Band theory Fermi-Dirac statistics

Band theory is a one-electron, independent particle theory, which assumes that the electrons are distributed amongst a set of available stationary states following the Fermi-Dirac statistics. The states are given by solutions of the Schrodinger equation... 

The next advance came from the application of Fermi-Dirac statistics to the electrons in metals, which led to the band theory of a quasi-continu-ous series of energy levels, and to the concept of Brillouin zones, which is of special value for alloys. This sets the stage for a detailed study of the electronic factor in catalysis on metals. 

We now need to know how the probability of finding an electron of specified energy varies across the permitted band. In the first and simplest version of the Electron Band Theory, electrons were assumed to move in a field of uniform positive potential (i.e. ion cores were neglected), and mutual electrostatic repulsion was ignored. Application of the Schrodinger equation and Fermi-Dirac statistics leads to the conclusion that a collection of N electrons at the absolute zero occupies the N/2 lowest levels, those at the maximum being said to be at the Fermi surface Ef. 

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