Page 467 - Numerical Methods for Chemical Engineering

456 9 Fourier analysis

are not themselves scattered before they strike the detector. The measured intensity is then
N N

2
2
I tot r D , t; q = E tot r D , t; q =|E s (θ)| e −iϕ j e iϕ m

j=1 m=1
N N N N

2 i(ϕ m −ϕ j ) 2 −i(q·R mj )
=|E s (θ)| e =|E s (θ)| e (9.111)
j=1 m=1 j=1 m=1
The structure factor – the ratio of time-averaged scattered intensity at q = 0 to that in the
decoherent limit q → 0 – is therefore
7 N N 8
I tot (q) 1 −i(q·R mj )
S(q) = = e (9.112)
I tot (q → 0) N
j=1 m=1

Applying Fourier analysis

We now show that this structure factor is closely related to the Fourier transform of the corre-
lation function of the electron density ρ(r). If we know the exact positions {R 1 , R 2 ,..., R N }
of each electron (we neglect quantum effects), the density function is merely a sum of Dirac
delta functions:
N

ρ(r) = δ(r − R j ) (9.113)
j=1
Let us consider the autocorrelation function of ρ(r),
1 '
C ρ,ρ (s) = ρ(r + s)ρ(r)dr
(2π) d/2
d (9.114)
1 '
2 −i(q·s)
C ρ,ρ (q) = ρ(q)ρ(−q) =|ρ(q)| = C ρ,ρ (s)e ds
(2π) d/2
d

Substituting for C ρ,ρ (s) and ρ(r) in the expression for C ρ,ρ (q),
1 ' ' −i(q·s)
C ρ,ρ (q) = ρ(r + s)ρ(r)e drds
(2π) d
d d

N
N
1 −i(q·s)
' '
= δ(r + s − R m ) δ(r − R j ) e drds
(2π) d m=1 j=1
d d

 
N
N
1  ' ' −i(q·s) 
= d δ(r + s − R m )δ(r − R j )e drds
(2π)
m=1 j=1  
d
d
(9.115)
From the Dirac delta functions, the integral is nonzero only if
(9.116)
r = R j r + s = R m ⇒ s = R m − R j = R mj