Page 469 - Numerical Methods for Chemical Engineering

458 9 Fourier analysis

Now, if q · (l 1 a + l 2 b + l 3 c) = m2π, m =±1, ±2,... , then
n 0 n 0 n 0 N cells
−im2π
C ρ,ρ (q ) = e = (1) = (9.124)
(2π) d/2 (2π) d/2 (2π) d/2
l 1 ,l 2 ,l 3 l 1 ,l 2 ,l 3

Thus, C ρ,ρ (q) has sharp peaks at all q such that q · (l 1 a + l 2 b + l 3 c) is a multiple of
2π. If we define the reciprocal lattice vectors α, β, γ to satisfy
α · a = 2π α · b = 0 α · c = 0
β · a = 0 β · b = 2π β · c = 0 (9.125)
γ · a = 0 γ · b = 0 γ · c = 2π

these peaks occur in q-space at

q = n 1 α + n 2 β + n 3 γ (9.126)

since

q · (l 1 a + l 2 b + l 3 c) = (n 1 α + n 2 β + n 3 γ) · (l 1 a + l 2 b + l 3 c)

= (n 1 l 1 + n 2 l 2 + n 3 l 3 )(2π) (9.127)

Above, we have assumed that the periodicity ρ(r + l 1 a + l 2 b + l 3 c) = ρ(r) extends
throughout the entire sample; however, often (unless we go to great lengths to anneal
the sample or crystallize it slowly from a single nucleation site) the sample contains many
different crystal domains, with the crystal lattices in each domain isotropically oriented at
random. In such a case, we measure a powder spectrum, and obtain not the full structure
factor S(q) but rather merely an isotropically-averaged structure factor S(q). When S(q)is
nonzero (it usually appears as a circular “halo” at some scattering angle θ), this signifies
that the material has structural periodicity on a length scale

2π
σ len ≈ (9.128)
q
Smaller q-values denote structures on larger length scales. Hence, when X-ray scattering
(for a Cu-Kα 1 source, λ is 1.54 Å) is used to measure the ˚angstrom-scale periodicity of
crystals, we measure S(q) at large values of q, and thus also at large scattering angles θ
(Figure 9.10). By contrast, when we attempt to probe longer nanometer-scale O(10 −9 m)
structure, we must measure the scattering at small angles. Thus, when X-ray scattering is
performed to probe the atomic-scale structure of crystals, it is known as WAXS (wide angle
X-ray scattering). When it is used to probe the structure on the order of tens of nanometers
of materials such as self-assembled block copolymers and micellar emulsions, it is known
as SAXS (small angle X-ray scattering). At very small θ, the scattered intensity is buried
within the intensity of the incident beam, resulting in an effective upper limit of resolution
for SAXS of O(100 nm). With special facilities, this technique can be extended to longer
length scales, as at the USAXS (ultra small angle X-ray scattering) facility at Argonne
National Laboratories.