Ph.D. thesis (pdf) - dirac

62 Experimental techniques and observables
Icoh(Q,t)
S(Q)
S is (Q)exp(−2W)
Icoh(Q,t)/S(Q)
a) b)
1
f Q
log t
log t
Iinc(Q,t)
1
c)
exp(−2W)
log t
Figure 4.2: Illustration of the time evolution of the intermediate scattering functions
in a solid. a) The coherent intermediate scattering function, b) The normalized
coherent intermediate scattering function, c) The incoherent intermediate scattering
function, which is intrinsically normalized.
Where ω s is the frequency of the modes s with wave vector Q and e s is the corresponding
polarization vector. The summation over i gives a delta function in the
wave vector dependence δ(Q), meaning that only the modes with wave vector Q
will contribute. For a disordered state one will measure the ensemble average over
the different inherent structures. The terms 〈n s 〉 is the bose factor, which gives the
occupation number of the mode. The bose factor is given by
〈n s 〉 =
( ( ) ω
−1 ( ( ) ω
−1
exp − 1)
, 〈n s + 1〉 = exp − 1)
+ 1 (4.3.32)
k B T
k B T
which in the classical limit → 0 gives
〈n s 〉 = 〈n s + 1〉 = k BT
ω , (4.3.33)
from which it is seen that the intensity of the one-phonon contribution (after normalization
with the Debye Waller factor) is proportional to the temperature T.