Ph.D. thesis (pdf) - dirac

54 Experimental techniques and observables
follows that
b i b j = b 2 for j ≠ i and b i b j = b 2 for j = i. (4.3.5)
consequently
∑
b i b j
i,j
= ∑ b 2 + ∑
i≠j j
= ∑ b 2 + ∑
i,j j
b 2 (4.3.6)
⎛
⎝b 2 − ∑ j
b 2 ⎞
⎠ (4.3.7)
where the second equality follows from adding and subtraction the term ∑ j b2 .
Based on this, one defines the coherent and the incoherent scattering cross sections
as
(
σ coh = 4πb 2 and σ inc = 4π b 2 − b 2) . (4.3.8)
Using these expressions it is possible to separate the sum in equation 4.3.4 in two
terms.
where
( ∂ 2 )
σ
= σ coh Q out 1 ∑
∂Ω∂E
coh
4π Q in 2π
∂ 2 (
σ ∂ 2 ) (
∂Ω∂E = σ ∂ 2 )
σ
+
∂Ω∂E
coh
∂Ω∂E
inc
i,j
and
( ∂ 2 )
σ
= σ inc Q out 1 ∑
∂Ω∂E
inc
4π Q in 2π
j
∫ ∞
−∞
∫ ∞
−∞
(4.3.9)
〈exp(−iQ ·r i (0))exp(iQ · r j (t))〉exp(−iωt)dt.
(4.3.10)
〈exp(−iQ ·r j (0))exp(iQ ·r j (t))〉exp(−iωt)dt,
(4.3.11)
In the case of X-ray scattering we replace the scattering length by the form factor,
|f(Q)|, which is the same for all the molecules of a given species. A sample consisting
of only one species therefore only gives rise to coherent X-ray scattering. The