Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 59
in the lattice. In the case of the disordered solid this position is less well defined.
However, it is in both cases possible to describe the time dependent position of the
particle by a sum of a time dependent and a time independent term:
r i (t) = r i,eq + u i (t) (4.3.23)
where u i (t) has a finite maximum value corresponding to the furthest distance of
the particle from its “equilibrium” position, r i . For the incoherent intermediate
scattering this gives
I inc (Q, t) = 〈exp(−iQ · {r i,eq + u i (0)})exp(iQ · {r i,eq + u i (t)})〉
= 〈exp(−iQ · u i (0))exp(iQ ·u i (t))〉exp(−iQ · {r i,eq − r i,eq })
= 〈exp(−iQ · u i (0))exp(iQ ·u i (t))〉. (4.3.24)
Here we again see that the incoherent signal holds no information of the structure
of the system, as it depends only on the dynamic part, u i (t), of the particle position
r i (t).
The coherent intermediate scattering function following from equation 4.3.23 is given
by
I coh (Q, t) =
1 ∑
〈exp(−iQ · {r i,eq + u i (0)})exp(iQ · {r j,eq + u j (t)})〉
N
i,j
= 1 ∑
〈exp(−iQ · {r i,eq − r j,eq })exp(−iQ ·u i (0))exp(iQ · u j (t))〉.
N
i,j
One can now define a structure factor corresponding to the structure of given by the
“equilibrium” position of the particles. We call this structure factor the inherent
structure structure factor S is,coh (Q)
S is,coh (Q) = 1 N
∑
〈exp(−iQ · {r i,eq − r j,eq })〉 = ∑
i,j
i
〈exp(−iQ · r ′ i,eq)〉 (4.3.25)
where r i,eq ′ = r i,eq − r j,eq and the 〈 〉 denote ensemble average. Note that inherent
structure structure factor differs from the actual structure factor of the system,
S is,coh (Q) ≠ I(Q, t = 0) = S(Q) because
u i (0) ≠ u j (0) and therefore exp(−iQ · u i (0))exp(iQ · u j (0)) ≠ 1. (4.3.26)
There is a difference between the static structure factor S coh (Q) which describes the
particles in a snapshot where they will be displaced from their equilibrium position