2_1_lecture_4

2_1_lecture_4 - 1 EE2 Fall 2007 Class 4 slides October 9,...

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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 Fermi-Dirac 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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2_1_lecture_4 - 1 EE2 Fall 2007 Class 4 slides October 9,...

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