23_562ln08

23_562ln08 - MIT OpenCourseWare http://ocw.mit.edu 5.62...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 5.62 Spring 2008 Lecture #23 Page 1 Phonons: 1D Linear Chain of Atoms An atomic description of the vibrations of a solid . This will be a more realistic description than Einsteins or Debyes, which are continuum models. Debyes model works well at low T (T 3 law) and high T, but not so well at intermediate T. Linear chain in 1D of atoms connected by harmonic springs of force constant k; atoms are labeled as (n2), (n1), n, (n+1), (n+2) ,.... Lattice positions or equilibrium positions of atoms along the chain are labeled as (n1)a, na, (n+1)a, where a is the lattice spacing N total number of atoms in chain instantaneous displacement of n th atom from equilibrium position Plot of x(na,t) vs. position along lattice at some t = t 1 represents a collective vibrational mode, a phonon , of the solid x n displacement x(na,t) t = t 1 longitudinal phonon displacements parallel to direction of motion (top) transverse phonon displacements perpendicular to direction of motion. Phonons of many 's can be supported revised 04/06/084/6/08 2:58 PM 5.62 Spring 2008 Lecture #23 Page 2 x(na,t) = 4a x(na,t) = 2a l = 2a is the shortest physically sensible wavelength; in the continuum model, there is no limit to the shortness of l. wavevector magnitude k = 2 ! " = q [Reserve k for force constant and for wavevector of photons, but not phonons .] GOAL: want to determine allowed values of or q but phonons are unlike photons, for which ! = c " 2 #! = $ = cq (photons) PROBLEM: for phonons, in general cq especially at large q (short wavelength), does not vary linearly with q. revised 04/06/084/6/08 2:58 PM 5.62 Spring 2008 Lecture #23 Page 3 DISPERSION RELATION GOAL: for phonons, find allowed values of q and find the relationship between and q....
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23_562ln08 - MIT OpenCourseWare http://ocw.mit.edu 5.62...

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