Page 468 - Numerical Methods for Chemical Engineering

Scattering theory 457

therefore,
N N
1 −i(q·R mj )
C ρ,ρ (q) = e (9.117)
(2π) d
j=1 m=1
The structure factor thus is related to the time-averaged autocorrelation function of the
electron density:
7 N N 8
1 −i(q·R mj ) NS(q) 2
C ρ,ρ (q) = e = =| ρ(q) | (9.118)
(2π) d (2π) d
j=1 m=1
The autocorrelation function is obtained from an inverse Fourier transform:
1 ' 1 ' NS(q) i(q·s)
C ρ,ρ (s) = ρ(r + s)ρ(r) dr = e dq (9.119)
(2π) d/2 (2π) d/2 (2π) d
d d

Scattering peaks from samples with periodic structure

When C ρ,ρ (s) has a peak at a particular s , it means that when we have an electron at

r , we tend to have another electron at r + s . For example, let us consider the case of a

periodic lattice (e.g. a crystal) with lattice vectors a, b, c such that
ρ(r + l 1 a + l 2 b + l 3 c) = ρ(r) l m = 0, ±1, ±2,... (9.120)
C ρ,ρ (s) then has intense peaks at every s = l 1 a + l 2 b + l 3 c, as can be seen by par-
titioning the sample volume into unit cells α ,α = 1, 2,..., N cells , each of volume
|a · (b × c)|:

1 N cells '

C ρ,ρ (l 1 a + l 2 b + l 3 c) = ρ(r + l 1 a + l 2 b + l 3 c)ρ(r) dr
(2π) d/2
α=1
α
'
N cells
= ρ(r)ρ(r) dr (9.121)
(2π) d/2
α
This periodicity ρ(r + l 1 a + l 2 b + l 3 c) = ρ(r) also yields peaks of S(q) and C ρ,ρ (q)
in q-space. Let us assume that C ρ,ρ (s) consists of infinitely-narrow peaks about each
l 1 a + l 2 b + l 3 c:

C ρ,ρ (s) = n 0 δ[s − (l 1 a + l 2 b + l 3 c)] (9.122)
l 1 ,l 2 ,l 3
In practice, peaks have finite widths, but, assuming this “sharp-peak” limit, the Fourier
transform of the autocorrelation function is
(
1 ' −i(q·s)
C ρ,ρ (q) = n 0 δ[s − (l 1 a + l 2 b + l 3 c)] e ds
(2π) d/2
l 1 ,l 2 ,l 3
d

n 0 −i[q·(l 1 a+l 2 b+l 3 c)]
= e (9.123)
(2π) d/2
l 1 ,l 2 ,l 3