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Unformatted text preview: 1 2 Quantum Theory of Conduction Band Theory of Solids Summary 3 4 In an infinitely large, perfect, crystal calculations show that the free electrons suffer no scattering. In real crystals, electrons scatter off lattice imperfections and the thermal vibrations of the lattice ions. The average distance between collisions is called the mean free path , . 5 Mean Free Path Consider a box of length L and crosssectional area A that contains n ions per unit volume. Suppose each ion presents a cross sectional area of size a . What is the probability of a collision between an electron and an ion within the box? A L Pr = a ( n AL ) A = naL 6 Mean Free Path A collision is guaranteed to occur when Pr = 1 , that is, when L = , the mean free path. This yields A L = 1 na = v where <v> is the average speed of the electron and is the average time between collisions. 7 Resistance For a wire of length L and cross sectional area A , the electrical resistance, R , can be written as A L where , the resistivity , is inversely proportional to the mean free path = C / . 8 Classically, lattice ions are modeled as spheres of crosssectional area a = r 2 . In the quantum theory, we model ions as points vibrating in three dimensions with an average cross section of a = r 2 where < r 2 > is the average oscillation amplitude of the ions. 9 Lets model the ions as simple harmonic oscillators with potential energy U = 1 2 Kr 2 = 1 2 M 2 r 2 If we assume the equipartition theorem holds, then the average potential energy of a ion is given by U = 1 2 M 2 r 2 = kT r 2 = 2 kT / M 2 10...
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This note was uploaded on 02/17/2010 for the course PHY PHY3101 taught by Professor Prosper during the Spring '10 term at FSU.
 Spring '10
 PROSPER

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