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Phys 460 Lecture 9

( pdf version - 6 slides/page )
Monday, September 25, 2006
Lecturer: Richard Martin 		No new homework today

Reading:
Kittel, Chapt. 4

Vibrations of atoms in crystals
Outline

  1. From previous lectures:
    • Typical Crystal Structures
    • Diffraction, Fourier Analysis, and the Reciprocal Lattice
    • Crystal binding, elastic waves
  2. Last Lecture: Vibrations of atoms in a crystal, part I
    • Linear chain of masses and springs
    • Dispersion Curves omega as function of k
    • Importance of Reciprocal Lattice, Brillouin Zone
    • Linear chain with two atoms per cell
      • Two modes - acoustic and optic
      • Limit for small k, K at BZ boundary
  3. Continuing examples in higher dimensions
    • Form for central forces in general crystals
    • Force = Fs = Sum i phii'' (DRs - DRs+i)
    • The direction of motion can be chosen by symmetry in high symmetry cases
    • We need only the vector component of the force along the direction of motion, so the formulas simplify
    • Homework for Na (bcc structure) - data in Fig. 11, Ch. 4 of Kittel
  4. General results
    • Vibration waves specified by k inside the BZ
    • In 3 dimensions, there are 3N phonon dispersion curves, where N = # atoms per cell
    • 3 acoustic modes - 3 types of sound waves in each direction
    • 3N -3 optic modes
  5. Quantization of vibration waves
    • Quantized units of vibration are called phonons
    • Act like particles - each has quantized energy hbar omega
    • k can be interpreted as a momemtum
      • But k is NOT real momentum
      • k is conserved plus or minus a reciprocal lattice vector
    • Detected experimentally (see below) by creation or destruction of quantized units, i.e., phonons
    • Later we will see they transport energy just like an ordinary gas of particles (like molecules in a gas)
  6. Experimental observation by inelastic scattering
    • Inelastic scattering creates or destroys a phonon at wavevector k
    • Thus the scattering wave has Delta k = kin - kout = k + G (or -k + G) where G is a recip. lat. vec.
    • This is differnt from Bragg diffraction which is only a recip. lat. vec. G
    • The the scattering wave has Delta E = kout - Ein = hbar omega (or - hbar omega)
    • Neutrons (and very recently by electrons and X-rays)
    • Resolve energy loss or gain of scattering particle when is diffracted from the vibrating crystal
    • By using triple axis spectrometer, resolve both the k and the energy (hbar omega) of the phonon
Email clarification questions and corrections to rmartin@uiuc.edu
Email questions on solving problems to xin2@.uiuc.edu