Paper:Scattering Curves of Ordered Mesoscopic Materials

This is a summary/discussion of the results from:

• S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. Scattering Curves of Ordered Mesoscopic Materials Phys. Chem. B 2005, 109 (4), 1347–1360 doi: 10.1021/jp0467494

Mathematics

Equation (1) describes the general scattered intensity from particles (phase 1) in a matrix (phase 2):

$I(q)=(b_{1}-b_{2})^{2}\rho _{N}\left(\left\langle F^{2}(q)\right\rangle +\left\langle F(q)\right\rangle ^{2}[\left\langle Z(q)\right\rangle -1]\right)$

The b₁ and b₂ are the scattering lengths, which basically describes how strongly each material "scatters" the x-rays. So the $(b_{1}-b_{2})$ is the scattering contrast. The F(q) is the Fourier transform of the particle form (related to the "Form Factor") and Z(q) is the lattice factor that describes the spatial distribution of the particles (related the "Structure Factor").

Equation (30) (with Equation (2)) recast this slightly:

${\begin{alignedat}{2}&I(q)=\left(b_{1}-b_{2}\right)^{2}\rho _{N}P(q)S(q)\\&P(q)=F^{2}(q)\\&S(q)=1+\beta (q)\left(Z_{0}(q)-1\right)G(q)\\\end{alignedat}}$

Where P(q) is the form factor and S(q) is the structure factor. G(q) is a Debye-Waller factor for thermal disorder:

$G(q)=e^{-\sigma _{a}^{2}a^{2}q^{2}}$

Z₀ is the lattice factor computed from a sum over reciprocal space peaks (Miller indices {hkl}):

$Z_{0}(q)={\frac {(2\pi )^{d-1}c}{nv_{d}\Omega _{d}q^{d-1}}}\sum _{\{hkl\}}m_{hkl}f_{hkl}^{2}L_{hkl}(q-q_{hkl})$

where the pre-factor is affected by the dimensionality, d, which also influences the projected volume $v_{d}$, the solid angle $\Omega _{d}$, and the lattice type, which influences the number of particles per unit cell, n. The sum over peaks {hkl} requires knowing the multiplicities ($m_{hkl}$), symmetry factors ($f_{hkl}$) and peak positions ($q_{hkl}$) for the given lattice type (BCC, FCC, etc.).