LatticeVibrationsCrystalsLect24ME501F2011

LatticeVibrationsCrystalsLect24ME501F2011 - Purdue...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lecture 24: Statistical Thermodynamic Analysis of Lattice Vibrations in Crystals Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana Lucht@purdue.edu , 765-494-5623 (Phone) November 7, 2011
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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics • The crystal lattice. • The Einstein theory of the heat capacity of crystalline solids. • The Debye theory of the heat capacity of crystalline solids.
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Patterned Border Template 3 Purdue University School of Mechanical Engineering The Crystal Lattice • Consider the crystal lattice of a solid as a large number N atoms hooked together by springs: McQuarrie, Statistical Mechanics , 1976
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Patterned Border Template 4 Purdue University School of Mechanical Engineering The Crystal Lattice • The N atoms in the crystal lattice are inherently distinguishable rather than indistinguishable because we can associate each atom with a particular lattice site. Therefore Maxwell-Boltzmann (MB) statistics rather than corrected Maxwell-Boltzmann statistics will apply to the lattice. From our previous discussion of MB and CMB statistics we found: , , ! ! ln ln ln j N j mMB j j j j j j g WN N g N N N   (ln ) (ln ) ln 1 ln 0 jj mMB mCMB j j gg dW d N d N NN      • Differentiating these expressions to find d(ln W m ) gives us: 2 ln ln ln ln j j mMB mCMB j j j j j gN g g WW N N g N N     , ln ln j mCMB j j j g N N 
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Patterned Border Template 5 Purdue University School of Mechanical Engineering The Crystal Lattice • At this point the analysis with undetermined multipliers to find the most probable distribution is exactly the same for MB and CMB statistics. The result of this analysis is:      e x p/ e x () exp / jj B B j B j gk T T NN N ZT T    • Thermodynamic properties that do not involve the entropy S will be the same for both MB and CMB statistics. The entropy will be different for the two statistical models, however, as is evident from the Boltzmann relations for the two models: , , ln ln ln ln j MB B m MB B j j j j CMB B m CMB B j j j g Sk W k N N N N g W k N N N    
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Patterned Border Template 6 Purdue University School of Mechanical Engineering The Crystal Lattice • Returning to the crystal lattice, a normal coordinate analysis of the crystal lattice will result in 3N-6 vibrational degrees of freedom. The vibrational frequencies of these normal modes are given by: 1 1,2,.
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This note was uploaded on 12/26/2011 for the course ME 501 taught by Professor Na during the Fall '10 term at Purdue University-West Lafayette.

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LatticeVibrationsCrystalsLect24ME501F2011 - Purdue...

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