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

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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 .
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5.62 Spring 2008 Lecture #23 Page 1 Phonons: 1–D Linear Chain of Atoms An atomic description of the vibrations of a solid . This will be a more realistic description than Einstein’s or Debye’s, which are continuum models. Debye’s 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 (n–2), (n–1), n, (n+1), (n+2) , .... Lattice positions or equilibrium positions of atoms along the chain are labeled as (n–1)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 λ 's can be supported revised 04/06/084/6/08 2:58 PM
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5.62 Spring 2008 Lecture #23 Page 2 x(na,t) λ = 4a x(na,t) λ = 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 λ or q but phonons are unlike photons, for which ! = c " 2 #! = $ = cq (photons) PROBLEM: for phonons, in general ω cq especially at large q (short wavelength), ω does not vary linearly with q. revised 04/06/084/6/08 2:58 PM
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5.62 Spring 2008 Lecture #23 Page 3 DISPERSION RELATION GOAL: for phonons, find allowed values of q and find the relationship between ω and q.
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