Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 61
(see also section 4.3.8). Combining all this we get
(
lim I inc(Q, t) = exp − Q2 〈u 2 )
〉
. (4.3.29)
t→∞ 3
We use the above expression in order to calculate the mean square displacement
from measured elastic intensities in chapter 7. The analogous result in the coherent
case reads
(
lim I coh(Q, t) = S is (Q)exp − Q2 〈u 2 )
〉
. (4.3.30)
t→∞ 3
The incoherent intermediate scattering function goes from I inc (Q, t = 0) = 1 to
I inc (Q, t = ∞) = exp(−2W). The coherent intermediate scattering function goes
from I coh (Q, t = 0) = S coh (Q) to I coh (Q, t = ∞) = exp(−2W)S is,coh (Q) (see figure
4.2). The inherent structure factor, S is,coh , contains only structural information,
the Debye Waller factor, exp(−2W), contains only dynamical information, while
the structure factor, S coh (Q) depends on both on the structure and the dynamics
of the system.
The long time limit of the normalized coherent intermediate scattering function is
called the nonergodicity factor, f Q , (figure 4.2), because it measures the relative
intensity of the correlations which survives in the long time limit. It has a nontrivial
Q-dependence. It also has a non-trivial dependence of the structure because
it contains both dynamical and structural information. If the structure is somehow
changed, then the nonergodicity factor also changes even if the amplitude of the
vibrations is kept constant.
One phonon scattering
In the approximation where only the harmonic part of the forces in a solid are taken
into account then the displacements of each particle can be described by the sum
of displacements due to the normal modes of the system. Each normal mode is
associated with an eigen-vector and a frequency. The harmonic modes give rise to
harmonic oscillations in the time dependence of I coh (Q, t) and hence to delta peaks
in its Fourier transform S coh (Q, ω). The contribution to the coherent dynamical
structure factor is for plane waves in the one-phonon approximation given by
S coh,inel (Q, ω) =
exp(−2W)
4πMN
∑
exp(−iQ · r i ) (Q · e s) 2
i,s
ω s
∫ ∞
−∞
[exp(−i(Q · r i − ω s t))〈n s + 1〉
+exp(i(Q · r i − ω s t))〈n s 〉] exp(−iωt)dt (4.3.31)