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Unformatted text preview: 1 EE2 Fall 2007 Class 4 slides October 9, 2007 2 Outline Review of momentum space Distribution of states in momentum space FermiDirac statistics Distribution of states in energy Distribution of electrons in energy Determination of Fermi energy Average of functions of dynamic variables Free electron model 3 Momentum space 4 If we consider an interval between p and p + dp and choose the interval dp to be infinitesimally small so that the value of p is the same within the interval and equal to p, we saw that the number of states having p within this interval is equal to Z(p) dp where Zp) is equal to (8 Vp 2 / h 3 ) dp Z(p) is called the density of states in the momentum space. 5 Plot of distribution of momentum states 6 Momentum Space 7 In the sphere of radius p all states lying on the surface of the sphere have the same magnitude of momentum p and hence the same value of kinetic energy p 2 /2m. The sphere represents a constant energy surface in the momentum space. We can therefore express Z(p) dp as equal to Z(E) dE where Z(E) dE is the number of states with energy between E and E+dE. Z(E) is called the density of states in energy and is determined by transforming Z(p) dp which is a function p to a function of energy E by expressing p =(2mE) 1/2 and pdp= mdE Then, The density of momentum states is proportional to E 1/2 . 8 Distribution of States in Energy 9 Fermi Dirac statistics The probability that a quantum state will be occupied by an electron is given by the Fermi function....
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 Spring '08
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