Conservation of Momentum in Electrical Conductivity

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Abstract
In the theory of the electrical conductivity of metals, the conservation theorem of Peierls establishes the invariance of the sum of the electron wave numbers and a function of the lattice vibrations. This is shown to be accompanied by other related conservation theorems based on the periodicity of the system. One of these refers to the difference between Peierls' integral and the total momentum of the system. The connection between Peierls' integral and the integral of momentum has usually been obscured by the use of electron states that do not represent a definite value of the momentum. When the system is treated as a whole it can be shown that the transfer of momentum from the electrons to the lattice need not be a process involving one electron at a time, but may involve a large number. One can say that the umklapprozesse involve a whole group of electrons rather than a single one. As a result one can understand the establishment of a steady state in a conducting metal without considering the umklapprozesse of single electrons as introduced by Peierls.

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