Ph.D. thesis (pdf) - dirac

98 High Q collective modes
does influence the nonergodicity factor. This leads to a relatively larger systematic
error in the nonergodicity factor as compared to measurements at ambient pressure.
Fitting
The data are fitted by use of a damped harmonic oscillator for the inelastic signal
and a delta function for the elastic line
(
S(Q, ω) = S(Q) f(Q)δ(ω) + [1 − f(Q)] 1 Ω 2 )
Γ(Q)
π (ω 2 − Ω 2 (Q)) 2 + ω 2 Γ 2 . (6.1.1)
(Q)
The normalization of the functions ensures that the integral ∫ S(Q, ω)dω = S(Q)
and ∫ ∆ω
−∆ω
S(Q, ω)dω/S(Q) = f(Q). The wavelength studied is 2π/Q, Ω gives the
frequency of the mode in question, and Γ denotes the damping/attenuation of the
sound mode.
The above function is symmetric in ω. Detailed balance is obtained by multiplying
with the factor ω/(k B T)/(1 − exp(−ω/(k B T))). Note that the sample gain of
energy is taken as positive energy in this case. This is done to follow the convention
in the IXS community, however it is the opposite of the convention in neutron
scattering. The neutron convention is used in section 4.3 and in chapter 8. The last
point of the analysis is the convolution with the resolution function. The resolution
is obtained experimentally at the beamline by a measurement on a plexiglass sample.
The complete function used in the fitting procedure is hence
∫
I(Q, ω) = A
R(ω − ω ′ ) ( f(Q)δ(ω ′ )+ (6.1.2)
)
dω ′ .
ω ′ β
1 − exp(−ω ′ β) [1 − f(Q)]1 π
Ω 2 Γ(Q)
(ω ′2 − Ω 2 (Q)) 2 + ω ′2 Γ 2 (Q)
Where A is a factor that contains S(Q) as well as the total number of scatterers,
scattering length etc., see section 4.3.2. The quality of the fits is generally very
convincing, this is illustrated in figure 6.2.
6.2 Sound speed and attenuation
6.2.1 PIB
Figure 6.3 shows the dispersion of the longitudinal sound modes in the Q-range
2 nm −1 to 20 nm −1 measured at ambient pressure and at 300 MPa, for the PIB680 at