Ph.D. thesis (pdf) - dirac

8.1. Time of flight 147
g(ω)/ω 2 [arb. unit]
1.6
1.4
1.2
1
0.8
0.6
Patm
0.41GPa
0.8GPa
1.4GPa
0.4
0.2
0
0 5 10 15
ω [meV]
Figure 8.2: Spectra of PIB3580 at different pressures. The full lines give g(ω)/ω 2
obtained from S(Q, ω) measured at a fixed Q value (Q=1.7 Å −1 ). The dots give
g(ω)/ω 2 based on the S(ω) calculated by summing over all angles. (See the text for
details). The results are in agreement, not only the shape and position of the boson
peak, but also the pressure dependence of the intensity found by the two methods
is the same.
The Q-dependence
From equation 8.1.2 it follows that only the Debye Waller factor and the Q 2 -term
are Q-depend. A Q independent quantity can therefore be found by dividing by
these
S inel,inc (Q, ω)
(
exp
− 〈u2 〉Q 2
3
) = 1
Q 2 2M
n(ω)
g(ω). (8.1.4)
ω
In figure 8.3 we illustrate the above at 3 different Q values, with the Debye Waller
factor taken from the measured elastic intensity. The three curves should overlap
according to equation 8.1.4, but this is clearly not the case. However, the shape is the
same and the curves can be brought to overlap by adjusting the Debye Waller factor
(figure 8.4). Additionally we find that the pressure dependence of the intensity
is the same at all Q-values. We therefore conclude that the measured frequency
dependence shown in figure 8.2 gives a correct representation of the vibrational
density of states.
Other techniques for extracting DOS from equation 8.1.2 exploit the Q dependence.
Rewriting equation 8.1.2 to
( )
Sinc (Q, ω)ω
ln
Q 2 = − 〈u2 〉Q 2
n(ω) 3
+ ln(g(ω)) (8.1.5)
it is seen that the left hand side can be obtained directly from the raw data. Fitting