Ph.D. thesis (pdf) - dirac

58 Experimental techniques and observables
S(Q) =
=
=
=
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
∫ ∞
−∞
S(Q, ω)dω
1
2π
∫ ∞
−∞
I(Q, t) 1
2π
I(Q, t)exp(−iωt)dt dω
∫ ∞
−∞
exp(−iωt)dω dt
I(Q, t)δ(t)dt = I(Q, t = 0). (4.3.19)
The coherent static structure factor holds information of the structure of the system,
it is in fact the space Fourier transform of the pair correlation function. The
incoherent structure factor on the other hand, does not hold any information as it
is always equal to one:
S inc (Q) = I inc (Q, t = 0) = 〈exp(−iQ ·r j (0))exp(iQ · r j (0))〉 = 1. (4.3.20)
Long time limit
Consider now the case where I(Q, t) has a finite value in its long time limit. It can
then be expressed as a sum of a time independent and a time dependent term
I(Q, t) = I ∞ (Q) + I t (Q, t) where I t (Q, t) → 0 for t → ∞. (4.3.21)
Fourier transforming this to get the dynamical structure factor yields
S(Q, ω) =
∫
1 ∞
2π −∞
= I ∞ (Q)δ(ω) + 1
2π
I ∞ (Q)exp(−iωt)dt + 1
2π
∫ ∞
−∞
∫ ∞
−∞
I t (Q, t)exp(−iωt)dt
I t (Q, t)exp(−iωt)dt. (4.3.22)
From this, it is seen that the dynamical structure factor will have a peak at ω = 0
and that the intensity of this peak is given by the long time value of the intermediate
scattering function, I ∞ (Q). Note that the second term does not have strictly zero
intensity at ω = 0.
4.3.7 Simple model - solid
Consider a solid, disordered or crystalline. There is no diffusion in the system, which
means that the particles are essentially vibrating (harmonic or not) around a fixed
position in space. In the case of a crystal, this position is the equilibrium position