Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 57
4.3.5 Correlation functions
The intermediate scattering function is defined as 8
I coh (Q, t) = 1 N
∑
〈exp(−iQ ·r i (0))exp(iQ · r j (t))〉 (4.3.15)
i,j
and its time Fourier transform is called the dynamical structure factor
S coh (Q, ω) = 1 ∫ ∞
I coh (Q, t)exp(−iωt)dt. (4.3.16)
2π −∞
Equivalent definitions are used for the incoherent scattering yielding the following
expression for the total scattering cross section.
∂ 2 σ
∂Ω∂E = N Q out
Q in
σ coh
4π S coh(Q, ω) + N Q out
Q in
σ inc
4π S inc(Q, ω). (4.3.17)
For X-ray scattering it follows from equation 4.3.12 that the cross section is given
by
( ∂ 2 )
σ
= N
∂Ω∂E
Xray
4.3.6 General Results - limiting behavior
( ) 2
e 2
4πǫ 0 m ec 2 (ei · e f ) 2 |f(Q)| 2 Qout
Q in
S coh (Q, ω) (4.3.18)
In this work we study the temperature dependence of the elastic and the integrated
scattered intensity both in the coherent and the incoherent case. In the following
section we therefore spend some time on the details of the information that can be
extracted from these quantities. Particularly the differences between the coherent
and the incoherent cases are considered. Results that hold for both the coherent
and the incoherent case are given without the subscript coh or inc.
Short time limit
The static structure factor, S(Q), is defined as the integral over energy of the dynamic
structure factor. Combining this definition with equation 4.3.16, it is seen
that the static structure factor is equal to the intermediate scattering function at
time zero, S(Q) = I(Q, t = 0):
8 The correlation functions which we introduce in this section are also described in [Lovesey,
1984] and [Squires, 1978]. Solids are not so treated in this frame in introductory textbooks, because
the spatial crystalline solids leads to a simpler description. Relevant references on the liquid state
include [Egelstaff, 1994; Hansen and McDonald, 1986; Boon and Yip, 1980].