Lecture13 - Phys 436 Modern Physics Lecture 13 Feb 7 2013...

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Phys 436 – Modern Physics Lecture 13 Feb. 7, 2013 Instructor: David Cooke Einstein solid Bose-­૒Einstein staHsHcs Debye specific heat Phonons 1
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The real world Temperature (K) C V [J/mole K] 25 3R = 3 N A k B (Dulong-­૒PeHt Law) 0 Last Hme we saw that a real solid only agreed with the predicHons of a classical distribuHon in the high T limit. Energies are discrete and not conHnuous! We’ll learn how to deal with this today. 2
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Einstein solid Einstein imagined a 3D solid as a collecHon of harmonic oscillators, each with energy ! ! i ( n i + 1/ 2) Each atom has 3 degrees of freedom (x, y, z) What is the internal energy of N discrete linear oscillators? 3
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How to calculate averages < E >= E i exp ! " E i ( ) i = 1 N # exp ! " E i ( ) i = 1 N # = 1 Z E i exp ! " E i ( ) i = 1 N # The PARTITION FUNCTION, Z < E >= ! 1 Z " " # Z ( # , E 1 , E 2 ,...) < E >= ! " " # ln( Z ) So once we know the parHHon funcHon, we know the energy! 4
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Summing over discrete states E n = ! ! i ( n i + 1/ 2) i " Each oscillator indexed by i has an energy indexed by n Z = ...
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