5. atomic dynamics in amorphous solids crystalline solids phonons in the reciprocal lattice
Post on 22-Dec-2015
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333
42
2345
2
D
B
D
BDebye
nk
v
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
2
1
2
1 2n = 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
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:
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 SPECTROSCOPY
BOSONPEAK
]1),([
)()()(
Tn
gCIR
Competition between increasing g() anddecreasing Bose-Einstein factor ???
RAMAN SPECTROSCOPY
BOSONPEAK
Martin & Brenig theory: a peak in the coupling coefficient C() due to elastoacoustic disorder ??
RAMAN SPECTROSCOPY
BOSONPEAK
]1),([
)()()(
Tn
gCIR
2/)()(]1),([/)( gCTnII Rred
[Sokolov et al. 1994]
The Boson Peak is a peak in C() g() / 2 !!!