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

5. ATOMIC DYNAMICS IN AMORPHOUS SOLIDS

Crystalline solids phonons in the reciprocal lattice

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

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 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 !!!

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),([

)()()(

Tn

gCIR

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