5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS Crystalline solids phonons in the reciprocal lattice
Feb 08, 2016
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