5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS Crystalline solids  phonons in the reciprocal lattice.

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Presentation transcript:

5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS Crystalline solids  phonons in the reciprocal lattice

C p (T) = C Debye T 3 2 Crystalline solids  Debye Theory g(  ) =  2 / 2  2 v D 3

ATOMIC DYNAMICS Hamiltonian for lattice vibrations:  Eq. of motion: n = 1, …, N  = 1, …, r i = x, y, z If: Dynamical matrix D has 3Nr real eigenvalues  j 2 and corresponding eigenvectors u n  i (j) In periodic crystals: q  only 3r curves  j (q) : 3 acoustic branches  j (q  0)  0 3(r-1) optic branches  j (q  0)  const.

Dispersion relations  (q) in amorphous solids

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:

COMPUTER SIMULATIONS

EXPERIMENTAL TECHNIQUES

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)

RAMAN SPECTROSCOPY BOSON PEAK Competition between increasing g(  ) and decreasing Bose-Einstein factor ???

RAMAN SPECTROSCOPY BOSON PEAK Martin & Brenig theory: a peak in the coupling coefficient C(  ) due to elastoacoustic disorder ??

RAMAN SPECTROSCOPY BOSON PEAK [Sokolov et al. 1994] The Boson Peak is a peak in C(  ) g(  ) /  2 !!!

Brillouin scattering: Experimental set-up

BRILLOUIN SCATTERING: ethanol

INELASTIC NEUTRON SCATTERING

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

Damped Harmonic Oscillator INELASTIC X-RAY SCATTERING