Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 65
Incoherent - diffusion
The intermediate scattering function is for a classical system given by
I inc (Q, t) = 〈exp(−iQ · r i (0))exp(iQ · r i (t))〉 = 〈exp(iQ · r(t))〉 (4.3.39)
where the definition of the displacement r(t) is r(t) = r i (t)−r i (0). If the probability
function of r(t) is a Gaussian and if the system is isotropic (average displacement is
the same in all directions) then it follows that
( −Q 2 〈r 2 )
〉(t)
I inc (Q, t) = exp
. (4.3.40)
6
To arrive at this equation it was assumed that the system was classical, Gaussian
and isotropic, but no particular assumptions about the time dependence of 〈r 2 〉(t)
were included. This means that the equation also could describe the incoherent
intermediate scattering function in a solid. The mean squared displacement, 〈r 2 〉(t),
in a solid has a finite value in its long time limit. This leads to a plateau in I inc (Q, t)
and the above result is equivalent to equation 4.3.29. The physical explanation
difference of the factor 6 vs 3 in the denominator arises from the difference of the
definition of 〈r 2 〉(t) which is the mean squared displacement from a position at t = 0
while 〈u 2 〉 is the mean distance from an equilibrium position in the solid.
If the time evolution is given by diffusion then one has 〈r 2 (t)〉 = 2Dt which leads to
an exponential decay of the intermediate scattering function:
( −Q 2 )
Dt
I inc (Q, t) = exp , (4.3.41)
3
and correspondingly a Lorentzian line shape of the dynamical structure factor.
4.3.9 Glasses and viscous liquids
One of the most characteristic features of the dynamics on highly viscous liquids is
the separation of time scales which was also discussed in section 2.5. This means
that the correlation functions on short time scales are similar to those of solids while
they on long times decay to zero as in liquids. When describing and interpreting the
dynamics in viscous liquids and glasses it is therefore useful to “lend” and generalize
both the description used for solids and hydrodynamics.