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 / ) VL π = . In the sphere, the number of states within k and k+dk is 22 2 1 4 (2 ) kkV kk N V ∆∆ ∆= = , 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 1 () Nk d k DE VE E d E ππ == = . 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( / ) ii B PE A E kT =− , in which the constant A can be determined by normalization over all quantum states ()1 i All QS = . For an open system exchanging energy with the outside, the probability becomes ( , ) exp[ ( )/ ] i i B PE N A E N kT µ , where N i is the particle number, chemical potential 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 iB Ek T nf 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 For phonons, the Pauli exclusion principle is no longer applicable.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 hk T e ν , which is called Bose-Einstein distribution. For molecules, similar results exist / 1 1 T E e , 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 T ± , we can ignore the 1 ± term in the denominator. Both distributions reduce to the Boltzmann distribution function,
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This note was uploaded on 01/12/2011 for the course ME 305 taught by Professor Wright,j during the Spring '10 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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

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