Ph.D. thesis (pdf) - dirac

68 Experimental techniques and observables
In the above we ignore the non-zero contribution of the second term in equation
4.3.22. Another approach which we shall actually use in chapter 6 is to separate
the elastic and the inelastic part of the intensity by fitting the total result to a
sum of two functions. This latter approach requires some assumption regarding the
dynamics of the system in order to choose a proper function for the inelastic part of
the signal.
The experimental intermediate scattering function, I exp (Q, t) found from the inverse
Fourier transform of the experimental dynamic structure factor, decays to zero at
t ≈ ∆ω, where ∆ω is the width of the resolution function. This effect can in principle
be corrected by deconvolution with the resolution function
I(Q, t) =
I(Q, t)
R(t)
where R(t) =
∫ ∞
−∞
R(ω)exp(iωt)dt (4.3.47)
but we can never get any reliable information about dynamics slower than 1/(∆ω).
The resolution function therefore determines the timescale of the experiment. Figure
4.3.10 illustrates how two different dynamical structure factors which cannot be
distinguished with a given experiment, because the difference is on a timescale which
is not accessed with the given resolution.
sample
S(Q, ω)
experimental
S(Q, ω)
sample
I(Q, t)
experimental
I(Q, t)
sample
S(Q, ω)
experimental
S(Q, ω)
sample
I(Q, t)
experimental
I(Q, t)
Figure 4.3: The figure illustrates two different dynamical structure factors which
cannot be distinguished with a given experiment, because the difference is on a
timescale which is not accessed with the given resolution. The left figure shows the
dynamical structure factor at fixed Q as a function of temperature. The right figures
show the intermediate scattering function at fixed Q as a function of logarithm of
the time. Upper figures: The dynamical structure factor of the sample has a a delta
function at zero energy transfer and the actual intermediate scattering function of
the sample never decays to zero. Lower figures: There is a slight broadening of
the central peak seen in dynamical structure factor and the intermediate scattering
function decays to zero at long times.