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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 - PowerPoint PPT Presentation
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5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS
Crystalline solids phonons in the reciprocal lattice
333
42
2345
2
D
B
D
BDebye
nkv
kC
333
113
TLD vvv
Cp(T) = CDebye T 3
2
Crystalline solids Debye Theory
g() = 2 / 22vD3
ATOMIC DYNAMICS
Hamiltonian for lattice vibrations:
Eq. of motion:
inin
inin
ininin
in
sssMH
21
21 2
n = 1, …, N = 1, …, r i = x, y, z
inin
ininin ssM
If:
)exp(1)( tiuM
ts inin
inin
ininin uDu
2
Dynamical matrix D has 3Nr real eigenvalues j2
and corresponding eigenvectors uni (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.
)exp( niin Rqicu
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 !
S k
g dSVg 3)2(
)(For crystals:
COMPUTER SIMULATIONS
EXPERIMENTAL TECHNIQUES
RAMAN SPECTROSCOPY
In amorphous solids, there is a breakdown of theRaman 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 SPECTROSCOPYBOSONPEAK
]1),([)()()(
TngCIR
Competition between increasing g() anddecreasing Bose-Einstein factor ???
RAMAN SPECTROSCOPYBOSONPEAK
Martin & Brenig theory: a peak in the coupling coefficient C() due to elastoacoustic disorder ??
RAMAN SPECTROSCOPYBOSONPEAK
]1),([)()()(
TngCIR
2/)()(]1),([/)( gCTnII Rred
[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
INELASTIC NEUTRON SCATTERING
INELASTIC NEUTRON SCATTERING
INELASTIC NEUTRON SCATTERING
INELASTIC NEUTRON SCATTERING
RAMAN SCATTERING
]1),([)()()(
TngCIR
The Boson Peak is a peak in C() g() / 2 !!!
Damped Harmonic Oscillator
INELASTIC X-RAY SCATTERING
INELASTIC X-RAY SCATTERING
INELASTIC X-RAY SCATTERING
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