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Phys 460 Lecture 8

( pdf version - 6 slides/page )
Wednesday, September 20, 2006
Lecturer: Richard Martin 		Homework 4

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. Why are vibrations important?
    • Atoms vibrate because of
      • Thermal motion
      • Quantum motion - uncertainty principle - even at T=0
    • Illustrates key idea of solid state physics
      • Importance of Brillouin Zone
      • Transformation from classical vibrations to quantized particles of vibration
    • We will consider simple examples to show the effects
  3. Displacements of atoms and harmonic forces
    • Define vector displacement DeltaRn = un for each atom n (use interger n to avoid confusion with the imaginary number i)
    • Energy increases quadratically E = E0 + (1/2)Sumnj un Dnj uj
    • Force on atom n = Fn = - dE/dRi = Sumj Dnj uj
    • D's are called "Force Constants"
  4. Example : Central Forces
    • Energy per cell = E = (1/2N)Sum nj phi(|Rn - Rn+j |)
    • To second order E = E0 + (1/4)Sumnj phij'' |un - un+j|2,
      phij'' = phi''(|Rn - Rn+j |) = second derivative of phi evaluated at |Rn - Rn+j |
    • Force = Fn = Sum j phij'' (un - un+j)
  5. Linear chain: Rn = na - all motion un is along a line
    • Assume nearest neighbor forces only:
      j = +1 and j = -1 and phi1'' = phi-1'' = phi''
    • Fn = phi'' Sum j (un+1 - un) = phi'' (un+1 + un-1 - 2un)
    • Like springs: Fn = K (un+1 + un-1 - 2un)
    • Solution: un(t) = exp(ik(na) - i omega t)
    • Leads to: M omega2 = phi'' [exp(ika) + exp(-ika) - 2]
      and omega2= (phi''/M) [cos(ka) - 2] = (phi''/ M) sin2(ka/2)
  6. Role of Brillouin Zone
    • Vibrations completely described by k inside BZ
    • Periodic for k outside BZ
    • Vibration is identical (not independent) for k and k+G where G is any recip. lattice vector
  7. Simplest example in higher dimensions - simple cubic with central forces
    • Longitudinal motion - same as linear chain
    • Transverse - unstable with only nearest-neighbor central forces
    • Stable with second-nearest-neighbor central forces
  8. Crystals with central forces actually occur in close packed structures or with long range forces, e.g., Coulomb
    • Longitudinal and transverse modes of close packed structures stable with only nearest-neighbor central forces
    • Homework example
Email clarification questions and corrections to rmartin@uiuc.edu
Email questions on solving problems to xin2@.uiuc.edu