrical conduction in ferromagnets. The physical meaning of these currents is discussed, and they are shown to be analogous to spin-dependent currents in Dirac's theory of the electron. They may be regarded formally as resulting from a non-commutability of the components of the coördinates. It is shown that such currents, proportional to the acceleration, are caused by every accelerating mechanism, including scattering mechanisms. A classical transport theory including the anomalous currents is derived, valid for scattering mechanisms for which the momentum transfer per collision is small, and a very simple problem carried through by way of example. A formal theory including the anomalous transport currents is given for the general case of arbitrary scattering mechanism and overlapping bands. Finally, a critique is given of some recent theories of the spontaneous Hall effect in ferromagnetics.