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lecture10 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Fall 2004 – Lecture 10 1 2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 10 Review on previous lectures In above figure, we can find the volume of one state is 3 1 (2 / ) V L π = . In the sphere, the number of states within k and k+dk is 2 2 2 1 4 (2 ) k k Vk k N V π π = = , in which V=L 3 is the crystal volume. The density of states is defined as the number of quantum states per unit interval of energy and per unit volume 2 2 2 2 1 ( ) 2 2 N k k k dk D E V E E dE π π = = = . A factor that considers polarization of waves may be added (electron, spin up and down, thus a factor of 2, photon, two polarizations, phonons, 3 polarizations) For an energy level E i , the Boltzmann factor for its occupying probability is ( ) exp( / ) i i B P E A E k T = , in which the constant A can be determined by normalization over all quantum states ( ) 1 i All QS P E = . For an open system exchanging energy with the outside, the probability becomes ( , ) exp[ ( ) / ] i i i i B P E N A E N k T µ = , where N i is the particle number, chemical potential µ is the criteria for the equilibrium state of mass exchanging process with the outside, just as pressure for mechanical equilibrium and temperature for thermal equilibrium. k k x k y k z
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2.57 Fall 2004 – Lecture 10 2 For electrons at a quantum state with energy E, the Fermi-Dirac distribution gives the average number of electrons as ( )/ 1 ( ) 1 i B E k T n f E e µ = = + . For phonons, the Pauli exclusion principle is no longer applicable. And the occupancy of the quantum state is changed into / 1 ( ) 1 B h k T n f e ν ν = = , which is called Bose-Einstein distribution. For molecules, similar results exist ( )/ 1 ( ) 1 i B E k T n f E e µ = = , where µ is again the chemical potential of the boson gas. The Bose-Einstein distribution changes the “plus one” in the denominator of the Femi- Dirac distribution into minus one. When B E k T µ ± , we can ignore the 1 ± term in the denominator. Both distributions reduce to the Boltzmann distribution function,
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