Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 67
is highly controversial and we shall not enter in this discussion. Suggestions are that
it is due to disorder, fast relaxation or that it is a signature of the fact that the
normal modes of the systems are not plane waves. [Ruocco et al., 2000; Ruffle et al.,
2003; Matic et al., 2004]
One of our main interest in the current study of the coherent dynamical structure
factor is the nonergodicity factor. The nonergodicity factor is in principle the long
time limit of the intermediate scattering function but it is operationally defined
as the ratio of the central peak over the total coherent intensity. The intensity
of the central peak is associated with the correlations that decay slower than the
Brillouin frequency considered. It follows from equation 4.3.38 that this ratio in the
hydrodynamic limit will be given by
f q = 1 − v2 T
vs
2 . (4.3.43)
The difference between the adiabatic and isothermal sound speeds is small in viscous
liquids close to the glass transition. However, the relevant adiabatic sound speed
is the sound speed at the Brillouin frequency, while the isothermal sound speed is
the low frequency equilibrium sound speed. The generalization of this result in the
viscoelastic liquid (still in the low Q limit) is therefore
f q = 1 − v2 0
v∞
2 . (4.3.44)
We apply this interpretation of the measured f Q in section 6.3.3.
4.3.10 Long time limit and the resolution function
The measured intensity in a scattering experiment includes a convolution with the
experimental resolution function, R(ω). The delta function in equation 4.3.22 therefore
becomes a broadened central peak in the experimental result
∫ ∞
S exp (Q, ω) = R(ω) ⊗ I ∞ (Q)δ(ω) + R(ω) ⊗ 1 I t (Q, t)exp(−iωt)dt
2π −∞
S exp (Q, ω) = R(ω)I ∞ (Q) + R(ω) ⊗ 1 ∫ ∞
I t (Q, t)exp(−iωt)dt. (4.3.45)
2π
R(ω) is normalized and the long time limit of the intermediate scattering function,
I ∞ (Q), can therefore be found from the integral over the central peak
I ∞ (Q) =
∫ ∆ω
−∆ω
−∞
S exp (Q, ω)dω. (4.3.46)