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5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS

5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS. Crystalline solids  phonons in the reciprocal lattice. Crystalline solids  Debye Theory. g (  ) =  2 / 2 2 v D 3. C p ( T ) = C Debye T 3. 2. ATOMIC DYNAMICS. Hamiltonian for lattice vibrations:. n = 1, …, N  = 1, …, r

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5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS

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  1. 5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS Crystallinesolids phonons in thereciprocallattice

  2. Crystallinesolids DebyeTheory g() =  2 / 22vD3 Cp(T) = CDebyeT 3 2

  3. ATOMIC DYNAMICS Hamiltonian for lattice vibrations: n = 1, …, N  = 1, …, r i = x, y, z  Eq. of motion: If: Dynamical matrix D has 3Nr real eigenvalues j2 and corresponding eigenvectors uni (j) • In periodic crystals: q  only3rcurves j(q) : • 3 acoustic branches j(q 0)  0 • 3(r-1) optic branches j(q 0)  const.

  4. Dispersion relations (q) in amorphous solids

  5. Does exist a quantity which can describe sensibly phonon modes in amorphous solids? YES: the vibrational density of states (VDOS): g()·d= number of states with frequencies between  and d ! For crystals:

  6. COMPUTER SIMULATIONS

  7. EXPERIMENTAL TECHNIQUES

  8. RAMAN SPECTROSCOPY • In amorphous solids, there is a breakdown of the • Raman selection rules in crystals for the wavevector • ALL vibrational modes contribute to Raman scattering (first-order scattering), in contrast to the case of crystals (second-order scattering due to selction rules)

  9. RAMAN SPECTROSCOPY BOSON PEAK Competitionbetweenincreasingg() and decreasing Bose-Einstein factor ???

  10. RAMAN SPECTROSCOPY BOSON PEAK Martin & Brenigtheory: a peak in thecoupling coefficientC() duetoelastoacousticdisorder ??

  11. RAMAN SPECTROSCOPY BOSON PEAK [Sokolov et al. 1994] TheBosonPeakis a peak in C() g() / 2!!!

  12. Brillouin scattering: Experimental set-up

  13. BRILLOUIN SCATTERING: ethanol

  14. INELASTIC NEUTRON SCATTERING

  15. INELASTIC NEUTRON SCATTERING

  16. INELASTIC NEUTRON SCATTERING

  17. INELASTIC NEUTRON SCATTERING

  18. INELASTIC NEUTRON SCATTERING

  19. RAMAN SCATTERING The Boson Peak is a peak in C() g() / 2!!!

  20. INELASTIC X-RAY SCATTERING Damped Harmonic Oscillator

  21. INELASTIC X-RAY SCATTERING

  22. INELASTIC X-RAY SCATTERING

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